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-19,7 +19,7 @@ Imagine you've conducted an experiment involving measurements from 20 animals. I If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below. -![Empirical and theoretical distributions](/Users/lenovo1/Desktop/mtech/courses/stats/Images/fig1.png) +![Empirical and theoretical distributions](dice rolling probability.png) But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.) @@ -34,10 +34,7 @@ That's a lot of theory, let's dive into some data now. ```{r} #Change the path according to your PC -path = "/Users/lenovo1/Desktop/mtech/courses/stats/Datasets/data/" -fileN ="aflsmall.Rdata" -file_path = paste(path,fileN, sep = "") -load(file_path) +load("aflsmall.Rdata") library(lsr) who() ``` @@ -47,7 +44,7 @@ As you can see there are multiple variables of different class and size. Let's take a look at afl.margins variable. ```{r} -print (afl.margins) +print(afl.margins) ``` This output doesn’t make it easy to get a sense of what the data is actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data. @@ -88,11 +85,9 @@ As you've already seen in previous classes, the mean of a set of observations is Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins ```{r} -#Don't forget to uncomment the following before running -#afl.mean5 = -#afl.mean = -#afl.mean5 -#afl.mean +mean(afl.margins) # average margin +mean(afl.margins[1:5]) # mean of the margin from the first 5 games + ``` **Median** @@ -105,15 +100,14 @@ Probably you mentally arranged these numbers in ascending order first and then f Now try finding out the median for afl.margins. ```{r} -#afl.median = -#afl.median +median(afl.margins) ``` **Difference between Mean and Median** Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data. -![Difference between mean and median](/Users/lenovo1/Desktop/mtech/courses/stats/Images/Fig2.png) +![Difference between mean and median](pic2.png) *Fig 5.2 from Learning Statistics with R by D. Navarro* **Some key points** @@ -151,11 +145,13 @@ mean(x = dataset, trim = .1) So far we've seen how to find the mean and median but what about mode. The **mode** of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the _lsr_ package has a function called modeOf() that does this. -Try to find out the mode for the variable afl.finalists +Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins ```{r} #afl.mode = #afl.mode +modeOf(x = afl.margins) +maxFreq(x = afl.margins) ``` So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability. @@ -180,6 +176,7 @@ That is why there is something called the interquartile range (IQR) which is lik Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range. ```{r} #Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR() + ``` IQR can simply be thought as the range spanned by the “middle half” of the data. diff --git a/Module 3/Notebooks/Module3_Nb1.nb.html b/Module 3/Notebooks/Module3_Nb1.nb.html index 880ac831..350b7d0b 100644 --- a/Module 3/Notebooks/Module3_Nb1.nb.html +++ b/Module 3/Notebooks/Module3_Nb1.nb.html @@ -221,17 +221,6 @@

Descriptive Statistics: Central and Variability mea 
#Initial packages
 install.packages("lsr")
- -
trying URL 'https://cran.rstudio.com/bin/macosx/big-sur-arm64/contrib/4.1/lsr_0.5.tgz'
-Content type 'application/x-gzip' length 214898 bytes (209 KB)
-==================================================
-downloaded 209 KB
- - -

-The downloaded binary packages are in
-    /var/folders/m1/c7s91v016q36bkqr9zkl051r0000gq/T//Rtmp47b7Y9/downloaded_packages
-

In this notebook, we’ll take a look at how to explore a dataset.

@@ -243,7 +232,7 @@

Describing data

What exactly is a distribution?

If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below.

- +

Empirical and theoretical distributions

But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.)

@@ -256,22 +245,16 @@
But why is it needed?

Loading the Australian Football League Dataset

- + 
#Change the path according to your PC
-path = "/Users/lenovo1/Desktop/mtech/courses/stats/Datasets/data/"
-fileN ="aflsmall.Rdata"
-file_path = paste(path,fileN, sep = "")
-load(file_path)
+load("aflsmall.Rdata")
 library(lsr)
 who()
- + 
   -- Name --      -- Class --   -- Size --
    afl.finalists   factor        400       
-   afl.margins     numeric       176       
-   file_path       character       1       
-   fileN           character       1       
-   path            character       1
+ afl.margins numeric 176 @@ -279,16 +262,18 @@
But why is it needed?

Let’s take a look at afl.margins variable.

- -
print (afl.margins)
+ +
print(afl.margins)
+
- -
  [1]  56  31  56   8  32  14  36  56  19   1   3 104  43  44  72   9  28  25  27  55  20  16  16   7  23  40  48  64  22  55  95  15  49  52
- [35]  50  10  65  12  39  36   3  26  23  20  43 108  53  38   4   8   3  13  66  67  50  61  36  38  29   9  81   3  26  12  36  37  70   1
- [69]  35  12  50  35   9  54  47   8  47   2  29  61  38  41  23  24   1   9  11  10  29  47  71  38  49  65  18   0  16   9  19  36  60  24
-[103]  25  44  55   3  57  83  84  35   4  35  26  22   2  14  19  30  19  68  11  75  48  32  36  39  50  11   0  63  82  26   3  82  73  19
-[137]  33  48   8  10  53  20  71  75  76  54  44   5  22  94  29   8  98   9  89   1 101   7  21  52  42  21 116   3  44  29  27  16   6  44
-[171]   3  28  38  29  10  10
+ +
  [1]  56  31  56   8  32  14  36  56  19   1   3 104  43  44  72   9  28  25  27  55  20  16  16   7  23  40  48
+ [28]  64  22  55  95  15  49  52  50  10  65  12  39  36   3  26  23  20  43 108  53  38   4   8   3  13  66  67
+ [55]  50  61  36  38  29   9  81   3  26  12  36  37  70   1  35  12  50  35   9  54  47   8  47   2  29  61  38
+ [82]  41  23  24   1   9  11  10  29  47  71  38  49  65  18   0  16   9  19  36  60  24  25  44  55   3  57  83
+[109]  84  35   4  35  26  22   2  14  19  30  19  68  11  75  48  32  36  39  50  11   0  63  82  26   3  82  73
+[136]  19  33  48   8  10  53  20  71  75  76  54  44   5  22  94  29   8  98   9  89   1 101   7  21  52  42  21
+[163] 116   3  44  29  27  16   6  44   3  28  38  29  10  10
@@ -301,7 +286,7 @@
But why is it needed?
hist (afl.margins)
-

+

@@ -329,13 +314,18 @@
Measures of Central Tendency

Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins

- -
#Don't forget to uncomment the following before running
-#afl.mean5 = 
-#afl.mean = 
-#afl.mean5
-#afl.mean
+ +
mean(afl.margins)      # average margin
+ +
[1] 35.30114
+ + +
mean(afl.margins[1:5]) # mean of the margin from the first 5 games
+ + +
[1] 36.6
+

Median

@@ -344,15 +334,17 @@
Measures of Central Tendency

Now try finding out the median for afl.margins.

- -
#afl.median = 
-#afl.median
+ +
median(afl.margins)
+ +
[1] 30.5
+

Difference between Mean and Median

Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data.

-

Difference between mean and median Fig 5.2 from Learning Statistics with R by D. Navarro

+

Difference between mean and median Fig 5.2 from Learning Statistics with R by D. Navarro

Some key points

  • If data is nominal scale, then it’s probably best to use the mode instead of mean or median.

    • @@ -383,13 +375,23 @@
    Measures of Central Tendency

    Mode

    So far we’ve seen how to find the mean and median but what about mode. The mode of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the lsr package has a function called modeOf() that does this.

    -

    Try to find out the mode for the variable afl.finalists

    +

    Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins

    - + 
    #afl.mode = 
-#afl.mode
    +#afl.mode +modeOf(x = afl.margins) + +
    [1] 3
    + + +
    maxFreq(x = afl.margins)
    + + +
    [1] 8
    +

    So far we’ve just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let’s take a look at some of the measures of variability.

    @@ -411,8 +413,9 @@
    Measures of variability

    Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.

    - -
    #Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR()
    + +
    #Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR() 
+
    @@ -478,14 +481,10 @@
    Quick cheat sheet: When to use what?

    Let’s try out summarizing a dataframe as well.

    - -
    load("clinicaltrial.Rdata")
    + +
    load("clinicaltrial.Rdata")
+#Check the name of the variable in the environment which contains the dataframe and try summarizing it
    - -
    Warning in readChar(con, 5L, useBytes = TRUE) :
-  cannot open compressed file 'clinicaltrial.Rdata', probable reason 'No such file or directory'
-Error in readChar(con, 5L, useBytes = TRUE) : cannot open the connection
    -

    The psych package also has a function called describe() for dataframes. Don’t forget to check it out too!

    @@ -493,27 +492,26 @@
    Quick cheat sheet: When to use what?

    For instance, run describeBy( x=clin.trial, group=clin.trial$therapy )

    - -
    #describeBy(clin.trial)
-describeBy( x=clin.trial, group=clin.trial$therapy )
    + +
    describeBy( x=clin.trial, group=clin.trial$therapy )
    
  Descriptive statistics by group 
 group: no.therapy
    - +
    - -
    --------------------------------------------------------------------------------------------------------- 
+
+
    ------------------------------------------------------------------------------------ 
 group: CBT
    
 
-
+
 
    
 
+
    
+
+
+
    ------------------------------------------------------------------------------------ 
+clin.trial$therapy: CBT
    
+
+
+
    
+
+
    
+
+
+
    #Also try replacing describe in FUN above with summary
    
 
 
 
@@ -541,17 +562,19 @@ 
    Quick cheat sheet: When to use what?
    
            data = clin.trial,
            FUN = mean)
    - +
    - + 
    #1 mood.gain by drug/therapy combination
 #2 data is in the clin.trial data frame
-#3 print out group means
    +#3 print out group means + +#Try interchanging the positions of drug and therapy above @@ -560,7 +583,7 @@
    Quick cheat sheet: When to use what?
    -
    ---
title: "Descriptive Statistics: Central and Variability measures"
output: html_notebook
---
```{r}
#Initial packages
install.packages("lsr")
```

In this notebook, we'll take a look at how to explore a dataset.

Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what **descriptive statistics** is all about.

#### Describing data

Imagine you've conducted an experiment involving measurements from 20 animals. If you wanted to report the outcome of your experiment to an audience, you wouldn’t want to talk through each and every data point. Instead, you report a summary, such as ‘The 20 animals had an average weight of 15 grams’, thus saving your audience valuable time and mental energy. This notebook focuses on such summaries of numerical information including distributions, measures of central tendency and measures of variability. 

##### What exactly is a distribution?

If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below.

![Empirical and theoretical distributions](/Users/lenovo1/Desktop/mtech/courses/stats/Images/fig1.png)

But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.)

Apart from *looking* at how a data is distributed, the most important descriptive statistics for numerical data are those measuring the location of a frequency distribution and its spread. The location tells us something about the average or *typical* individual—where the observations are centered. The spread tells us how variable the measurements are from individual to individual—how widely scattered the observations are around the center. The proportion is the most important descriptive statistic for a categorical variable, measuring the fraction of observations in a given category. 

##### But why is it needed?
The importance of calculating some sort of a centre of a distribution seems obvious. How else do we address questions like “Which species is larger?” or “Which drug yielded the greatest response?” The importance of describing distribution spread is less obvious but no less crucial, at least in biology. In some fields of science, variability around a central value is instrument noise or measurement error, but in biology much of the variability signifies real differences among individuals. Different individuals respond differently to treatments, and this variability begs measurement. (Adapted from Chapter 3, Whitlock & Schluter, 2015)

That's a lot of theory, let's dive into some data now.

**Loading the Australian Football League Dataset**

```{r}
#Change the path according to your PC
path = "/Users/lenovo1/Desktop/mtech/courses/stats/Datasets/data/"
fileN ="aflsmall.Rdata"
file_path = paste(path,fileN, sep = "")
load(file_path)
library(lsr)
who()
```

As you can see there are multiple variables of different class and size. 

Let's take a look at afl.margins variable.

```{r}
print (afl.margins)
```

This output doesn’t make it easy to get a sense of what the data is actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data.

Let's try to plot it.

**Frequency distribution**

```{r}
hist (afl.margins)
```

As you can see, different margins in a sample will have different measurements. We can see this variability with a **frequency distribution**. The frequency of a specific measurement in a sample is the number of observations having a particular value of the measurement. The frequency distribution shows how often each value of the variable occurs in the sample. 

Therefore, here we have plotted a histogram for the afl.margins variable which gives the frequency distribution of the different margin values.

**Skewness**

If you observe the graph, you will find that it is not entirely symmetrical. A measure of such asymmetry is called **Skewness**. If the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel), then we say that the data are _negatively skewed_. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are _positively skewed_.

`psych` package contains a `skew()` function that you can use to calculate skewness. 

Try finding the skewness for the above data for afl.margins using skew() function and also try to guess whether this data is positively or negatively skewed.

```{r}
library(psych)
#Try finding skewness of afl.margins here
```

Although such a graphical representation gives a 'gist' of the data but it is useful to find some "summary" statistics as well.

##### Measures of Central Tendency
In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you’d like to know something about the “average” or “middle” of your data lies. The two most commonly used measures are the mean, median and mode.

**Mean**

As you've already seen in previous classes, the mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values.

Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins
```{r}
#Don't forget to uncomment the following before running
#afl.mean5 = 
#afl.mean = 
#afl.mean5
#afl.mean
```

**Median**

The second measure is the median. It is just the middle value of a set of observations. 
*Try : Guess the median for 56, 31, 56, 8 and 32 *

Probably you mentally arranged these numbers in ascending order first and then found the middle value. If there were a list of numbers like this `8, 14, 31, 32, 56, 56` . You will then find the average of middle 2 values. 

Now try finding out the median for afl.margins.

```{r}
#afl.median = 
#afl.median
```

**Difference between Mean and Median**

Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data.

![Difference between mean and median](/Users/lenovo1/Desktop/mtech/courses/stats/Images/Fig2.png)
*Fig 5.2 from Learning Statistics with R by D. Navarro*

**Some key points**

- If data is nominal scale, then it’s probably best to use the mode instead of mean or median.

- If your data are ordinal scale, you’re more likely to want to use the median than the mean.

- For interval and ratio scale data, either mean or median is generally acceptable. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values.

*You can read more about this in Section 5.1.4, Learning Statistics with R by D. Navarro*

Now let's take a look at some more data:

` -100,2,3,4,5,6,7,8,9,10`

If you observed such data in real life, you will probably think that -100 is an **_outlier_**, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely but you don’t always get such cut-and-dried examples. For instance, you might get this instead:

` -15,2,3,4,5,6,7,8,9,12`

The `-15` looks a bit suspicious, but not anywhere near as much as `-100` did. In this case, it’s a
little trickier. It might be a legitimate observation, it might not. In such situations, the mean might give you an error as it is highly sensitive to one or two extreme values, and is thus not considered to be a robust measure.

In such situations, one solution is to use the median or another is to use a **trimmed mean**. To calculate a trimmed mean, what you do is **discard** the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. This helps in taking the mean by excluding the outliers.

Let's try trimming the mean for above data.

```{r}
dataset <- c(-15,2,3,4,5,6,7,8,9,12)
mean(x = dataset, trim = .1)
#Try calculating 5% trimmed mean for above dataset
```

**Mode**

So far we've seen how to find the mean and median but what about mode. The **mode** of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the _lsr_ package has a function called modeOf() that does this. 

Try to find out the mode for the variable afl.finalists

```{r}
#afl.mode = 
#afl.mode
```

So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability.

##### Measures of variability

This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be?

**Range**

The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the `range()` function.

```{r}
#Find range of afl.margins here
```

But what about the earlier data we saw, ` -100,2,3,4,5,6,7,8,9,10`. Without removing the outlier, we'll get a range of 110 but without the outlier, we'll get a range of only 8.

**Inter-quartile Range (IQR)**
That is why there is something called the interquartile range (IQR) which is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. A 10% _quantile_ or _percentile_ of a data set is defined as the smallest number _x_ such that 10% of the data is less than _x_.

Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.
```{r}
#Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR() 
```

IQR can simply be thought as the range spanned by the “middle half” of the data.

**Variance**

In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (X~i~ - $\overline{X}$). ($\overline{X}$ is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we'll get the **variance** of the data. 

Try finding out the variance using `var()`.

```{r}
#Use var() for finding variance of afl.margins
```

_Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro_

**Standard Deviation**

But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the _root mean square deviation_ for interpreting the spread of data points. This is called _Standard Deviation_ and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package.

Try to find out the standard deviation of afl.margins.

```{r}
#Find out Std dev. here
```

##### Quick cheat sheet: When to use what?

- Range: 
  - Gives full spread of data. 
  - Very vulnerable to outliers

- Interquartile range: 
  - Gives the “middle half” of data
  - Robust, and complements the median nicely
  
- Variance:
  - Average squared deviation from the mean
  - It’s mathematically elegant but it’s completely uninterpretable

- Standard deviation:
  - Square root of the variance
  - Fairly elegant mathematically, and can be interpreted pretty well
  - Complements mean and is the most popular measure of variation

Now that we've learnt about the different methods of describing a data, it would've been awesome if R could summarize all of this for us together, right? 

There's indeed a function called `summary()` in R.

```{r}
#Check out what summary() does for afl.margins
```

Pretty cool, no?

Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)`

Let's try out summarizing a dataframe as well.

```{r}
load("clinicaltrial.Rdata")
#Check the name of the variable in the environment which contains the dataframe and try summarizing it
```

The `psych` package also has a function called `describe()` for dataframes. Don't forget to check it out too!

In fact, you can also describe these statistics group wise. 

For instance, run `describeBy( x=clin.trial, group=clin.trial$therapy )`

```{r}
describeBy( x=clin.trial, group=clin.trial$therapy )
```
Notice that, the output displays asterisks for factor variables, in order to draw your attention to the fact that the descriptive statistics that it has calculated won’t be very meaningful for those variables.

Another more general command for grouping is `by()` 

Try running the following chunk and compare the results with the `describeBy()` command above.

```{r}
by(data=clin.trial, INDICES=clin.trial$therapy, FUN=describe)
#Also try replacing describe in FUN above with summary
```

What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy.We can use `aggregate()` command.

```{r}
aggregate( formula = mood.gain ~ drug + therapy, 
           data = clin.trial,
           FUN = mean) 
#1 mood.gain by drug/therapy combination
#2 data is in the clin.trial data frame
#3 print out group means

#Try interchanging the positions of drug and therapy above
```

That's all for today!

    +
    ---
title: "Descriptive Statistics: Central and Variability measures"
output: html_notebook
---
```{r}
#Initial packages
install.packages("lsr")
```

In this notebook, we'll take a look at how to explore a dataset.

Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what **descriptive statistics** is all about.

#### Describing data

Imagine you've conducted an experiment involving measurements from 20 animals. If you wanted to report the outcome of your experiment to an audience, you wouldn’t want to talk through each and every data point. Instead, you report a summary, such as ‘The 20 animals had an average weight of 15 grams’, thus saving your audience valuable time and mental energy. This notebook focuses on such summaries of numerical information including distributions, measures of central tendency and measures of variability. 

##### What exactly is a distribution?

If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below.

![Empirical and theoretical distributions](dice rolling probability.png)

But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.)

Apart from *looking* at how a data is distributed, the most important descriptive statistics for numerical data are those measuring the location of a frequency distribution and its spread. The location tells us something about the average or *typical* individual—where the observations are centered. The spread tells us how variable the measurements are from individual to individual—how widely scattered the observations are around the center. The proportion is the most important descriptive statistic for a categorical variable, measuring the fraction of observations in a given category. 

##### But why is it needed?
The importance of calculating some sort of a centre of a distribution seems obvious. How else do we address questions like “Which species is larger?” or “Which drug yielded the greatest response?” The importance of describing distribution spread is less obvious but no less crucial, at least in biology. In some fields of science, variability around a central value is instrument noise or measurement error, but in biology much of the variability signifies real differences among individuals. Different individuals respond differently to treatments, and this variability begs measurement. (Adapted from Chapter 3, Whitlock & Schluter, 2015)

That's a lot of theory, let's dive into some data now.

**Loading the Australian Football League Dataset**

```{r}
#Change the path according to your PC
load("aflsmall.Rdata")
library(lsr)
who()
```

As you can see there are multiple variables of different class and size. 

Let's take a look at afl.margins variable.

```{r}
print(afl.margins)
```

This output doesn’t make it easy to get a sense of what the data is actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data.

Let's try to plot it.

**Frequency distribution**

```{r}
hist (afl.margins)
```

As you can see, different margins in a sample will have different measurements. We can see this variability with a **frequency distribution**. The frequency of a specific measurement in a sample is the number of observations having a particular value of the measurement. The frequency distribution shows how often each value of the variable occurs in the sample. 

Therefore, here we have plotted a histogram for the afl.margins variable which gives the frequency distribution of the different margin values.

**Skewness**

If you observe the graph, you will find that it is not entirely symmetrical. A measure of such asymmetry is called **Skewness**. If the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel), then we say that the data are _negatively skewed_. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are _positively skewed_.

`psych` package contains a `skew()` function that you can use to calculate skewness. 

Try finding the skewness for the above data for afl.margins using skew() function and also try to guess whether this data is positively or negatively skewed.

```{r}
library(psych)
#Try finding skewness of afl.margins here
```

Although such a graphical representation gives a 'gist' of the data but it is useful to find some "summary" statistics as well.

##### Measures of Central Tendency
In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you’d like to know something about the “average” or “middle” of your data lies. The two most commonly used measures are the mean, median and mode.

**Mean**

As you've already seen in previous classes, the mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values.

Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins
```{r}
mean(afl.margins)      # average margin
mean(afl.margins[1:5]) # mean of the margin from the first 5 games

```

**Median**

The second measure is the median. It is just the middle value of a set of observations. 
*Try : Guess the median for 56, 31, 56, 8 and 32 *

Probably you mentally arranged these numbers in ascending order first and then found the middle value. If there were a list of numbers like this `8, 14, 31, 32, 56, 56` . You will then find the average of middle 2 values. 

Now try finding out the median for afl.margins.

```{r}
median(afl.margins)
```

**Difference between Mean and Median**

Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data.

![Difference between mean and median](pic2.png)
*Fig 5.2 from Learning Statistics with R by D. Navarro*

**Some key points**

- If data is nominal scale, then it’s probably best to use the mode instead of mean or median.

- If your data are ordinal scale, you’re more likely to want to use the median than the mean.

- For interval and ratio scale data, either mean or median is generally acceptable. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values.

*You can read more about this in Section 5.1.4, Learning Statistics with R by D. Navarro*

Now let's take a look at some more data:

` -100,2,3,4,5,6,7,8,9,10`

If you observed such data in real life, you will probably think that -100 is an **_outlier_**, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely but you don’t always get such cut-and-dried examples. For instance, you might get this instead:

` -15,2,3,4,5,6,7,8,9,12`

The `-15` looks a bit suspicious, but not anywhere near as much as `-100` did. In this case, it’s a
little trickier. It might be a legitimate observation, it might not. In such situations, the mean might give you an error as it is highly sensitive to one or two extreme values, and is thus not considered to be a robust measure.

In such situations, one solution is to use the median or another is to use a **trimmed mean**. To calculate a trimmed mean, what you do is **discard** the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. This helps in taking the mean by excluding the outliers.

Let's try trimming the mean for above data.

```{r}
dataset <- c(-15,2,3,4,5,6,7,8,9,12)
mean(x = dataset, trim = .1)
#Try calculating 5% trimmed mean for above dataset
```

**Mode**

So far we've seen how to find the mean and median but what about mode. The **mode** of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the _lsr_ package has a function called modeOf() that does this. 

Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins

```{r}
#afl.mode = 
#afl.mode
modeOf(x = afl.margins)
maxFreq(x = afl.margins)
```

So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability.

##### Measures of variability

This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be?

**Range**

The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the `range()` function.

```{r}
#Find range of afl.margins here
```

But what about the earlier data we saw, ` -100,2,3,4,5,6,7,8,9,10`. Without removing the outlier, we'll get a range of 110 but without the outlier, we'll get a range of only 8.

**Inter-quartile Range (IQR)**
That is why there is something called the interquartile range (IQR) which is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. A 10% _quantile_ or _percentile_ of a data set is defined as the smallest number _x_ such that 10% of the data is less than _x_.

Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.
```{r}
#Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR() 

```

IQR can simply be thought as the range spanned by the “middle half” of the data.

**Variance**

In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (X~i~ - $\overline{X}$). ($\overline{X}$ is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we'll get the **variance** of the data. 

Try finding out the variance using `var()`.

```{r}
#Use var() for finding variance of afl.margins
```

_Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro_

**Standard Deviation**

But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the _root mean square deviation_ for interpreting the spread of data points. This is called _Standard Deviation_ and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package.

Try to find out the standard deviation of afl.margins.

```{r}
#Find out Std dev. here
```

##### Quick cheat sheet: When to use what?

- Range: 
  - Gives full spread of data. 
  - Very vulnerable to outliers

- Interquartile range: 
  - Gives the “middle half” of data
  - Robust, and complements the median nicely
  
- Variance:
  - Average squared deviation from the mean
  - It’s mathematically elegant but it’s completely uninterpretable

- Standard deviation:
  - Square root of the variance
  - Fairly elegant mathematically, and can be interpreted pretty well
  - Complements mean and is the most popular measure of variation

Now that we've learnt about the different methods of describing a data, it would've been awesome if R could summarize all of this for us together, right? 

There's indeed a function called `summary()` in R.

```{r}
#Check out what summary() does for afl.margins
```

Pretty cool, no?

Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)`

Let's try out summarizing a dataframe as well.

```{r}
load("clinicaltrial.Rdata")
#Check the name of the variable in the environment which contains the dataframe and try summarizing it
```

The `psych` package also has a function called `describe()` for dataframes. Don't forget to check it out too!

In fact, you can also describe these statistics group wise. 

For instance, run `describeBy( x=clin.trial, group=clin.trial$therapy )`

```{r}
describeBy( x=clin.trial, group=clin.trial$therapy )
```
Notice that, the output displays asterisks for factor variables, in order to draw your attention to the fact that the descriptive statistics that it has calculated won’t be very meaningful for those variables.

Another more general command for grouping is `by()` 

Try running the following chunk and compare the results with the `describeBy()` command above.

```{r}
by(data=clin.trial, INDICES=clin.trial$therapy, FUN=describe)
#Also try replacing describe in FUN above with summary
```

What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy.We can use `aggregate()` command.

```{r}
aggregate( formula = mood.gain ~ drug + therapy, 
           data = clin.trial,
           FUN = mean) 
#1 mood.gain by drug/therapy combination
#2 data is in the clin.trial data frame
#3 print out group means

#Try interchanging the positions of drug and therapy above
```

That's all for today!

    @@ -607,7 +630,7 @@
    Quick cheat sheet: When to use what?
    From 6718529ad1445a6db4c7b15eec53775823e4ffe3 Mon Sep 17 00:00:00 2001 From: Arjun Date: Mon, 6 Sep 2021 17:54:00 +0000 Subject: [PATCH 05/55] Added a note on derivation of variance based estimates. Added Bessel's correction --- Module 3/Notebooks/Module3_Nb1.Rmd | 73 ++++++++++++++++++-- Module 3/Notebooks/Module3_Nb1.nb.html | 92 ++++++++++++++++++++------ 2 files changed, 141 insertions(+), 24 deletions(-) diff --git a/Module 3/Notebooks/Module3_Nb1.Rmd b/Module 3/Notebooks/Module3_Nb1.Rmd index fa9e6d53..ea61e465 100644 --- a/Module 3/Notebooks/Module3_Nb1.Rmd +++ b/Module 3/Notebooks/Module3_Nb1.Rmd @@ -150,22 +150,32 @@ Say, you want to bet your money on the outcome of a match. You may want to find ```{r} #afl.mode = #afl.mode -modeOf(x = afl.margins) -maxFreq(x = afl.margins) +modeOf(afl.margins) +maxFreq(afl.margins) ``` So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability. + + +```{r} +mean(afl.margins) +``` + ##### Measures of variability This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be? + +```{r} +plot(afl.margins) +``` **Range** The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the `range()` function. ```{r} -#Find range of afl.margins here +#Find range of afl.margins here using the range function ``` But what about the earlier data we saw, ` -100,2,3,4,5,6,7,8,9,10`. Without removing the outlier, we'll get a range of 110 but without the outlier, we'll get a range of only 8. @@ -175,16 +185,26 @@ That is why there is something called the interquartile range (IQR) which is lik Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range. ```{r} -#Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR() +#Use the functions quantile(x = afl.margins, prob = 0.25) for 25% quantile and IQR() +quantile(x = afl.margins, prob = 0.2) ``` IQR can simply be thought as the range spanned by the “middle half” of the data. + +```{r} +quantile( x = afl.margins, probs = c(.25,.75) ) +# try using IQR() here + +``` + + **Variance** In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (X~i~ - $\overline{X}$). ($\overline{X}$ is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we'll get the **variance** of the data. + Try finding out the variance using `var()`. ```{r} @@ -193,6 +213,9 @@ Try finding out the variance using `var()`. _Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro_ +Also note that the division is by N-1 for variance for a sample! Why is that not N? +This we will discuss later. + **Standard Deviation** But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the _root mean square deviation_ for interpreting the spread of data points. This is called _Standard Deviation_ and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package. @@ -200,7 +223,7 @@ But what does this variance signify? It is very difficult to interpret the squar Try to find out the standard deviation of afl.margins. ```{r} -#Find out Std dev. here +#Find out Std dev. here using sd() ``` ##### Quick cheat sheet: When to use what? @@ -221,19 +244,59 @@ Try to find out the standard deviation of afl.margins. - Square root of the variance - Fairly elegant mathematically, and can be interpreted pretty well - Complements mean and is the most popular measure of variation + + +##### Derivation for deviation based variance estimates +Mean absolute deviation; Variance; SD + +Say, the sample is sample = [20,30,40]; +the mean of the sample is then 30 + +The deviation of each sample data point from the mean is: +deviation = [20-30, 30-30, 40-30]; + = [-10, 0, 10] + +absolute deviation = [10,0,10] +mean of absolute deviation = (10+0+10)/3 + +square of the deviation = [-10^2, 0^2, 10^2]; +mean of the squared deviations = variance = (100+0+100)/3 + +root of mean squared deviation = standard deviation = sqrt(variance) + + +##### Bessel's correction +While calculating variance and standard deviation of the sample, we are always +trying to estimate the variance and standard deviation of the population. +Remember the heights of students example! + +Now since the sample variance and standard deviation is biased and less than +that of the population, we divide by N-1 instead of N to inflate the estimates. + +So the variance after Bessel's correction should be +mean of the squared deviations = variance = (100+0+100)/(3-1) + + +##### Summary function Now that we've learnt about the different methods of describing a data, it would've been awesome if R could summarize all of this for us together, right? There's indeed a function called `summary()` in R. ```{r} #Check out what summary() does for afl.margins +summary(afl.margins) + ``` Pretty cool, no? Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)` + + +##### Summarizing dataframes + Let's try out summarizing a dataframe as well. ```{r} diff --git a/Module 3/Notebooks/Module3_Nb1.nb.html b/Module 3/Notebooks/Module3_Nb1.nb.html index 350b7d0b..ee6d420b 100644 --- a/Module 3/Notebooks/Module3_Nb1.nb.html +++ b/Module 3/Notebooks/Module3_Nb1.nb.html @@ -378,16 +378,9 @@
    Measures of Central Tendency

    Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins

    - -
    #afl.mode = 
-#afl.mode
-modeOf(x = afl.margins)
    - - -
    [1] 3
    - - -
    maxFreq(x = afl.margins)
    + +
    maxFreq(afl.margins)
+
    [1] 8
    @@ -395,16 +388,36 @@
    Measures of Central Tendency

    So far we’ve just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let’s take a look at some of the measures of variability.

    + + + +
    mean(afl.margins)
    + + +
    [1] 35.30114
    + + +
    Measures of variability

    This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be?

    + + + +
    plot(afl.margins)
    + + +

    + + +

    Range

    The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the range() function.

    - -
    #Find range of afl.margins here
    + +
    #Find range of afl.margins here using the range function
    @@ -413,13 +426,26 @@
    Measures of variability

    Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.

    - + 
    #Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR() 
-
    +quantile(x = afl.margins, prob = 0.2) + +
    20% 
+ 10
    +

    IQR can simply be thought as the range spanned by the “middle half” of the data.

    + + + +
    quantile( x = afl.margins, probs = c(.25,.75) )
+# try using IQR() here
+
    + + +

    Variance

    In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (Xi - \(\overline{X}\)). (\(\overline{X}\) is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we’ll get the variance of the data.

    Try finding out the variance using var().

    @@ -431,13 +457,14 @@
    Measures of variability

    Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro

    +

    Also note that the division is by N-1 for variance for a sample! Why is that not N? This we will discuss later.

    Standard Deviation

    But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the root mean square deviation for interpreting the spread of data points. This is called Standard Deviation and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package.

    Try to find out the standard deviation of afl.margins.

    - -
    #Find out Std dev. here
    + +
    #Find out Std dev. here using sd()
    @@ -467,17 +494,44 @@
    Quick cheat sheet: When to use what?
  • Complements mean and is the most popular measure of variation
+ +
+
Derivation for deviation based variance estimates
+

Mean absolute deviation; Variance; SD

+

Say, the sample is sample = [20,30,40]; the mean of the sample is then 30

+

The deviation of each sample data point from the mean is: deviation = [20-30, 30-30, 40-30]; = [-10, 0, 10]

+

absolute deviation = [10,0,10] mean of absolute deviation = (10+0+10)/3

+

square of the deviation = [-10^2, 0^2, 10^2]; mean of the squared deviations = variance = (100+0+100)/3

+

root of mean squared deviation = standard deviation = sqrt(variance)

+
+
+
Bessel’s correction
+

While calculating variance and standard deviation of the sample, we are always trying to estimate the variance and standard deviation of the population.

+

Remember the heights of students example!

+

Now since the sample variance and standard deviation is biased and less than that of the population, we divide by N-1 instead of N to inflate the estimates.

+

So the variance after Bessel’s correction should be mean of the squared deviations = variance = (100+0+100)/(3-1)

+
+
+
Summary function

Now that we’ve learnt about the different methods of describing a data, it would’ve been awesome if R could summarize all of this for us together, right?

There’s indeed a function called summary() in R.

- -
#Check out what summary() does for afl.margins
+ +
#Check out what summary() does for afl.margins
+summary(afl.margins)
+ +
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
+   0.00   12.75   30.50   35.30   50.50  116.00
+

Pretty cool, no?

Also try it out for other kinds of variables like afl.finalists or as.character(afl.finalists)

+
+
+
Summarizing dataframes

Let’s try out summarizing a dataframe as well.

@@ -583,7 +637,7 @@
Quick cheat sheet: When to use what?
-
---
title: "Descriptive Statistics: Central and Variability measures"
output: html_notebook
---
```{r}
#Initial packages
install.packages("lsr")
```

In this notebook, we'll take a look at how to explore a dataset.

Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what **descriptive statistics** is all about.

#### Describing data

Imagine you've conducted an experiment involving measurements from 20 animals. If you wanted to report the outcome of your experiment to an audience, you wouldn’t want to talk through each and every data point. Instead, you report a summary, such as ‘The 20 animals had an average weight of 15 grams’, thus saving your audience valuable time and mental energy. This notebook focuses on such summaries of numerical information including distributions, measures of central tendency and measures of variability. 

##### What exactly is a distribution?

If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below.

![Empirical and theoretical distributions](dice rolling probability.png)

But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.)

Apart from *looking* at how a data is distributed, the most important descriptive statistics for numerical data are those measuring the location of a frequency distribution and its spread. The location tells us something about the average or *typical* individual—where the observations are centered. The spread tells us how variable the measurements are from individual to individual—how widely scattered the observations are around the center. The proportion is the most important descriptive statistic for a categorical variable, measuring the fraction of observations in a given category. 

##### But why is it needed?
The importance of calculating some sort of a centre of a distribution seems obvious. How else do we address questions like “Which species is larger?” or “Which drug yielded the greatest response?” The importance of describing distribution spread is less obvious but no less crucial, at least in biology. In some fields of science, variability around a central value is instrument noise or measurement error, but in biology much of the variability signifies real differences among individuals. Different individuals respond differently to treatments, and this variability begs measurement. (Adapted from Chapter 3, Whitlock & Schluter, 2015)

That's a lot of theory, let's dive into some data now.

**Loading the Australian Football League Dataset**

```{r}
#Change the path according to your PC
load("aflsmall.Rdata")
library(lsr)
who()
```

As you can see there are multiple variables of different class and size. 

Let's take a look at afl.margins variable.

```{r}
print(afl.margins)
```

This output doesn’t make it easy to get a sense of what the data is actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data.

Let's try to plot it.

**Frequency distribution**

```{r}
hist (afl.margins)
```

As you can see, different margins in a sample will have different measurements. We can see this variability with a **frequency distribution**. The frequency of a specific measurement in a sample is the number of observations having a particular value of the measurement. The frequency distribution shows how often each value of the variable occurs in the sample. 

Therefore, here we have plotted a histogram for the afl.margins variable which gives the frequency distribution of the different margin values.

**Skewness**

If you observe the graph, you will find that it is not entirely symmetrical. A measure of such asymmetry is called **Skewness**. If the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel), then we say that the data are _negatively skewed_. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are _positively skewed_.

`psych` package contains a `skew()` function that you can use to calculate skewness. 

Try finding the skewness for the above data for afl.margins using skew() function and also try to guess whether this data is positively or negatively skewed.

```{r}
library(psych)
#Try finding skewness of afl.margins here
```

Although such a graphical representation gives a 'gist' of the data but it is useful to find some "summary" statistics as well.

##### Measures of Central Tendency
In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you’d like to know something about the “average” or “middle” of your data lies. The two most commonly used measures are the mean, median and mode.

**Mean**

As you've already seen in previous classes, the mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values.

Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins
```{r}
mean(afl.margins)      # average margin
mean(afl.margins[1:5]) # mean of the margin from the first 5 games

```

**Median**

The second measure is the median. It is just the middle value of a set of observations. 
*Try : Guess the median for 56, 31, 56, 8 and 32 *

Probably you mentally arranged these numbers in ascending order first and then found the middle value. If there were a list of numbers like this `8, 14, 31, 32, 56, 56` . You will then find the average of middle 2 values. 

Now try finding out the median for afl.margins.

```{r}
median(afl.margins)
```

**Difference between Mean and Median**

Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data.

![Difference between mean and median](pic2.png)
*Fig 5.2 from Learning Statistics with R by D. Navarro*

**Some key points**

- If data is nominal scale, then it’s probably best to use the mode instead of mean or median.

- If your data are ordinal scale, you’re more likely to want to use the median than the mean.

- For interval and ratio scale data, either mean or median is generally acceptable. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values.

*You can read more about this in Section 5.1.4, Learning Statistics with R by D. Navarro*

Now let's take a look at some more data:

` -100,2,3,4,5,6,7,8,9,10`

If you observed such data in real life, you will probably think that -100 is an **_outlier_**, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely but you don’t always get such cut-and-dried examples. For instance, you might get this instead:

` -15,2,3,4,5,6,7,8,9,12`

The `-15` looks a bit suspicious, but not anywhere near as much as `-100` did. In this case, it’s a
little trickier. It might be a legitimate observation, it might not. In such situations, the mean might give you an error as it is highly sensitive to one or two extreme values, and is thus not considered to be a robust measure.

In such situations, one solution is to use the median or another is to use a **trimmed mean**. To calculate a trimmed mean, what you do is **discard** the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. This helps in taking the mean by excluding the outliers.

Let's try trimming the mean for above data.

```{r}
dataset <- c(-15,2,3,4,5,6,7,8,9,12)
mean(x = dataset, trim = .1)
#Try calculating 5% trimmed mean for above dataset
```

**Mode**

So far we've seen how to find the mean and median but what about mode. The **mode** of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the _lsr_ package has a function called modeOf() that does this. 

Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins

```{r}
#afl.mode = 
#afl.mode
modeOf(x = afl.margins)
maxFreq(x = afl.margins)
```

So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability.

##### Measures of variability

This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be?

**Range**

The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the `range()` function.

```{r}
#Find range of afl.margins here
```

But what about the earlier data we saw, ` -100,2,3,4,5,6,7,8,9,10`. Without removing the outlier, we'll get a range of 110 but without the outlier, we'll get a range of only 8.

**Inter-quartile Range (IQR)**
That is why there is something called the interquartile range (IQR) which is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. A 10% _quantile_ or _percentile_ of a data set is defined as the smallest number _x_ such that 10% of the data is less than _x_.

Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.
```{r}
#Use the functions quantile(x = afl.margins, prob = 0.2) for 20% quantile and IQR() 

```

IQR can simply be thought as the range spanned by the “middle half” of the data.

**Variance**

In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (X~i~ - $\overline{X}$). ($\overline{X}$ is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we'll get the **variance** of the data. 

Try finding out the variance using `var()`.

```{r}
#Use var() for finding variance of afl.margins
```

_Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro_

**Standard Deviation**

But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the _root mean square deviation_ for interpreting the spread of data points. This is called _Standard Deviation_ and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package.

Try to find out the standard deviation of afl.margins.

```{r}
#Find out Std dev. here
```

##### Quick cheat sheet: When to use what?

- Range: 
  - Gives full spread of data. 
  - Very vulnerable to outliers

- Interquartile range: 
  - Gives the “middle half” of data
  - Robust, and complements the median nicely
  
- Variance:
  - Average squared deviation from the mean
  - It’s mathematically elegant but it’s completely uninterpretable

- Standard deviation:
  - Square root of the variance
  - Fairly elegant mathematically, and can be interpreted pretty well
  - Complements mean and is the most popular measure of variation

Now that we've learnt about the different methods of describing a data, it would've been awesome if R could summarize all of this for us together, right? 

There's indeed a function called `summary()` in R.

```{r}
#Check out what summary() does for afl.margins
```

Pretty cool, no?

Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)`

Let's try out summarizing a dataframe as well.

```{r}
load("clinicaltrial.Rdata")
#Check the name of the variable in the environment which contains the dataframe and try summarizing it
```

The `psych` package also has a function called `describe()` for dataframes. Don't forget to check it out too!

In fact, you can also describe these statistics group wise. 

For instance, run `describeBy( x=clin.trial, group=clin.trial$therapy )`

```{r}
describeBy( x=clin.trial, group=clin.trial$therapy )
```
Notice that, the output displays asterisks for factor variables, in order to draw your attention to the fact that the descriptive statistics that it has calculated won’t be very meaningful for those variables.

Another more general command for grouping is `by()` 

Try running the following chunk and compare the results with the `describeBy()` command above.

```{r}
by(data=clin.trial, INDICES=clin.trial$therapy, FUN=describe)
#Also try replacing describe in FUN above with summary
```

What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy.We can use `aggregate()` command.

```{r}
aggregate( formula = mood.gain ~ drug + therapy, 
           data = clin.trial,
           FUN = mean) 
#1 mood.gain by drug/therapy combination
#2 data is in the clin.trial data frame
#3 print out group means

#Try interchanging the positions of drug and therapy above
```

That's all for today!

+
---
title: "Descriptive Statistics: Central and Variability measures"
output: html_notebook
---
```{r}
#Initial packages
install.packages("lsr")
```

In this notebook, we'll take a look at how to explore a dataset.

Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what **descriptive statistics** is all about.

#### Describing data

Imagine you've conducted an experiment involving measurements from 20 animals. If you wanted to report the outcome of your experiment to an audience, you wouldn’t want to talk through each and every data point. Instead, you report a summary, such as ‘The 20 animals had an average weight of 15 grams’, thus saving your audience valuable time and mental energy. This notebook focuses on such summaries of numerical information including distributions, measures of central tendency and measures of variability. 

##### What exactly is a distribution?

If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below.

![Empirical and theoretical distributions](dice rolling probability.png)

But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.)

Apart from *looking* at how a data is distributed, the most important descriptive statistics for numerical data are those measuring the location of a frequency distribution and its spread. The location tells us something about the average or *typical* individual—where the observations are centered. The spread tells us how variable the measurements are from individual to individual—how widely scattered the observations are around the center. The proportion is the most important descriptive statistic for a categorical variable, measuring the fraction of observations in a given category. 

##### But why is it needed?
The importance of calculating some sort of a centre of a distribution seems obvious. How else do we address questions like “Which species is larger?” or “Which drug yielded the greatest response?” The importance of describing distribution spread is less obvious but no less crucial, at least in biology. In some fields of science, variability around a central value is instrument noise or measurement error, but in biology much of the variability signifies real differences among individuals. Different individuals respond differently to treatments, and this variability begs measurement. (Adapted from Chapter 3, Whitlock & Schluter, 2015)

That's a lot of theory, let's dive into some data now.

**Loading the Australian Football League Dataset**

```{r}
#Change the path according to your PC
load("aflsmall.Rdata")
library(lsr)
who()
```

As you can see there are multiple variables of different class and size. 

Let's take a look at afl.margins variable.

```{r}
print(afl.margins)
```

This output doesn’t make it easy to get a sense of what the data is actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data.

Let's try to plot it.

**Frequency distribution**

```{r}
hist (afl.margins)
```

As you can see, different margins in a sample will have different measurements. We can see this variability with a **frequency distribution**. The frequency of a specific measurement in a sample is the number of observations having a particular value of the measurement. The frequency distribution shows how often each value of the variable occurs in the sample. 

Therefore, here we have plotted a histogram for the afl.margins variable which gives the frequency distribution of the different margin values.

**Skewness**

If you observe the graph, you will find that it is not entirely symmetrical. A measure of such asymmetry is called **Skewness**. If the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel), then we say that the data are _negatively skewed_. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are _positively skewed_.

`psych` package contains a `skew()` function that you can use to calculate skewness. 

Try finding the skewness for the above data for afl.margins using skew() function and also try to guess whether this data is positively or negatively skewed.

```{r}
library(psych)
#Try finding skewness of afl.margins here
```

Although such a graphical representation gives a 'gist' of the data but it is useful to find some "summary" statistics as well.

##### Measures of Central Tendency
In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you’d like to know something about the “average” or “middle” of your data lies. The two most commonly used measures are the mean, median and mode.

**Mean**

As you've already seen in previous classes, the mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values.

Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins
```{r}
mean(afl.margins)      # average margin
mean(afl.margins[1:5]) # mean of the margin from the first 5 games

```

**Median**

The second measure is the median. It is just the middle value of a set of observations. 
*Try : Guess the median for 56, 31, 56, 8 and 32 *

Probably you mentally arranged these numbers in ascending order first and then found the middle value. If there were a list of numbers like this `8, 14, 31, 32, 56, 56` . You will then find the average of middle 2 values. 

Now try finding out the median for afl.margins.

```{r}
median(afl.margins)
```

**Difference between Mean and Median**

Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data.

![Difference between mean and median](pic2.png)
*Fig 5.2 from Learning Statistics with R by D. Navarro*

**Some key points**

- If data is nominal scale, then it’s probably best to use the mode instead of mean or median.

- If your data are ordinal scale, you’re more likely to want to use the median than the mean.

- For interval and ratio scale data, either mean or median is generally acceptable. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values.

*You can read more about this in Section 5.1.4, Learning Statistics with R by D. Navarro*

Now let's take a look at some more data:

` -100,2,3,4,5,6,7,8,9,10`

If you observed such data in real life, you will probably think that -100 is an **_outlier_**, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely but you don’t always get such cut-and-dried examples. For instance, you might get this instead:

` -15,2,3,4,5,6,7,8,9,12`

The `-15` looks a bit suspicious, but not anywhere near as much as `-100` did. In this case, it’s a
little trickier. It might be a legitimate observation, it might not. In such situations, the mean might give you an error as it is highly sensitive to one or two extreme values, and is thus not considered to be a robust measure.

In such situations, one solution is to use the median or another is to use a **trimmed mean**. To calculate a trimmed mean, what you do is **discard** the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. This helps in taking the mean by excluding the outliers.

Let's try trimming the mean for above data.

```{r}
dataset <- c(-15,2,3,4,5,6,7,8,9,12)
mean(x = dataset, trim = .1)
#Try calculating 5% trimmed mean for above dataset
```

**Mode**

So far we've seen how to find the mean and median but what about mode. The **mode** of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the _lsr_ package has a function called modeOf() that does this. 

Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins

```{r}
#afl.mode = 
#afl.mode
modeOf(afl.margins)
maxFreq(afl.margins)
```

So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability.



```{r}
mean(afl.margins)
```

##### Measures of variability

This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be?


```{r}
plot(afl.margins)
```
**Range**

The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the `range()` function.

```{r}
#Find range of afl.margins here using the range function
```

But what about the earlier data we saw, ` -100,2,3,4,5,6,7,8,9,10`. Without removing the outlier, we'll get a range of 110 but without the outlier, we'll get a range of only 8.

**Inter-quartile Range (IQR)**
That is why there is something called the interquartile range (IQR) which is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. A 10% _quantile_ or _percentile_ of a data set is defined as the smallest number _x_ such that 10% of the data is less than _x_.

Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.
```{r}
#Use the functions quantile(x = afl.margins, prob = 0.25) for 25% quantile and IQR() 
quantile(x = afl.margins, prob = 0.2)

```

IQR can simply be thought as the range spanned by the “middle half” of the data.


```{r}
quantile( x = afl.margins, probs = c(.25,.75) )
# try using IQR() here

```


**Variance**

In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (X~i~ - $\overline{X}$). ($\overline{X}$ is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we'll get the **variance** of the data. 


Try finding out the variance using `var()`.

```{r}
#Use var() for finding variance of afl.margins
```

_Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro_

Also note that the division is by N-1 for variance for a sample! Why is that not N? 
This we will discuss later. 

**Standard Deviation**

But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the _root mean square deviation_ for interpreting the spread of data points. This is called _Standard Deviation_ and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package.

Try to find out the standard deviation of afl.margins.

```{r}
#Find out Std dev. here using sd()
```

##### Quick cheat sheet: When to use what?

- Range: 
  - Gives full spread of data. 
  - Very vulnerable to outliers

- Interquartile range: 
  - Gives the “middle half” of data
  - Robust, and complements the median nicely
  
- Variance:
  - Average squared deviation from the mean
  - It’s mathematically elegant but it’s completely uninterpretable

- Standard deviation:
  - Square root of the variance
  - Fairly elegant mathematically, and can be interpreted pretty well
  - Complements mean and is the most popular measure of variation
  
  
##### Derivation for deviation based variance estimates 
Mean absolute deviation; Variance; SD
 
Say, the sample is sample = [20,30,40]; 
the mean of the sample is then 30

The deviation of each sample data point from the mean is:
deviation = [20-30, 30-30, 40-30];
          = [-10, 0, 10]

absolute deviation = [10,0,10] 
mean of absolute deviation = (10+0+10)/3 

square of the deviation = [-10^2, 0^2, 10^2];
mean of the squared deviations = variance = (100+0+100)/3

root of mean squared deviation = standard deviation = sqrt(variance)


##### Bessel's correction
While calculating variance and standard deviation of the sample, we are always
trying to estimate the variance and standard deviation of the population. 

Remember the heights of students example! 

Now since the sample variance and standard deviation is biased and less than
that of the population, we divide by N-1 instead of N to inflate the estimates.

So the variance after Bessel's correction should be 
mean of the squared deviations = variance = (100+0+100)/(3-1)

  
##### Summary function
Now that we've learnt about the different methods of describing a data, it would've been awesome if R could summarize all of this for us together, right? 

There's indeed a function called `summary()` in R.

```{r}
#Check out what summary() does for afl.margins
summary(afl.margins)

```

Pretty cool, no?

Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)`



##### Summarizing dataframes 

Let's try out summarizing a dataframe as well.

```{r}
load("clinicaltrial.Rdata")
#Check the name of the variable in the environment which contains the dataframe and try summarizing it
```

The `psych` package also has a function called `describe()` for dataframes. Don't forget to check it out too!

In fact, you can also describe these statistics group wise. 

For instance, run `describeBy( x=clin.trial, group=clin.trial$therapy )`

```{r}
describeBy( x=clin.trial, group=clin.trial$therapy )
```
Notice that, the output displays asterisks for factor variables, in order to draw your attention to the fact that the descriptive statistics that it has calculated won’t be very meaningful for those variables.

Another more general command for grouping is `by()` 

Try running the following chunk and compare the results with the `describeBy()` command above.

```{r}
by(data=clin.trial, INDICES=clin.trial$therapy, FUN=describe)
#Also try replacing describe in FUN above with summary
```

What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy.We can use `aggregate()` command.

```{r}
aggregate( formula = mood.gain ~ drug + therapy, 
           data = clin.trial,
           FUN = mean) 
#1 mood.gain by drug/therapy combination
#2 data is in the clin.trial data frame
#3 print out group means

#Try interchanging the positions of drug and therapy above
```

That's all for today!

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z;o{ohzQnwyQ#CB3Z<^HjIHm7Xg(-;*AMn`oG8JbU|3Ce~cv)@1M({hHLZUHFQ8Fx-8mhfl^PQ+2o2Q{v{@jmLfz#|rzT{r&vpbSJb)c$^bv z@0$DHTcCGx#^sZHU$zMCl2h=-`!@wT3z&E5uF?_sy$UWZ%9vEn<03A3+Tr#VuvzR~ z>t97v_OxigLbvVND-% Date: Mon, 6 Sep 2021 19:19:27 +0000 Subject: [PATCH 08/55] Set right figure paths Added correlate() function --- Module 3/Notebooks/Module3_Nb2.Rmd | 18 +- Module 3/Notebooks/Module3_Nb2.nb.html | 726 +++++++++++++++++++++++++ 2 files changed, 738 insertions(+), 6 deletions(-) create mode 100644 Module 3/Notebooks/Module3_Nb2.nb.html diff --git a/Module 3/Notebooks/Module3_Nb2.Rmd b/Module 3/Notebooks/Module3_Nb2.Rmd index 508c8cee..5d850e5b 100644 --- a/Module 3/Notebooks/Module3_Nb2.Rmd +++ b/Module 3/Notebooks/Module3_Nb2.Rmd @@ -65,8 +65,6 @@ _Reference: Chapter 5, Winter B._ So far we have focused entirely on how to construct descriptive statistics for a single variable. We haven’t talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables. ```{r} -setwd("/Users/lenovo1/Desktop/mtech/courses/stats/Datasets/data/") - #Let's load some data load( "parenthood.Rdata" ) who(TRUE) @@ -132,7 +130,7 @@ The correlation coefficient (or Pearson's correlation coefficient) between two v Look at the plots for different _r_ values: -![Correlation plots](/Users/lenovo1/Desktop/mtech/courses/stats/Images/fig4.png) +![Correlation plots](fig 4.png) ##### Covariance @@ -170,7 +168,7 @@ What did you find? It really depends on what you want to use the data for, and on how strong the correlations in your field tend to be. -![Correlation coefficient interpretation table](/Users/lenovo1/Desktop/mtech/courses/stats/Images/fig5.png) +![Correlation coefficient interpretation table](fig 5.png) Now let's take a look at this data called "Anscombe's Quartet" @@ -187,7 +185,12 @@ Were the correlation coefficients same? Now try plotting them. ```{r} -scatterplot() +scatterplot(x = X1, y = Y1,regLine = FALSE, smooth = FALSE) +scatterplot(x = X2, y = Y2,regLine = FALSE, smooth = FALSE) +scatterplot(x = X3, y = Y3,regLine = FALSE, smooth = FALSE) +scatterplot(x = X4, y = Y4,regLine = FALSE, smooth = FALSE) + + ``` Therefore, remember to always look at the scatterplot before attaching any interpretation to the data! @@ -199,7 +202,6 @@ If we have to properly define the role of Pearson's coefficient, we can say that But let's take a look at another dataset and find correlation between its variables. ```{r} -setwd("/Users/lenovo1/Desktop/mtech/courses/stats/Datasets/data/") load( "effort.Rdata" ) effort cor( effort$hours, effort$grade ) @@ -230,6 +232,10 @@ Now the correlation coefficient we get is different from the Perason's correlati cor( effort$hours, effort$grade, method = "spearman") ``` +##### the correlate() function +Try using this function to find the relationship between several variables in a dataframe at once. + + ##### Handling missing values We've seen in earlier lectures that there could be missing values in data which are represented by `NA` in R. One easy way to remove them is using `na.rm = TRUE` as argument in many functions. diff --git a/Module 3/Notebooks/Module3_Nb2.nb.html b/Module 3/Notebooks/Module3_Nb2.nb.html new file mode 100644 index 00000000..01b7fc3f --- /dev/null +++ b/Module 3/Notebooks/Module3_Nb2.nb.html @@ -0,0 +1,726 @@ + + + + + + + + + + + + + +Descriptive Statistics: Scaling and Correlations + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + +

After taking a first look at our data in the last notebook, now we want to start looking at it more closely as per our needs and requirements.

+
+

Scaling

+

In simple terms, scaling refers to changing size of an object without affecting its shape.

+
+
Linear Transformation:
+

A linear transformation involves addition, subtraction, multiplication, or division with a constant value. For example, if you add 1 to the numbers 2, 4, and 6, the resulting numbers (3, 5, and 7) are a linear transformation of the original numbers. Linear transformations are useful, because they allow you to represent your data in a metric that is suitable to you and your audience.

+

Centering:

+

‘Centering’ is a particularly common linear transformation. This linear transfor- mation is frequently applied to continuous predictor variables. To center a predictor variable, subtract the mean of that predictor variable from each data point. As a result, each data point is expressed in terms of how much it is above the mean (positive score) or below the mean (negative score). Thus, subtracting the mean out of the variable expresses each data point as a mean-deviation score. The value zero now has a new meaning for this variable: it is at the ‘center’ of the variable’s distribution, namely, the mean.

+

Standardizing:

+

A second common linear transformation is ‘standardizing’ or ‘z–scoring’. For standardizing, the centered variable is divided by the standard deviation of the sample.

+

Let’s look at an example:

+

The following are response durations from a psycholinguistic experiment:

+

460ms 480ms 500ms 520ms 540ms

+

The mean of these five numbers is 500ms.

+

Centering these numbers results in the following:

+

− 40ms − 20ms 0ms +20ms + 40ms

+

The standard deviation (learnt in last notebook) for these numbers is ~32ms.

+

To ‘standardize’, we have to divide the centered data by the standard deviation. For example, the first point, –40ms, divided by 32ms, yields –1.3. Since each data point is divided by the same number, this change qualifies as a linear transformation.

+

As a result of standardization, you get the following numbers (rounded to one digit):

+

−1.3z − 0.6z 0z + 0.6z +1.3z

+

The raw response duration 460ms is –40ms (after centering), which corresponds to being 1.3 standard deviations below the mean. Thus, standardization involves re-expressing the data in terms of how many standard deviations they are away from the mean.

+
+
+
But why this extra effort?
+

Standardizing is a way of getting rid of a variable’s metric. In a situation with multiple variables, each variable may have a different standard deviation, but by dividing each variable by the respective standard deviation, it is possible to convert all variables into a scale of standard units. This sometimes may help in making variables comparable, for example, when assessing the relative impact of multiple predictors. For example, if you can imagine we have two questionnaires - one for extraversion where you scored 2 out of 10 and the other for grumpiness where you scored 35 out of 50, then it doesn’t make a lot of sense to try to compare your raw score of 2 on the extraversion questionnaire to your raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are “about” fundamentally different things, so this would be like comparing apples to oranges. But if you standardize them, they will still become comparable in some sense.

+

Let’s also examine the score of 35 out of 50 for grumpiness. Would this mean that you’re 70% grumpy? Instead of interpreting raw data this way, it would make more sense if we describe your grumpiness in terms of the overall distribution of the grumpiness of humans which is possible through standardisation i.e. where do you lie on the grumpiness spectrum of the all humans? ;)

+ + + +
#Try it out yourself
+#Define a vector with Grumpiness scores of you and your friends and find the z score for your self
+X =                        
+z = (X - mean(X)) / sd(X)
+ + + +

Reference: Chapter 5, Winter B.

+
+
+
+

Correlation

+

So far we have focused entirely on how to construct descriptive statistics for a single variable. We haven’t talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables.

+ + + +
#Let's load some data
+load( "parenthood.Rdata" )
+who(TRUE)
+ + +
   -- Name --     -- Class --   -- Size --
+   parenthood     data.frame    100 x 4   
+    $dan.sleep    numeric       100       
+    $baby.sleep   numeric       100       
+    $dan.grump    numeric       100       
+    $day          integer       100
+ + + + + + +
#Try describe() for the above dataframe
+ + + + + + +
#Let's also take a graphical look at the data 
+hist(parenthood$dan.sleep)
+ + +

+ + +

+#Try plotting for the other 2 variables
+
+ + + +

But we now want to take a look at the relationship between two variables. n order to visualize that, it is better to plot a scatter plot. (Plotting graphs will be covered in detail a separate notebook).

+

Brief note on Scatterplots:

+

In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don’t really have a clear opinion about what the causal relationship is (e.g., does A cause B, or does B cause A, or does some other variable C controls both A and B). If that’s the case, it doesn’t really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it’s conventional to plot the cause variable on the x-axis, and the effect variable on the y-axis.

+

Suppose our goal is to draw a scatterplot displaying the relationship between the amount of sleep that Dan gets (dan.sleep) and how grumpy she is the next day (dan.grump). Do you suspect a causal relationship here?

+

A simple way to plot these scatter plots is to use the scatterplot() function in the car package.

+

Let’s load the package and get started.

+ + + +
install.packages("car")
+ + +
Installing package into ‘/cloud/lib/x86_64-pc-linux-gnu-library/4.1’
+(as ‘lib’ is unspecified)
+trying URL 'http://package-proxy/focal/src/contrib/car_3.0-11.tar.gz'
+Content type 'application/x-tar' length 1569269 bytes (1.5 MB)
+==================================================
+downloaded 1.5 MB
+
+* installing *binary* package ‘car’ ...
+* DONE (car)
+
+The downloaded source packages are in
+    ‘/tmp/Rtmpj4OOz7/downloaded_packages’
+ + +
install.packages("Rcpp")
+ + +
Installing package into ‘/cloud/lib/x86_64-pc-linux-gnu-library/4.1’
+(as ‘lib’ is unspecified)
+trying URL 'http://package-proxy/focal/src/contrib/Rcpp_1.0.7.tar.gz'
+Content type 'application/x-tar' length 4189703 bytes (4.0 MB)
+==================================================
+downloaded 4.0 MB
+
+* installing *binary* package ‘Rcpp’ ...
+* DONE (Rcpp)
+
+The downloaded source packages are in
+    ‘/tmp/Rtmpj4OOz7/downloaded_packages’
+ + + + + + +
library(car)
+ + +
Loading required package: carData
+Registered S3 method overwritten by 'data.table':
+  method           from
+  print.data.table     
+
+Attaching package: ‘car’
+
+The following object is masked from ‘package:psych’:
+
+    logit
+ + +
scatterplot( dan.grump ~ dan.sleep, data = parenthood, regLine = FALSE, smooth = FALSE)
+ + +

+ + +
scatterplot
+ + +
function (x, ...) 
+{
+    UseMethod("scatterplot")
+}
+<bytecode: 0x5578de5c2560>
+<environment: namespace:car>
+ + + + + + +
#Plot a scatter plot for baby.sleep and dan.grump variables
+ + + +

Just by plain observation and comparison, you can see that the relationship is qualitatively the same in both cases: more sleep equals less grump! However, it’s also pretty obvious that the relationship between dan.sleep and dan.grump is stronger than the relationship between baby.sleep and dan.grump.

+

But what about the plot between baby.sleep and dan.sleep?

+ + + +
#Plot baby sleep and dan sleep here
+ + + +

Is the direction of this plot same as the earlier plots? What about strength?

+
+
Correlation coefficient
+

In order to to quantitatively represent the relationships of strength and direction we discussed above, we can use correlation coefficient.

+

The correlation coefficient (or Pearson’s correlation coefficient) between two variables X and Y (sometimes denoted rXY ) is a measure that varies from -1 to 1. When r = -1 it means that we have a perfect negative relationship, and when r = 1 it means we have a perfect positive relationship. When r = 0, there’s no relationship at all.

+

Look at the plots for different r values:

+
+ +

Correlation plots

+
+
+
+
Covariance
+

The covariance between two variables X and Y is a generalisation of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables:

+

\[\begin{align*} + + Cov (X, Y) = \frac{1}{N-1}\sum_{i=1}^{N} (X- \overline{X} ) (Y- \overline{Y} ) \\ + + \end{align*}\]

+

Covariance can be understood as an “average cross product” between X and Y . The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If it is positive, then the covariance is also positive; and if the relationship is negative then the covariance is also negative. But as it has weird units (try seeing for yourself), it si difficult to interpret and therefore we standardise the covariance, the exact same way that the z-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations.

+

This is what we call as the correlation coefficent, r:

+

\[\begin{align*} + + r~XY~ = \frac{Cov(X,Y)}{\sigma_{X} \sigma_{Y}} + +\end{align*}\]

+

This way, covariance properties are retained and it also becomes interpretable.

+

Now let’s check out how to code this using cor().

+ + + +
cor(x = parenthood$dan.sleep, y = parenthood$dan.grump)
+
+#Try giving the entire dataframe 'parenthood' as input in cor()
+ + + +

What did you find?

+
+
+
What does r = 0.4 mean?
+

It really depends on what you want to use the data for, and on how strong the correlations in your field tend to be.

+
+ +

Correlation coefficient interpretation table

+
+

Now let’s take a look at this data called “Anscombe’s Quartet”

+ + + +
load( "anscombesquartet.Rdata" )
+cor( X1, Y1 )
+ + +
[1] 0.8164205
+ + +
cor( X2, Y2 )
+ + +
[1] 0.8162365
+ + +
cor (X3, Y3)
+ + +
[1] 0.8162867
+ + +
cor (X4, Y4)
+ + +
[1] 0.8165214
+ + + +

Were the correlation coefficients same?

+

Now try plotting them.

+ + + +
scatterplot(x = X1, y = Y1,regLine = FALSE, smooth = FALSE)
+ + +

+ + +
scatterplot(x = X2, y = Y2,regLine = FALSE, smooth = FALSE)
+ + +

+ + +
scatterplot(x = X3, y = Y3,regLine = FALSE, smooth = FALSE)
+ + +

+ + +
scatterplot(x = X4, y = Y4,regLine = FALSE, smooth = FALSE)
+ + +

+ + +
NA
+NA
+ + + +

Therefore, remember to always look at the scatterplot before attaching any interpretation to the data!

+

If we have to properly define the role of Pearson’s coefficient, we can say that it actually measures the strength of the linear relationship between two variables. In other words, it gives a measure of the extent to which the data all tend to fall on a single, perfectly straight line.

+
+
+
Spearman’s Rank Order Correlation Coefficient
+

But let’s take a look at another dataset and find correlation between its variables.

+ + + +
load( "effort.Rdata" )
+effort
+ + +
+ +
+ + +
cor( effort$hours, effort$grade )
+ + +
[1] 0.909402
+ + + +

If you plot this -

+ + + +
scatterplot(effort$hours, effort$grade, regLine = TRUE, smooth = FALSE)
+ + +

+ + + +

The correlation r = 0.91 we get above doe snot represent the actual relationship the plot is depicting. What we’re looking for is something that captures the fact that there is a perfect ordinal relationship here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade.

+

If we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, let’s rank all 10 of the students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2).

+ + + +
hours.rank <- rank( effort$hours )   # rank students by hours worked
+grade.rank <- rank( effort$grade )   # rank students by grade received
+
+#Now try cor() function for these
+cor( hours.rank, grade.rank )
+ + + +

Now the correlation coefficient we get is different from the Perason’s correlation coefficient r we got earlier. This new correlation coefficient that we got is called ‘Spearman’s Correlation Coefficient’, denoted by \(\rho\).

+ + + +
#Execute this and compare with the correlation coefficient we got above
+cor( effort$hours, effort$grade, method = "spearman")
+ + + +
+
+
the correlate() function
+

Try using this function to find the relationship between several variables in a dataframe at once.

+
+
+
Handling missing values
+

We’ve seen in earlier lectures that there could be missing values in data which are represented by NA in R. One easy way to remove them is using na.rm = TRUE as argument in many functions.

+

But what if we have missing values in a dataframe where we have to find correlations across variables.

+

Let’s look at such a dataset.

+ + + +
load( "parenthood2.Rdata" )
+print( parenthood2 )
+ + +
+ +
+ + +
describe( parenthood2 )
+ + +
+ +
+ + +
#Check how many missing values are there for each variable - compare the values in 'n' with the number of days.
+ + + +

Now, let’s try finding correlations for this dataframe.

+ + + +
cor(parenthood2)
+ + +
           dan.sleep baby.sleep dan.grump day
+dan.sleep          1         NA        NA  NA
+baby.sleep        NA          1        NA  NA
+dan.grump         NA         NA         1  NA
+day               NA         NA        NA   1
+ + + +

In order top overcome this problem, we can use use as an argument in the cor() function. Try out the following.

+ + + +
cor(parenthood2, use = "complete.obs")
+ + +
             dan.sleep baby.sleep   dan.grump         day
+dan.sleep   1.00000000  0.6394985 -0.89951468  0.06132891
+baby.sleep  0.63949845  1.0000000 -0.58656066  0.14555814
+dan.grump  -0.89951468 -0.5865607  1.00000000 -0.06816586
+day         0.06132891  0.1455581 -0.06816586  1.00000000
+ + +
cor(parenthood2, use = "pairwise.complete.obs")
+ + +
             dan.sleep  baby.sleep    dan.grump          day
+dan.sleep   1.00000000  0.61472303 -0.903442442 -0.076796665
+baby.sleep  0.61472303  1.00000000 -0.567802669  0.058309485
+dan.grump  -0.90344244 -0.56780267  1.000000000  0.005833399
+day        -0.07679667  0.05830949  0.005833399  1.000000000
+ + + +

When we choose use = "complete.obs", R will completely ignore all cases (i.e., all rows in our parenthood2 data frame) that have any missing values at all. For eg., if you choose use = “complete.obs” R will ignore that row completely: that is, even when it’s trying to calculate the correlation between dan.sleep and dan.grump, observation 1 will be ignored, because the value of baby.sleep is missing for that observation.

+

Whereas when we set use = "pairwise.complete.obs" R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of parenthood2 is for baby.sleep R will only drop observation 1 when baby.sleep is one of the variables involved: and so R keeps observation 1 when trying to correlate dan.sleep and dan.grump.

+

The above operation can also be performed by another function called correlate() in lsr package.

+

Try it out.

+ + + +
#Try correlate() for parenthood2 here
+ + + +

Reference : Chapter 5, D. Navarro

+

That’s all folks!

+ +
+
+ +
---
title: "Descriptive Statistics: Scaling and Correlations"
output: html_notebook
---

After taking a first look at our data in the last notebook, now we want to start looking at it more closely as per our needs and requirements. 

#### Scaling

In simple terms, scaling refers to changing size of an object without affecting its shape.

##### Linear Transformation:

A linear transformation involves addition, subtraction, multiplication, or division with a constant value. For example, if you add 1 to the numbers 2, 4, and 6, the resulting numbers (3, 5, and 7) are a linear transformation of the original numbers. 
Linear transformations are useful, because they allow you to represent your data in a metric that is suitable to you and your audience.

**Centering:**

‘Centering’ is a particularly common linear transformation. This linear transfor- mation is frequently applied to continuous predictor variables. 
To center a predictor variable, subtract the mean of that predictor variable from each data point. As a result, each data point is expressed in terms of how much it is above the mean (positive score) or below the mean (negative score). Thus, subtracting the mean out of the variable expresses each data point as a mean-deviation score. The value zero now has a new meaning for this variable: it is at the ‘center’ of the variable’s distribution, namely, the mean.

**Standardizing:**

A second common linear transformation is ‘standardizing’ or ‘z–scoring’. For standardizing, the centered variable is divided by the standard deviation of the sample.

Let's look at an example: 

The following are response durations from a psycholinguistic experiment:

`460ms 480ms 500ms 520ms 540ms`

The mean of these five numbers is `500ms`. 

Centering these numbers results in the following:

`− 40ms − 20ms 0ms +20ms + 40ms`

The standard deviation (learnt in last notebook) for these numbers is `~32ms`. 

To ‘standardize’, we have to divide the centered data by the standard deviation. For example, the first point, `–40ms`, divided by `32ms`, yields `–1.3`. Since each data point is divided by the same number, this change qualifies as a linear transformation.

As a result of standardization, you get the following numbers (rounded to one digit):

`−1.3z − 0.6z 0z + 0.6z +1.3z`

The raw response duration `460ms` is `–40ms` (after centering), which corresponds to being `1.3` standard deviations below the mean. Thus, standardization involves re-expressing the data in terms of **how many standard deviations they are away from the mean**.

##### But why this extra effort?

Standardizing is a way of getting rid of a variable’s metric. In a situation with multiple variables, each variable may have a different standard deviation, but by dividing each variable by the respective standard deviation, it is possible to convert all variables into a scale of **standard units**. This sometimes may help in making variables comparable, for example, when assessing the relative impact of multiple predictors. For example, if you can imagine we have two questionnaires - one for extraversion where you scored 2 out of 10 and the other for grumpiness where you scored 35 out of 50, then it doesn’t make a lot of sense to try to compare your raw score of 2 on the extraversion questionnaire to your raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are “about” fundamentally different things, so this would be like comparing apples to oranges. But if you standardize them, they will still become comparable in some sense.

Let's also examine the score of 35 out of 50 for grumpiness. Would this mean that you're 70% grumpy? Instead of interpreting raw data this way, it would make more sense if we describe your grumpiness in terms of the overall distribution of the grumpiness of humans which is possible through  standardisation i.e. where do you lie on the grumpiness spectrum of the all humans? ;)

```{r}
#Try it out yourself
#Define a vector with Grumpiness scores of you and your friends and find the z score for your self
X =                        
z = (X - mean(X)) / sd(X)
```

_Reference: Chapter 5, Winter B._

#### Correlation

So far we have focused entirely on how to construct descriptive statistics for a single variable. We haven’t talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables.

```{r}
#Let's load some data
load( "parenthood.Rdata" )
who(TRUE)
```

```{r}
#Try describe() for the above dataframe
```


```{r}
#Let's also take a graphical look at the data 
hist(parenthood$dan.sleep)

#Try plotting for the other 2 variables

```

But we now want to take a look at the relationship between two variables. n order to visualize that, it is better to plot a **scatter plot.** (Plotting graphs will be covered in detail a separate notebook).

_Brief note on Scatterplots:_

In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don’t really have a clear opinion about what the causal relationship is (e.g., does A cause B, or does B cause A, or does some other variable C controls both A and B). If that’s the case, it doesn’t really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it’s conventional to plot the **cause** variable on the **x-axis**, and the **effect** variable on the **y-axis**. 

Suppose our goal is to draw a scatterplot displaying the relationship between the amount of sleep that Dan gets (dan.sleep) and how grumpy she is the next day (dan.grump). _Do you suspect a causal relationship here?_

A simple way to plot these scatter plots is to use the scatterplot() function in the car package. 

Let's load the package and get started.

```{r}
install.packages("car")
install.packages("Rcpp")
```


```{r}
library(car)
scatterplot( dan.grump ~ dan.sleep, data = parenthood, regLine = FALSE, smooth = FALSE)
scatterplot
```

```{r}
#Plot a scatter plot for baby.sleep and dan.grump variables
```


Just by plain observation and comparison, you can see that the relationship is qualitatively the same in both cases: more sleep equals less grump! However, it’s also pretty obvious that the relationship between dan.sleep and dan.grump is stronger than the relationship between baby.sleep and dan.grump. 

But what about the plot between baby.sleep and dan.sleep?

```{r}
#Plot baby sleep and dan sleep here
```

Is the direction of this plot same as the earlier plots? What about strength?

##### Correlation coefficient

In order to to quantitatively represent the relationships of strength and direction we discussed above, we can use correlation coefficient.

The correlation coefficient (or Pearson's correlation coefficient) between two variables X and Y (sometimes denoted _r~XY~_ ) is a measure that varies from -1 to 1. When _r_ = -1 it means that we have a perfect negative relationship, and when _r_ = 1 it means we have a perfect positive relationship. When _r_ = 0, there’s no relationship at all.

Look at the plots for different _r_ values:

![Correlation plots](fig 4.png)

##### Covariance

The covariance between two variables X and Y is a generalisation of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables:

 \begin{align*}
 
 Cov (X, Y) = \frac{1}{N-1}\sum_{i=1}^{N} (X- \overline{X} ) (Y- \overline{Y} )  \\
 
 \end{align*}
 
Covariance can be understood as an “average cross product” between X and Y . The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If it is positive, then the covariance is also positive; and if the relationship is negative then the covariance is also negative. But as it has weird units (try seeing for yourself), it si difficult to interpret and therefore we standardise the covariance, the exact same way that the z-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations. 

This is what we call as the correlation coefficent, _r_:

\begin{align*}

 r~XY~ = \frac{Cov(X,Y)}{\sigma_{X} \sigma_{Y}}

\end{align*}

This way, covariance properties are retained and it also becomes interpretable.

Now let's check out how to code this using cor().

```{r}
cor(x = parenthood$dan.sleep, y = parenthood$dan.grump)

#Try giving the entire dataframe 'parenthood' as input in cor()
```

What did you find?

##### What does r = 0.4 mean?

It really depends on what you want to use the data for, and on how strong the correlations in your field tend to be.

![Correlation coefficient interpretation table](fig 5.png)
 
Now let's take a look at this data called "Anscombe's Quartet"
 
```{r}
load( "anscombesquartet.Rdata" )
cor( X1, Y1 )
cor( X2, Y2 )
cor (X3, Y3)
cor (X4, Y4)
```

Were the correlation coefficients same?

Now try plotting them.

```{r}
scatterplot(x = X1, y = Y1,regLine = FALSE, smooth = FALSE)
scatterplot(x = X2, y = Y2,regLine = FALSE, smooth = FALSE)
scatterplot(x = X3, y = Y3,regLine = FALSE, smooth = FALSE)
scatterplot(x = X4, y = Y4,regLine = FALSE, smooth = FALSE)


```

Therefore, remember to always look at the scatterplot before attaching any interpretation to the data!

If we have to properly define the role of Pearson's coefficient, we can say that it actually measures the strength of the linear relationship between two variables. In other words, it gives a measure of the extent to which the data all tend to fall on a single, perfectly straight line.

##### Spearman's Rank Order Correlation Coefficient
 
But let's take a look at another dataset and find correlation between its variables.

```{r}
load( "effort.Rdata" )
effort
cor( effort$hours, effort$grade )
```

If you plot this - 

```{r}
scatterplot(effort$hours, effort$grade, regLine = TRUE, smooth = FALSE)
```

The correlation _r_ = 0.91 we get above doe snot represent the actual relationship the plot is depicting. What we’re looking for is something that captures the fact that there is a perfect **ordinal relationship** here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade.

If we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, let's rank all 10 of the students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2).

```{r}
hours.rank <- rank( effort$hours )   # rank students by hours worked
grade.rank <- rank( effort$grade )   # rank students by grade received

#Now try cor() function for these
cor( hours.rank, grade.rank )
```

Now the correlation coefficient we get is different from the Perason's correlation coefficient _r_ we got earlier. This new correlation coefficient that we got is called '**Spearman's Correlation Coefficient**', denoted by $\rho$.

```{r}
#Execute this and compare with the correlation coefficient we got above
cor( effort$hours, effort$grade, method = "spearman")
```

##### the correlate() function
Try using this function to find the relationship between several variables in a dataframe at once.


##### Handling missing values

We've seen in earlier lectures that there could be missing values in data which are represented by `NA` in R. One easy way to remove them is using `na.rm = TRUE` as argument in many functions.

But what if we have missing values in a dataframe where we have to find correlations across variables.

Let's look at such a dataset.

```{r}
load( "parenthood2.Rdata" )
print( parenthood2 )
describe( parenthood2 ) 
#Check how many missing values are there for each variable - compare the values in 'n' with the number of days.
```

Now, let's try finding correlations for this dataframe.

```{r}
cor(parenthood2)
```

In order top overcome this problem, we can use `use` as an argument in the cor() function. Try out the following.

```{r}
cor(parenthood2, use = "complete.obs")
cor(parenthood2, use = "pairwise.complete.obs")
```

When we choose `use = "complete.obs"`, R will completely ignore all cases (i.e., all rows in our parenthood2 data frame) that have any missing values at all. For eg., if you choose use = "complete.obs" R will ignore that row completely: that is, even when it’s trying to calculate the correlation between dan.sleep and dan.grump, observation 1 will be ignored, because the value of baby.sleep is missing for that observation.

Whereas when we set `use = "pairwise.complete.obs"` R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of parenthood2 is for baby.sleep R will only drop observation 1 when baby.sleep is one of the variables involved: and so R keeps observation 1 when trying to correlate dan.sleep and dan.grump.

The above operation can also be performed by another function called `correlate()` in `lsr` package.

Try it out.
```{r}
#Try correlate() for parenthood2 here
```

_Reference : Chapter 5, D. Navarro_

That's all folks!

+ + + +
+ + + + + + + + + + + + + + + + From b03352e621ceb7fb49f13e5e8d39626cf043d2d5 Mon Sep 17 00:00:00 2001 From: Arjun Date: Tue, 7 Sep 2021 04:13:10 +0000 Subject: [PATCH 09/55] Added scale() and associated histograms --- Module 3/Notebooks/Module3_Nb2.Rmd | 16 ++++++++++++ Module 3/Notebooks/Module3_Nb2.nb.html | 34 +++++++++++++++++++++++++- 2 files changed, 49 insertions(+), 1 deletion(-) diff --git a/Module 3/Notebooks/Module3_Nb2.Rmd b/Module 3/Notebooks/Module3_Nb2.Rmd index 5d850e5b..879883c4 100644 --- a/Module 3/Notebooks/Module3_Nb2.Rmd +++ b/Module 3/Notebooks/Module3_Nb2.Rmd @@ -58,6 +58,22 @@ X = z = (X - mean(X)) / sd(X) ``` +Using scale() to center and normalize +```{r} +load("aflsmall.Rdata") +afl.margins_c <- scale(afl.margins, scale = FALSE) +afl.margins_z <- scale(afl.margins) +``` + +Plotting the histogram +```{r} +hist(afl.margins) +hist(afl.margins_c) +hist(afl.margins_z) +``` + + + _Reference: Chapter 5, Winter B._ #### Correlation diff --git a/Module 3/Notebooks/Module3_Nb2.nb.html b/Module 3/Notebooks/Module3_Nb2.nb.html index 01b7fc3f..25dd7025 100644 --- a/Module 3/Notebooks/Module3_Nb2.nb.html +++ b/Module 3/Notebooks/Module3_Nb2.nb.html @@ -252,6 +252,38 @@
But why this extra effort?
+

Using scale() to center and normalize

+ + + +
load("aflsmall.Rdata")
+
+ + + +

Plotting the histogram

+ + + +
hist(afl.margins)
+ + +

+ + +
hist(afl.margins_c)
+ + +

+ + +
hist(afl.margins_z)
+ + +

+ + +

Reference: Chapter 5, Winter B.

@@ -659,7 +691,7 @@
Handling missing values
-
---
title: "Descriptive Statistics: Scaling and Correlations"
output: html_notebook
---

After taking a first look at our data in the last notebook, now we want to start looking at it more closely as per our needs and requirements. 

#### Scaling

In simple terms, scaling refers to changing size of an object without affecting its shape.

##### Linear Transformation:

A linear transformation involves addition, subtraction, multiplication, or division with a constant value. For example, if you add 1 to the numbers 2, 4, and 6, the resulting numbers (3, 5, and 7) are a linear transformation of the original numbers. 
Linear transformations are useful, because they allow you to represent your data in a metric that is suitable to you and your audience.

**Centering:**

‘Centering’ is a particularly common linear transformation. This linear transfor- mation is frequently applied to continuous predictor variables. 
To center a predictor variable, subtract the mean of that predictor variable from each data point. As a result, each data point is expressed in terms of how much it is above the mean (positive score) or below the mean (negative score). Thus, subtracting the mean out of the variable expresses each data point as a mean-deviation score. The value zero now has a new meaning for this variable: it is at the ‘center’ of the variable’s distribution, namely, the mean.

**Standardizing:**

A second common linear transformation is ‘standardizing’ or ‘z–scoring’. For standardizing, the centered variable is divided by the standard deviation of the sample.

Let's look at an example: 

The following are response durations from a psycholinguistic experiment:

`460ms 480ms 500ms 520ms 540ms`

The mean of these five numbers is `500ms`. 

Centering these numbers results in the following:

`− 40ms − 20ms 0ms +20ms + 40ms`

The standard deviation (learnt in last notebook) for these numbers is `~32ms`. 

To ‘standardize’, we have to divide the centered data by the standard deviation. For example, the first point, `–40ms`, divided by `32ms`, yields `–1.3`. Since each data point is divided by the same number, this change qualifies as a linear transformation.

As a result of standardization, you get the following numbers (rounded to one digit):

`−1.3z − 0.6z 0z + 0.6z +1.3z`

The raw response duration `460ms` is `–40ms` (after centering), which corresponds to being `1.3` standard deviations below the mean. Thus, standardization involves re-expressing the data in terms of **how many standard deviations they are away from the mean**.

##### But why this extra effort?

Standardizing is a way of getting rid of a variable’s metric. In a situation with multiple variables, each variable may have a different standard deviation, but by dividing each variable by the respective standard deviation, it is possible to convert all variables into a scale of **standard units**. This sometimes may help in making variables comparable, for example, when assessing the relative impact of multiple predictors. For example, if you can imagine we have two questionnaires - one for extraversion where you scored 2 out of 10 and the other for grumpiness where you scored 35 out of 50, then it doesn’t make a lot of sense to try to compare your raw score of 2 on the extraversion questionnaire to your raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are “about” fundamentally different things, so this would be like comparing apples to oranges. But if you standardize them, they will still become comparable in some sense.

Let's also examine the score of 35 out of 50 for grumpiness. Would this mean that you're 70% grumpy? Instead of interpreting raw data this way, it would make more sense if we describe your grumpiness in terms of the overall distribution of the grumpiness of humans which is possible through  standardisation i.e. where do you lie on the grumpiness spectrum of the all humans? ;)

```{r}
#Try it out yourself
#Define a vector with Grumpiness scores of you and your friends and find the z score for your self
X =                        
z = (X - mean(X)) / sd(X)
```

_Reference: Chapter 5, Winter B._

#### Correlation

So far we have focused entirely on how to construct descriptive statistics for a single variable. We haven’t talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables.

```{r}
#Let's load some data
load( "parenthood.Rdata" )
who(TRUE)
```

```{r}
#Try describe() for the above dataframe
```


```{r}
#Let's also take a graphical look at the data 
hist(parenthood$dan.sleep)

#Try plotting for the other 2 variables

```

But we now want to take a look at the relationship between two variables. n order to visualize that, it is better to plot a **scatter plot.** (Plotting graphs will be covered in detail a separate notebook).

_Brief note on Scatterplots:_

In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don’t really have a clear opinion about what the causal relationship is (e.g., does A cause B, or does B cause A, or does some other variable C controls both A and B). If that’s the case, it doesn’t really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it’s conventional to plot the **cause** variable on the **x-axis**, and the **effect** variable on the **y-axis**. 

Suppose our goal is to draw a scatterplot displaying the relationship between the amount of sleep that Dan gets (dan.sleep) and how grumpy she is the next day (dan.grump). _Do you suspect a causal relationship here?_

A simple way to plot these scatter plots is to use the scatterplot() function in the car package. 

Let's load the package and get started.

```{r}
install.packages("car")
install.packages("Rcpp")
```


```{r}
library(car)
scatterplot( dan.grump ~ dan.sleep, data = parenthood, regLine = FALSE, smooth = FALSE)
scatterplot
```

```{r}
#Plot a scatter plot for baby.sleep and dan.grump variables
```


Just by plain observation and comparison, you can see that the relationship is qualitatively the same in both cases: more sleep equals less grump! However, it’s also pretty obvious that the relationship between dan.sleep and dan.grump is stronger than the relationship between baby.sleep and dan.grump. 

But what about the plot between baby.sleep and dan.sleep?

```{r}
#Plot baby sleep and dan sleep here
```

Is the direction of this plot same as the earlier plots? What about strength?

##### Correlation coefficient

In order to to quantitatively represent the relationships of strength and direction we discussed above, we can use correlation coefficient.

The correlation coefficient (or Pearson's correlation coefficient) between two variables X and Y (sometimes denoted _r~XY~_ ) is a measure that varies from -1 to 1. When _r_ = -1 it means that we have a perfect negative relationship, and when _r_ = 1 it means we have a perfect positive relationship. When _r_ = 0, there’s no relationship at all.

Look at the plots for different _r_ values:

![Correlation plots](fig 4.png)

##### Covariance

The covariance between two variables X and Y is a generalisation of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables:

 \begin{align*}
 
 Cov (X, Y) = \frac{1}{N-1}\sum_{i=1}^{N} (X- \overline{X} ) (Y- \overline{Y} )  \\
 
 \end{align*}
 
Covariance can be understood as an “average cross product” between X and Y . The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If it is positive, then the covariance is also positive; and if the relationship is negative then the covariance is also negative. But as it has weird units (try seeing for yourself), it si difficult to interpret and therefore we standardise the covariance, the exact same way that the z-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations. 

This is what we call as the correlation coefficent, _r_:

\begin{align*}

 r~XY~ = \frac{Cov(X,Y)}{\sigma_{X} \sigma_{Y}}

\end{align*}

This way, covariance properties are retained and it also becomes interpretable.

Now let's check out how to code this using cor().

```{r}
cor(x = parenthood$dan.sleep, y = parenthood$dan.grump)

#Try giving the entire dataframe 'parenthood' as input in cor()
```

What did you find?

##### What does r = 0.4 mean?

It really depends on what you want to use the data for, and on how strong the correlations in your field tend to be.

![Correlation coefficient interpretation table](fig 5.png)
 
Now let's take a look at this data called "Anscombe's Quartet"
 
```{r}
load( "anscombesquartet.Rdata" )
cor( X1, Y1 )
cor( X2, Y2 )
cor (X3, Y3)
cor (X4, Y4)
```

Were the correlation coefficients same?

Now try plotting them.

```{r}
scatterplot(x = X1, y = Y1,regLine = FALSE, smooth = FALSE)
scatterplot(x = X2, y = Y2,regLine = FALSE, smooth = FALSE)
scatterplot(x = X3, y = Y3,regLine = FALSE, smooth = FALSE)
scatterplot(x = X4, y = Y4,regLine = FALSE, smooth = FALSE)


```

Therefore, remember to always look at the scatterplot before attaching any interpretation to the data!

If we have to properly define the role of Pearson's coefficient, we can say that it actually measures the strength of the linear relationship between two variables. In other words, it gives a measure of the extent to which the data all tend to fall on a single, perfectly straight line.

##### Spearman's Rank Order Correlation Coefficient
 
But let's take a look at another dataset and find correlation between its variables.

```{r}
load( "effort.Rdata" )
effort
cor( effort$hours, effort$grade )
```

If you plot this - 

```{r}
scatterplot(effort$hours, effort$grade, regLine = TRUE, smooth = FALSE)
```

The correlation _r_ = 0.91 we get above doe snot represent the actual relationship the plot is depicting. What we’re looking for is something that captures the fact that there is a perfect **ordinal relationship** here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade.

If we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, let's rank all 10 of the students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2).

```{r}
hours.rank <- rank( effort$hours )   # rank students by hours worked
grade.rank <- rank( effort$grade )   # rank students by grade received

#Now try cor() function for these
cor( hours.rank, grade.rank )
```

Now the correlation coefficient we get is different from the Perason's correlation coefficient _r_ we got earlier. This new correlation coefficient that we got is called '**Spearman's Correlation Coefficient**', denoted by $\rho$.

```{r}
#Execute this and compare with the correlation coefficient we got above
cor( effort$hours, effort$grade, method = "spearman")
```

##### the correlate() function
Try using this function to find the relationship between several variables in a dataframe at once.


##### Handling missing values

We've seen in earlier lectures that there could be missing values in data which are represented by `NA` in R. One easy way to remove them is using `na.rm = TRUE` as argument in many functions.

But what if we have missing values in a dataframe where we have to find correlations across variables.

Let's look at such a dataset.

```{r}
load( "parenthood2.Rdata" )
print( parenthood2 )
describe( parenthood2 ) 
#Check how many missing values are there for each variable - compare the values in 'n' with the number of days.
```

Now, let's try finding correlations for this dataframe.

```{r}
cor(parenthood2)
```

In order top overcome this problem, we can use `use` as an argument in the cor() function. Try out the following.

```{r}
cor(parenthood2, use = "complete.obs")
cor(parenthood2, use = "pairwise.complete.obs")
```

When we choose `use = "complete.obs"`, R will completely ignore all cases (i.e., all rows in our parenthood2 data frame) that have any missing values at all. For eg., if you choose use = "complete.obs" R will ignore that row completely: that is, even when it’s trying to calculate the correlation between dan.sleep and dan.grump, observation 1 will be ignored, because the value of baby.sleep is missing for that observation.

Whereas when we set `use = "pairwise.complete.obs"` R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of parenthood2 is for baby.sleep R will only drop observation 1 when baby.sleep is one of the variables involved: and so R keeps observation 1 when trying to correlate dan.sleep and dan.grump.

The above operation can also be performed by another function called `correlate()` in `lsr` package.

Try it out.
```{r}
#Try correlate() for parenthood2 here
```

_Reference : Chapter 5, D. Navarro_

That's all folks!

+
---
title: "Descriptive Statistics: Scaling and Correlations"
output: html_notebook
---

After taking a first look at our data in the last notebook, now we want to start looking at it more closely as per our needs and requirements. 

#### Scaling

In simple terms, scaling refers to changing size of an object without affecting its shape.

##### Linear Transformation:

A linear transformation involves addition, subtraction, multiplication, or division with a constant value. For example, if you add 1 to the numbers 2, 4, and 6, the resulting numbers (3, 5, and 7) are a linear transformation of the original numbers. 
Linear transformations are useful, because they allow you to represent your data in a metric that is suitable to you and your audience.

**Centering:**

‘Centering’ is a particularly common linear transformation. This linear transfor- mation is frequently applied to continuous predictor variables. 
To center a predictor variable, subtract the mean of that predictor variable from each data point. As a result, each data point is expressed in terms of how much it is above the mean (positive score) or below the mean (negative score). Thus, subtracting the mean out of the variable expresses each data point as a mean-deviation score. The value zero now has a new meaning for this variable: it is at the ‘center’ of the variable’s distribution, namely, the mean.

**Standardizing:**

A second common linear transformation is ‘standardizing’ or ‘z–scoring’. For standardizing, the centered variable is divided by the standard deviation of the sample.

Let's look at an example: 

The following are response durations from a psycholinguistic experiment:

`460ms 480ms 500ms 520ms 540ms`

The mean of these five numbers is `500ms`. 

Centering these numbers results in the following:

`− 40ms − 20ms 0ms +20ms + 40ms`

The standard deviation (learnt in last notebook) for these numbers is `~32ms`. 

To ‘standardize’, we have to divide the centered data by the standard deviation. For example, the first point, `–40ms`, divided by `32ms`, yields `–1.3`. Since each data point is divided by the same number, this change qualifies as a linear transformation.

As a result of standardization, you get the following numbers (rounded to one digit):

`−1.3z − 0.6z 0z + 0.6z +1.3z`

The raw response duration `460ms` is `–40ms` (after centering), which corresponds to being `1.3` standard deviations below the mean. Thus, standardization involves re-expressing the data in terms of **how many standard deviations they are away from the mean**.

##### But why this extra effort?

Standardizing is a way of getting rid of a variable’s metric. In a situation with multiple variables, each variable may have a different standard deviation, but by dividing each variable by the respective standard deviation, it is possible to convert all variables into a scale of **standard units**. This sometimes may help in making variables comparable, for example, when assessing the relative impact of multiple predictors. For example, if you can imagine we have two questionnaires - one for extraversion where you scored 2 out of 10 and the other for grumpiness where you scored 35 out of 50, then it doesn’t make a lot of sense to try to compare your raw score of 2 on the extraversion questionnaire to your raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are “about” fundamentally different things, so this would be like comparing apples to oranges. But if you standardize them, they will still become comparable in some sense.

Let's also examine the score of 35 out of 50 for grumpiness. Would this mean that you're 70% grumpy? Instead of interpreting raw data this way, it would make more sense if we describe your grumpiness in terms of the overall distribution of the grumpiness of humans which is possible through  standardisation i.e. where do you lie on the grumpiness spectrum of the all humans? ;)

```{r}
#Try it out yourself
#Define a vector with Grumpiness scores of you and your friends and find the z score for your self
X =                        
z = (X - mean(X)) / sd(X)
```

Using scale() to center and normalize
```{r}
load("aflsmall.Rdata")
afl.margins_c <- scale(afl.margins, scale = FALSE)
afl.margins_z <- scale(afl.margins)
```

Plotting the histogram
```{r}
hist(afl.margins)
hist(afl.margins_c)
hist(afl.margins_z)
```



_Reference: Chapter 5, Winter B._

#### Correlation

So far we have focused entirely on how to construct descriptive statistics for a single variable. We haven’t talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables.

```{r}
#Let's load some data
load( "parenthood.Rdata" )
who(TRUE)
```

```{r}
#Try describe() for the above dataframe
```


```{r}
#Let's also take a graphical look at the data 
hist(parenthood$dan.sleep)

#Try plotting for the other 2 variables

```

But we now want to take a look at the relationship between two variables. n order to visualize that, it is better to plot a **scatter plot.** (Plotting graphs will be covered in detail a separate notebook).

_Brief note on Scatterplots:_

In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don’t really have a clear opinion about what the causal relationship is (e.g., does A cause B, or does B cause A, or does some other variable C controls both A and B). If that’s the case, it doesn’t really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it’s conventional to plot the **cause** variable on the **x-axis**, and the **effect** variable on the **y-axis**. 

Suppose our goal is to draw a scatterplot displaying the relationship between the amount of sleep that Dan gets (dan.sleep) and how grumpy she is the next day (dan.grump). _Do you suspect a causal relationship here?_

A simple way to plot these scatter plots is to use the scatterplot() function in the car package. 

Let's load the package and get started.

```{r}
install.packages("car")
install.packages("Rcpp")
```


```{r}
library(car)
scatterplot( dan.grump ~ dan.sleep, data = parenthood, regLine = FALSE, smooth = FALSE)
scatterplot
```

```{r}
#Plot a scatter plot for baby.sleep and dan.grump variables
```


Just by plain observation and comparison, you can see that the relationship is qualitatively the same in both cases: more sleep equals less grump! However, it’s also pretty obvious that the relationship between dan.sleep and dan.grump is stronger than the relationship between baby.sleep and dan.grump. 

But what about the plot between baby.sleep and dan.sleep?

```{r}
#Plot baby sleep and dan sleep here
```

Is the direction of this plot same as the earlier plots? What about strength?

##### Correlation coefficient

In order to to quantitatively represent the relationships of strength and direction we discussed above, we can use correlation coefficient.

The correlation coefficient (or Pearson's correlation coefficient) between two variables X and Y (sometimes denoted _r~XY~_ ) is a measure that varies from -1 to 1. When _r_ = -1 it means that we have a perfect negative relationship, and when _r_ = 1 it means we have a perfect positive relationship. When _r_ = 0, there’s no relationship at all.

Look at the plots for different _r_ values:

![Correlation plots](fig 4.png)

##### Covariance

The covariance between two variables X and Y is a generalisation of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables:

 \begin{align*}
 
 Cov (X, Y) = \frac{1}{N-1}\sum_{i=1}^{N} (X- \overline{X} ) (Y- \overline{Y} )  \\
 
 \end{align*}
 
Covariance can be understood as an “average cross product” between X and Y . The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If it is positive, then the covariance is also positive; and if the relationship is negative then the covariance is also negative. But as it has weird units (try seeing for yourself), it si difficult to interpret and therefore we standardise the covariance, the exact same way that the z-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations. 

This is what we call as the correlation coefficent, _r_:

\begin{align*}

 r~XY~ = \frac{Cov(X,Y)}{\sigma_{X} \sigma_{Y}}

\end{align*}

This way, covariance properties are retained and it also becomes interpretable.

Now let's check out how to code this using cor().

```{r}
cor(x = parenthood$dan.sleep, y = parenthood$dan.grump)

#Try giving the entire dataframe 'parenthood' as input in cor()
```

What did you find?

##### What does r = 0.4 mean?

It really depends on what you want to use the data for, and on how strong the correlations in your field tend to be.

![Correlation coefficient interpretation table](fig 5.png)
 
Now let's take a look at this data called "Anscombe's Quartet"
 
```{r}
load( "anscombesquartet.Rdata" )
cor( X1, Y1 )
cor( X2, Y2 )
cor (X3, Y3)
cor (X4, Y4)
```

Were the correlation coefficients same?

Now try plotting them.

```{r}
scatterplot(x = X1, y = Y1,regLine = FALSE, smooth = FALSE)
scatterplot(x = X2, y = Y2,regLine = FALSE, smooth = FALSE)
scatterplot(x = X3, y = Y3,regLine = FALSE, smooth = FALSE)
scatterplot(x = X4, y = Y4,regLine = FALSE, smooth = FALSE)


```

Therefore, remember to always look at the scatterplot before attaching any interpretation to the data!

If we have to properly define the role of Pearson's coefficient, we can say that it actually measures the strength of the linear relationship between two variables. In other words, it gives a measure of the extent to which the data all tend to fall on a single, perfectly straight line.

##### Spearman's Rank Order Correlation Coefficient
 
But let's take a look at another dataset and find correlation between its variables.

```{r}
load( "effort.Rdata" )
effort
cor( effort$hours, effort$grade )
```

If you plot this - 

```{r}
scatterplot(effort$hours, effort$grade, regLine = TRUE, smooth = FALSE)
```

The correlation _r_ = 0.91 we get above doe snot represent the actual relationship the plot is depicting. What we’re looking for is something that captures the fact that there is a perfect **ordinal relationship** here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade.

If we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, let's rank all 10 of the students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2).

```{r}
hours.rank <- rank( effort$hours )   # rank students by hours worked
grade.rank <- rank( effort$grade )   # rank students by grade received

#Now try cor() function for these
cor( hours.rank, grade.rank )
```

Now the correlation coefficient we get is different from the Perason's correlation coefficient _r_ we got earlier. This new correlation coefficient that we got is called '**Spearman's Correlation Coefficient**', denoted by $\rho$.

```{r}
#Execute this and compare with the correlation coefficient we got above
cor( effort$hours, effort$grade, method = "spearman")
```

##### the correlate() function
Try using this function to find the relationship between several variables in a dataframe at once.


##### Handling missing values

We've seen in earlier lectures that there could be missing values in data which are represented by `NA` in R. One easy way to remove them is using `na.rm = TRUE` as argument in many functions.

But what if we have missing values in a dataframe where we have to find correlations across variables.

Let's look at such a dataset.

```{r}
load( "parenthood2.Rdata" )
print( parenthood2 )
describe( parenthood2 ) 
#Check how many missing values are there for each variable - compare the values in 'n' with the number of days.
```

Now, let's try finding correlations for this dataframe.

```{r}
cor(parenthood2)
```

In order top overcome this problem, we can use `use` as an argument in the cor() function. Try out the following.

```{r}
cor(parenthood2, use = "complete.obs")
cor(parenthood2, use = "pairwise.complete.obs")
```

When we choose `use = "complete.obs"`, R will completely ignore all cases (i.e., all rows in our parenthood2 data frame) that have any missing values at all. For eg., if you choose use = "complete.obs" R will ignore that row completely: that is, even when it’s trying to calculate the correlation between dan.sleep and dan.grump, observation 1 will be ignored, because the value of baby.sleep is missing for that observation.

Whereas when we set `use = "pairwise.complete.obs"` R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of parenthood2 is for baby.sleep R will only drop observation 1 when baby.sleep is one of the variables involved: and so R keeps observation 1 when trying to correlate dan.sleep and dan.grump.

The above operation can also be performed by another function called `correlate()` in `lsr` package.

Try it out.
```{r}
#Try correlate() for parenthood2 here
```

_Reference : Chapter 5, D. Navarro_

That's all folks!

From 8c22e9dc9afd7aa403c49e695cf9f2dd26aa4179 Mon Sep 17 00:00:00 2001 From: Arjun Date: Sat, 11 Sep 2021 18:24:13 +0000 Subject: [PATCH 10/55] Added load dataset command --- Module 3/Notebooks/Plotting.Rmd | 2 ++ 1 file changed, 2 insertions(+) diff --git a/Module 3/Notebooks/Plotting.Rmd b/Module 3/Notebooks/Plotting.Rmd index 904ee23c..6537dee1 100644 --- a/Module 3/Notebooks/Plotting.Rmd +++ b/Module 3/Notebooks/Plotting.Rmd @@ -91,6 +91,8 @@ _Reference: Chapter 2, Whitlock and Schluter_ To examine data for single variable, we show its frequency distribution. The _frequency distribution_ of a variable is the number of occurrences of all values in the data. ```{r} +#Let's load aflsmall.Rdata +load("aflsmall.Rdata") #Here's a frequency table for afl.finalists table(afl.finalists) ``` From 7db764e64a33197929063b8821a28205e3b53bf8 Mon Sep 17 00:00:00 2001 From: Arjun Date: Mon, 20 Sep 2021 11:49:43 +0000 Subject: [PATCH 11/55] I added a few lines to illustrate committing to Git --- Module 3/Notebooks/Distributions.Rmd | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/Module 3/Notebooks/Distributions.Rmd b/Module 3/Notebooks/Distributions.Rmd index 00bd9618..b841db32 100644 --- a/Module 3/Notebooks/Distributions.Rmd +++ b/Module 3/Notebooks/Distributions.Rmd @@ -3,6 +3,13 @@ title: "Inferential Statistics: Probability & Distributions - 1" output: html_notebook --- + +You: made a change +I: added a new command +You: want to sync the new command + +### Follow this carefully + So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics. #### Pre-requisite: Probability @@ -104,7 +111,7 @@ Some basic terminology - We’ll let `N` denote the number of dice rolls in our Let's generate a binomial distribution in R: ```{r} -dbinom( n = 4, size = 20, prob = 1/6 ) +dbinom( x = 4, size = 20, prob = 1/6 ) ``` The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6. From 5551f1af9312c528e086690f05bedacd0f08a494 Mon Sep 17 00:00:00 2001 From: Arjun Date: Mon, 20 Sep 2021 20:01:33 +0000 Subject: [PATCH 12/55] Added normal and other distributions. --- Module 3/Notebooks/Distributions.Rmd | 117 ++++- Module 3/Notebooks/Distributions.nb.html | 588 +++++++++++++++++++++++ 2 files changed, 700 insertions(+), 5 deletions(-) create mode 100644 Module 3/Notebooks/Distributions.nb.html diff --git a/Module 3/Notebooks/Distributions.Rmd b/Module 3/Notebooks/Distributions.Rmd index b841db32..589493a3 100644 --- a/Module 3/Notebooks/Distributions.Rmd +++ b/Module 3/Notebooks/Distributions.Rmd @@ -19,6 +19,20 @@ Probability theory is a branch of mathematics that tells you how often different In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]* +Probability of heads when you toss a coin? +P(H) = 0.5 + +Frequentist view is also an objective view. +Coin - [H, T, H, T, H, T, H] +Prob - [1, 1/2, 2/3, 0.5, ... ] + + + + + +Coin - [1, 1, 1, 1, 0] +P'(H) - 0.7 + - - - - **A short note on Models** @@ -104,14 +118,18 @@ If any of these elementary events occurs, then E is also said to have occurred. *Refer to section 9.4.1, Navarro D., for the detailed example* -Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skulls, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`. +Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`. `X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N. +4 ~ Binomial(1/6, 20) + +5 ~ Binomial(1/2, 10) + Let's generate a binomial distribution in R: ```{r} -dbinom( x = 4, size = 20, prob = 1/6 ) +dbinom( x = 1, size = 20, prob = 1/6) ``` The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6. @@ -126,7 +144,7 @@ If we want to find the probability of obtaining an outcome smaller than or equal ```{r} #Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls -pbinom( q= 4, size = 20, prob = 1/6) +pbinom( q= 3, size = 20, prob = 1/6) #Practice - Find probability of getting 0-5 heads in 50 trials of coin flip ``` @@ -135,7 +153,7 @@ In other words, value of 4 is actually the 76.9th percentile of this binomial di Now let’s say we want to calculate the 75th percentile of the binomial distribution. ```{r} -qbinom( p = 0.75, size = 20, prob = 1/6 ) +qbinom( p = 0.566, size = 20, prob = 1/6 ) #Practice - Find the 40th percentile ``` @@ -152,6 +170,95 @@ hist(z, col = 'steelblue') All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. -End of part 1 + +##### Normal Distribution + +Most frequently encountered distribution. +Eg: heights of all students in the class, marks obtained in exams, etc + +Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution. + + + +```{r} +normal_distribution <- rnorm(10000, mean = 0, sd = 1) +histogram_normal_distribution <- hist(normal_distribution) +plot(histogram_normal_distribution$mids,histogram_normal_distribution$density) + +``` + +Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution + +The notation for a normal distribution is: X ∼ Normal(μ,σ) + + +dnorm tells you the probability of getting a particular outcome +```{r} +dnorm(x=0, mean=0, sd=1) + +``` +Cumulative normal distribution +```{r} +pnorm(2, mean = 0, sd = 1) +``` + +```{r} +qnorm(0.5 ,mean = , sd = 1) +``` + +##### Other useful distributions + +Some other distributions you may encounter include: +*1) t distribution* + +Looks like the normal distribution but has heavier tails. +Used when data looks like a normal distribution but the mean and SD are unknown. + +Use the following functions to visualize the t distribution: +dt(), pt(), qt() and rt() + +```{r} +t_distribution <- rt(10000, 3) +histogram_t_distribution <- hist(t_distribution) +plot(histogram_t_distribution$mids,histogram_t_distribution$density) +``` + + + + +*2) Chi square (χ2) distribution* + +All positive and heavily skewed to the left. +Used when data represents sum of squares of a normally distributed variables. + +Use the following functions to visualize the chi sq distribution: +dchisq(), pchisq(), qchisq(), rchisq(). + +```{r} +chisq_distribution <- rchisq(10000, 3, ncp = 0) +histogram_chisq_distribution <- hist(chisq_distribution) +plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density) +``` + + + + +3) F distribution + +This one looka a bit like the chi square distribution. But this distribution comes into picture when +one compares two chi sq distributions. + +Use the following functions to visualize the chi sq distribution: +df(), pf(), qf() and rf() + +```{r} +f_distribution <- rf(10000, 3, 5) +histogram_f_distribution <- hist(f_distribution) +plot(histogram_f_distribution$mids,histogram_f_distribution$density) +``` + + + +The End Reference - *Chapter 9, Navarro D.* diff --git a/Module 3/Notebooks/Distributions.nb.html b/Module 3/Notebooks/Distributions.nb.html new file mode 100644 index 00000000..862f186e --- /dev/null +++ b/Module 3/Notebooks/Distributions.nb.html @@ -0,0 +1,588 @@ + + + + + + + + + + + + + +Inferential Statistics: Probability & Distributions - 1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + +

You: made a change I: added a new command You: want to sync the new command

+
+

Follow this carefully

+

So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.

+
+

Pre-requisite: Probability

+
+
Difference between probability and statistics**
+

Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?

+

In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. [Chapter 9, Navarro D.]

+

Probability of heads when you toss a coin? P(H) = 0.5

+

Frequentist view is also an objective view. Coin - [H, T, H, T, H, T, H] Prob - [1, 1/2, 2/3, 0.5, … ]

+

Coin - [1, 1, 1, 1, 0] P’(H) - 0.7

+ ++++++ + + + + + + + + + + + + + + +
**Ashort note on Models**
Amodel is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. [Chapter 3, Winter B.]
+

But even though we know the models like P(heads) = 0.5, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?

+

We want to figure out which is the true model of the world. Is it P(heads) = 0.5 or is it P(heads) \(\ne\) 0.5?

+
+
+
What is probability really?
+

The frequentist view

+
+ +

Frequentist_graph

+
+

According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.

+

Advantages - It is objective: the probability of an event is necessarily grounded in the world. - It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.

+

But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?

+

The Bayesian view

+

Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this ‘degree of belief’?

+

One way is to use ‘rational gambling’. So a “subjective probability” will be operationalized in terms of what bets you’re willing to accept.

+

Advantage - You don’t need to be limited to those events that are repeatable.

+

Disadvantage - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an intelligent agent out there that believes in things.

+

In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).

+
+
+
Definitions
+

Refer to the example described in Section 9.3.1, Navarro D. for the following content.

+

Elementary event: Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.

+

Sample space: The set of all possible events (e.g., the wardrobe)

+

Probability: Numbers between 0 and 1.

+

For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.

+

If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).

+

Law of total probability: The probabilities of the elementary events need to add up to 1

+
+
+
+

Distributions

+

Let’s take a look at this and see what is a distribution.

+ + + +
pants <- data.frame(
+   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
+   label = c("X1", "X2", "X3", "X4", "X5"),
+   probability = c(0.5,0.3,0.1,0,0.1))
+
+pants
+ + + +

Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.

+ + + +
#Try plotting a bar graph of all the probabilities above
+ + + +

Let’s think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans” has occurred. Then what will be the probability of event E.

+

P(E) = P(X1) + P(X2) + P(X3)

+

If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:

+
+ +

Probability_rules

+
+
+
Binomial Distribution
+

Refer to section 9.4.1, Navarro D., for the detailed example

+

Some basic terminology - We’ll let N denote the number of dice rolls in our experiment; which is often referred to as the size parameter of our binomial distribution. We’ll also use θ to refer to the the probability that a single die comes up skull, a quantity that is usually called the success probability of the binomial. Finally, we’ll use X to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a random variable.

+

X ~ Binomial(θ, N) denotes X is generated randomly from a binomial distribution with parameters θ and N.

+

4 ~ Binomial(1/6, 20)

+

5 ~ Binomial(1/2, 10)

+

Let’s generate a binomial distribution in R:

+ + + +
dbinom( x = 3, size = 20, prob = 1/6)
+ + +
[1] 0.2378866
+ + + +

The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.

+

What if the dice is replaced by a coin in the above example? How will the probability change?

+ + + +
#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
+ + + +

There are different functions in R for different distributions as well as different ones for finding different quantity of interest.

+

If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use pbinom.

+ + + +
#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
+pbinom( q= 3, size = 20, prob = 1/6)
+ + +
[1] 0.5665456
+ + +
#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
+ + + +

In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.

+

Now let’s say we want to calculate the 75th percentile of the binomial distribution.

+ + + +
qbinom( p = 0.566, size = 20, prob = 1/6 )
+ + +
[1] 3
+ + +
#Practice - Find the 40th percentile
+ + + +

We’ve found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the rbinom function.

+ + + +
z <- rbinom( n = 100, size = 20, prob = 1/6 )
+z
+ + +
  [1] 2 5 3 1 6 4 2 5 4 4 5 3 3 4 0 3 3 3 4 3 3 1 3 2 2 6 3 5 2 8 2 5 4 4 2 2 1 6 2 3 4 2 3 3 4 4 3 1 2 3 2 1 2 1 4 4 6 3
+ [59] 4 6 2 5 7 3 2 5 5 5 4 2 4 0 1 2 5 5 2 6 3 3 3 1 3 3 3 3 2 3 5 3 3 3 4 3 2 3 3 5 6 3
+ + +
#Let's also plot this and see how it looks
+hist(z, col = 'steelblue')
+ + +

+ + + +

#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.

+

All these different functions d, p, q, n are also applicable to other distributions. E.g. dnorm, pnorm, qnorm, rnorm for Normal distribution.

+
+
+
Normal Distribution
+

Most frequently encountered distribution. Eg: heights of all students in the class, marks obtained in exams, etc

+

Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.

+ + + +
normal_distribution <- rnorm(10000, mean = 0, sd = 1) 
+histogram_normal_distribution <- hist(normal_distribution)
+ + +

+ + +
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density)
+
+ + +

+ + + +

Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution

+

The notation for a normal distribution is: X ∼ Normal(μ,σ)

+

dnorm tells you the probability of getting a particular outcome

+ + + +
dnorm(x=0, mean=0, sd=1)
+ + +
[1] 0.3989423
+ + + +

Cumulative normal distribution

+ + + +
pnorm(2, mean = 0, sd = 1)
+ + +
[1] 0.9772499
+ + + + + + +
qnorm(0.5 , mean = , sd = 1)
+ + +
[1] 0
+ + + +
+
+
Other useful distributions
+

Some other distributions you may encounter include: 1) t distribution

+

Looks like the normal distribution but has heavier tails. Used when data looks like a normal distribution but the mean and SD are unknown.

+

Use the following functions to visualize the t distribution: dt(), pt(), qt() and rt()

+ + + +
t_distribution <- rt(10000, 3)
+histogram_t_distribution <- hist(t_distribution)
+ + +

+ + +
plot(histogram_t_distribution$mids,histogram_t_distribution$density)
+ + +

+ + + +

2) Chi square (χ2) distribution

+

All positive and heavily skewed to the left.
+Used when data represents sum of squares of a normally distributed variables.

+

Use the following functions to visualize the chi sq distribution: dchisq(), pchisq(), qchisq(), rchisq().

+ + + +
chisq_distribution <- rchisq(10000, 3, ncp = 0)
+histogram_chisq_distribution <- hist(chisq_distribution)
+ + +

+ + +
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density)
+ + +

+ + + +
    +
  1. F distribution
    2. +
+

This one looka a bit like the chi square distribution. But this distribution comes into picture when one compares two chi sq distributions.

+

Use the following functions to visualize the chi sq distribution: df(), pf(), qf() and rf()

+ + + +
f_distribution <- rf(10000, 3, 5)
+histogram_f_distribution <- hist(f_distribution)
+ + +

+ + +
plot(histogram_f_distribution$mids,histogram_f_distribution$density)
+ + +

+ + + +

The End

+

Reference - Chapter 9, Navarro D.

+ +
+
+
+ +
---
title: "Inferential Statistics: Probability & Distributions - 1"
output: html_notebook
---


You: made a change
I: added a new command
You: want to sync the new command

### Follow this carefully

So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.

#### Pre-requisite: Probability

##### Difference between probability and statistics**
Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?

In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]*

Probability of heads when you toss a coin? 
P(H) = 0.5 

Frequentist view is also an objective view. 
Coin - [H, T, H, T, H, T, H]
Prob - [1, 1/2, 2/3, 0.5, ... ]





Coin - [1, 1, 1, 1, 0]
P'(H) - 0.7

- - - -
**A short note on Models**

A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]*
- - - -

But even though we know the models like `P(heads) = 0.5`, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?

We want to figure out which is the true model of the world. Is it *P(heads) = 0.5* or is it *P(heads) $\ne$ 0.5*?

##### What is probability really?

**The frequentist view**

![Frequentist_graph](Fig4.png)

According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.

 *Advantages*
 -  It is objective: the probability of an event is necessarily grounded in the world.
 -  It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.

But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?

**The Bayesian view**

Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this 'degree of belief'? 

One way is to use 'rational gambling'. So a “subjective probability” will be operationalized in terms of what bets you're willing to accept.

 *Advantage*
 - You don’t need to be limited to those events that are repeatable.
 
 *Disadvantage*
 - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an **intelligent agent** out there that believes in things. 


In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).

##### Definitions

Refer to the example described in *Section 9.3.1, Navarro D.* for the following content.

**Elementary event:** Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.

**Sample space:** The set of all possible events (e.g., the wardrobe)

**Probability:** Numbers between 0 and 1.

For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.

If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).

**Law of total probability:** The probabilities of the elementary events need to add up to 1

#### Distributions

Let's take a look at this and see what is a distribution. 

```{r}
pants <- data.frame(
   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
   label = c("X1", "X2", "X3", "X4", "X5"),
   probability = c(0.5,0.3,0.1,0,0.1))

pants
```
Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.

```{r}
#Try plotting a bar graph of all the probabilities above
```
Let's think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans" has occurred. 
Then what will be the probability of event E.

P(E) = P(X1) + P(X2) + P(X3)

If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:

![Probability_rules](Fig5.png)

##### Binomial Distribution

*Refer to section 9.4.1, Navarro D., for the detailed example*

Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`.

`X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N.

4 ~ Binomial(1/6, 20)

5 ~ Binomial(1/2, 10)

Let's generate a binomial distribution in R:

```{r}
dbinom( x = 1, size = 20, prob = 1/6)
```
The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.

What if the dice is replaced by a coin in the above example? How will the probability change? 

```{r}
#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
```
There are different functions in R for different distributions as well as different ones for finding different quantity of interest.

If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use `pbinom`.

```{r}
#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
pbinom( q= 3, size = 20, prob = 1/6)

#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
```
In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.

Now let’s say we want to calculate the 75th percentile of the binomial distribution.

```{r}
qbinom( p = 0.566, size = 20, prob = 1/6 )

#Practice - Find the 40th percentile
```

We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.

```{r}
z <- rbinom( n = 100, size = 20, prob = 1/6 )
z
#Let's also plot this and see how it looks
hist(z, col = 'steelblue')
```
#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.

All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. 


##### Normal Distribution

Most frequently encountered distribution.
Eg: heights of all students in the class, marks obtained in exams, etc

Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.



```{r}
normal_distribution <- rnorm(10000, mean = 0, sd = 1) 
histogram_normal_distribution <- hist(normal_distribution)
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density)

```

Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution

The notation for a normal distribution is: X ∼ Normal(μ,σ) 


dnorm tells you the probability of getting a particular outcome
```{r}
dnorm(x=0, mean=0, sd=1)

```
Cumulative normal distribution
```{r}
pnorm(2, mean = 0, sd = 1)
```

```{r}
qnorm(0.5 ,mean = , sd = 1)
```

##### Other useful distributions

Some other distributions you may encounter include:
*1) t distribution*

Looks like the normal distribution but has heavier tails. 
Used when data looks like a normal distribution but the mean and SD are unknown.

Use the following functions to visualize the t distribution: 
dt(), pt(), qt() and rt()

```{r}
t_distribution <- rt(10000, 3)
histogram_t_distribution <- hist(t_distribution)
plot(histogram_t_distribution$mids,histogram_t_distribution$density)
```




*2) Chi square (χ2) distribution*

All positive and heavily skewed to the left.  
Used when data represents sum of squares of a normally distributed variables.

Use the following functions to visualize the chi sq distribution: 
dchisq(), pchisq(), qchisq(), rchisq().

```{r}
chisq_distribution <- rchisq(10000, 3, ncp = 0)
histogram_chisq_distribution <- hist(chisq_distribution)
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density)
```




3) F distribution

This one looka a bit like the chi square distribution. But this distribution comes into picture when 
one compares two chi sq distributions.

Use the following functions to visualize the chi sq distribution: 
df(), pf(), qf() and rf()

```{r}
f_distribution <- rf(10000, 3, 5)
histogram_f_distribution <- hist(f_distribution)
plot(histogram_f_distribution$mids,histogram_f_distribution$density)
```



The End

Reference - *Chapter 9, Navarro D.*

+ + + +
+ + + + + + + + + + + + + + + + From 90280efcf16f655db1fc5d9355f5eec6ee269dca Mon Sep 17 00:00:00 2001 From: Arjun Date: Tue, 21 Sep 2021 04:26:22 +0000 Subject: [PATCH 13/55] Added line plots, Shapiro Wilk test, added definitions for dnorm, qnorm etc, removed additions from yesterday's class, --- Module 3/Notebooks/Distributions.Rmd | 55 ++++++----- Module 3/Notebooks/Distributions.nb.html | 111 ++++++++++++----------- 2 files changed, 87 insertions(+), 79 deletions(-) diff --git a/Module 3/Notebooks/Distributions.Rmd b/Module 3/Notebooks/Distributions.Rmd index 589493a3..fd8eca5c 100644 --- a/Module 3/Notebooks/Distributions.Rmd +++ b/Module 3/Notebooks/Distributions.Rmd @@ -3,13 +3,6 @@ title: "Inferential Statistics: Probability & Distributions - 1" output: html_notebook --- - -You: made a change -I: added a new command -You: want to sync the new command - -### Follow this carefully - So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics. #### Pre-requisite: Probability @@ -19,21 +12,6 @@ Probability theory is a branch of mathematics that tells you how often different In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]* -Probability of heads when you toss a coin? -P(H) = 0.5 - -Frequentist view is also an objective view. -Coin - [H, T, H, T, H, T, H] -Prob - [1, 1/2, 2/3, 0.5, ... ] - - - - - -Coin - [1, 1, 1, 1, 0] -P'(H) - 0.7 - -- - - - **A short note on Models** A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]* @@ -179,11 +157,18 @@ Eg: heights of all students in the class, marks obtained in exams, etc Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution. +- `dnorm()` - For probability density +- `pnorm()` - For cumulative probability +- `qnorm()` - For quantile of +- `rnorm()` - For random number generation + + ```{r} normal_distribution <- rnorm(10000, mean = 0, sd = 1) histogram_normal_distribution <- hist(normal_distribution) -plot(histogram_normal_distribution$mids,histogram_normal_distribution$density) +plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=3) + ``` @@ -206,6 +191,20 @@ pnorm(2, mean = 0, sd = 1) qnorm(0.5 ,mean = , sd = 1) ``` +*Checking for normality using the Shapiro-Wilk Test* +```{r} +norm <- rnorm(500, mean = 0, sd = 1) +shapiro.test(norm) + +binom <- rbinom(100, 20, 1/6) +shapiro.test(binom) + +``` + + + + + ##### Other useful distributions Some other distributions you may encounter include: @@ -218,9 +217,9 @@ Use the following functions to visualize the t distribution: dt(), pt(), qt() and rt() ```{r} -t_distribution <- rt(10000, 3) +t_distribution <- rt(10000, 8) histogram_t_distribution <- hist(t_distribution) -plot(histogram_t_distribution$mids,histogram_t_distribution$density) +plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3) ``` @@ -237,7 +236,7 @@ dchisq(), pchisq(), qchisq(), rchisq(). ```{r} chisq_distribution <- rchisq(10000, 3, ncp = 0) histogram_chisq_distribution <- hist(chisq_distribution) -plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density) +plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3) ``` @@ -252,9 +251,9 @@ Use the following functions to visualize the chi sq distribution: df(), pf(), qf() and rf() ```{r} -f_distribution <- rf(10000, 3, 5) +f_distribution <- rf(10000, 5, 10) histogram_f_distribution <- hist(f_distribution) -plot(histogram_f_distribution$mids,histogram_f_distribution$density) +plot(histogram_f_distribution$mids,histogram_f_distribution$density, type="l", col="blue", lwd=3) ``` diff --git a/Module 3/Notebooks/Distributions.nb.html b/Module 3/Notebooks/Distributions.nb.html index 862f186e..2f1b7b04 100644 --- a/Module 3/Notebooks/Distributions.nb.html +++ b/Module 3/Notebooks/Distributions.nb.html @@ -215,9 +215,6 @@

Inferential Statistics: Probability & Distribut -

You: made a change I: added a new command You: want to sync the new command

-
-

Follow this carefully

So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.

Pre-requisite: Probability

@@ -225,31 +222,8 @@

Pre-requisite: Probability

Difference between probability and statistics**

Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?

In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. [Chapter 9, Navarro D.]

-

Probability of heads when you toss a coin? P(H) = 0.5

-

Frequentist view is also an objective view. Coin - [H, T, H, T, H, T, H] Prob - [1, 1/2, 2/3, 0.5, … ]

-

Coin - [1, 1, 1, 1, 0] P’(H) - 0.7

- ------ - - - - - - - - - - - - - - -
**Ashort note on Models**
Amodel is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. [Chapter 3, Winter B.]
+

A short note on Models

+

A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. [Chapter 3, Winter B.] - - - -

But even though we know the models like P(heads) = 0.5, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?

We want to figure out which is the true model of the world. Is it P(heads) = 0.5 or is it P(heads) \(\ne\) 0.5?

@@ -396,6 +370,12 @@
Binomial Distribution
Normal Distribution

Most frequently encountered distribution. Eg: heights of all students in the class, marks obtained in exams, etc

Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.

+
    +
  • dnorm() - For probability density
    • +
  • pnorm() - For cumulative probability
    • +
  • qnorm() - For quantile of
    • +
  • rnorm() - For random number generation
    • +
@@ -403,15 +383,18 @@
Normal Distribution
histogram_normal_distribution <- hist(normal_distribution) -

+

- -
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density)
-
+ +
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=3)
-

+

+ +
NA
+NA
+

Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution

@@ -448,6 +431,33 @@
Normal Distribution
+

Checking for normality using the Shapiro-Wilk Test

+ + + +
norm <- rnorm(500, mean = 0, sd = 1) 
+shapiro.test(norm)
+ + +

+    Shapiro-Wilk normality test
+
+data:  norm
+W = 0.99593, p-value = 0.2251
+ + +
binom <- rbinom(100, 20, 1/6)
+shapiro.test(binom)
+ + +

+    Shapiro-Wilk normality test
+
+data:  binom
+W = 0.95443, p-value = 0.001644
+ + +
Other useful distributions
@@ -456,18 +466,18 @@
Other useful distributions

Use the following functions to visualize the t distribution: dt(), pt(), qt() and rt()

- -
t_distribution <- rt(10000, 3)
+
+
t_distribution <- rt(10000, 8)
 histogram_t_distribution <- hist(t_distribution)

 
 
-

+

 
-
-
plot(histogram_t_distribution$mids,histogram_t_distribution$density)

+
+
plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3)

 
 
-

+

 
 
 
@@ -482,13 +492,13 @@ 
Other useful distributions

 histogram_chisq_distribution <- hist(chisq_distribution)
-

+

- -
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density)
+ +
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
-

+

@@ -499,18 +509,18 @@
Other useful distributions

Use the following functions to visualize the chi sq distribution: df(), pf(), qf() and rf()

- -
f_distribution <- rf(10000, 3, 5)
+
+
f_distribution <- rf(10000, 5, 10)
 histogram_f_distribution <- hist(f_distribution)

 
 
-

+

 
-
-
plot(histogram_f_distribution$mids,histogram_f_distribution$density)

+
+
plot(histogram_f_distribution$mids,histogram_f_distribution$density, type="l", col="blue", lwd=3)

 
 
-

+

 
 
 
@@ -519,9 +529,8 @@ 
Other useful distributions
- -
---
title: "Inferential Statistics: Probability & Distributions - 1"
output: html_notebook
---


You: made a change
I: added a new command
You: want to sync the new command

### Follow this carefully

So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.

#### Pre-requisite: Probability

##### Difference between probability and statistics**
Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?

In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]*

Probability of heads when you toss a coin? 
P(H) = 0.5 

Frequentist view is also an objective view. 
Coin - [H, T, H, T, H, T, H]
Prob - [1, 1/2, 2/3, 0.5, ... ]





Coin - [1, 1, 1, 1, 0]
P'(H) - 0.7

- - - -
**A short note on Models**

A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]*
- - - -

But even though we know the models like `P(heads) = 0.5`, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?

We want to figure out which is the true model of the world. Is it *P(heads) = 0.5* or is it *P(heads) $\ne$ 0.5*?

##### What is probability really?

**The frequentist view**

![Frequentist_graph](Fig4.png)

According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.

 *Advantages*
 -  It is objective: the probability of an event is necessarily grounded in the world.
 -  It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.

But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?

**The Bayesian view**

Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this 'degree of belief'? 

One way is to use 'rational gambling'. So a “subjective probability” will be operationalized in terms of what bets you're willing to accept.

 *Advantage*
 - You don’t need to be limited to those events that are repeatable.
 
 *Disadvantage*
 - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an **intelligent agent** out there that believes in things. 


In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).

##### Definitions

Refer to the example described in *Section 9.3.1, Navarro D.* for the following content.

**Elementary event:** Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.

**Sample space:** The set of all possible events (e.g., the wardrobe)

**Probability:** Numbers between 0 and 1.

For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.

If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).

**Law of total probability:** The probabilities of the elementary events need to add up to 1

#### Distributions

Let's take a look at this and see what is a distribution. 

```{r}
pants <- data.frame(
   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
   label = c("X1", "X2", "X3", "X4", "X5"),
   probability = c(0.5,0.3,0.1,0,0.1))

pants
```
Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.

```{r}
#Try plotting a bar graph of all the probabilities above
```
Let's think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans" has occurred. 
Then what will be the probability of event E.

P(E) = P(X1) + P(X2) + P(X3)

If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:

![Probability_rules](Fig5.png)

##### Binomial Distribution

*Refer to section 9.4.1, Navarro D., for the detailed example*

Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`.

`X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N.

4 ~ Binomial(1/6, 20)

5 ~ Binomial(1/2, 10)

Let's generate a binomial distribution in R:

```{r}
dbinom( x = 1, size = 20, prob = 1/6)
```
The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.

What if the dice is replaced by a coin in the above example? How will the probability change? 

```{r}
#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
```
There are different functions in R for different distributions as well as different ones for finding different quantity of interest.

If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use `pbinom`.

```{r}
#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
pbinom( q= 3, size = 20, prob = 1/6)

#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
```
In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.

Now let’s say we want to calculate the 75th percentile of the binomial distribution.

```{r}
qbinom( p = 0.566, size = 20, prob = 1/6 )

#Practice - Find the 40th percentile
```

We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.

```{r}
z <- rbinom( n = 100, size = 20, prob = 1/6 )
z
#Let's also plot this and see how it looks
hist(z, col = 'steelblue')
```
#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.

All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. 


##### Normal Distribution

Most frequently encountered distribution.
Eg: heights of all students in the class, marks obtained in exams, etc

Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.



```{r}
normal_distribution <- rnorm(10000, mean = 0, sd = 1) 
histogram_normal_distribution <- hist(normal_distribution)
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density)

```

Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution

The notation for a normal distribution is: X ∼ Normal(μ,σ) 


dnorm tells you the probability of getting a particular outcome
```{r}
dnorm(x=0, mean=0, sd=1)

```
Cumulative normal distribution
```{r}
pnorm(2, mean = 0, sd = 1)
```

```{r}
qnorm(0.5 ,mean = , sd = 1)
```

##### Other useful distributions

Some other distributions you may encounter include:
*1) t distribution*

Looks like the normal distribution but has heavier tails. 
Used when data looks like a normal distribution but the mean and SD are unknown.

Use the following functions to visualize the t distribution: 
dt(), pt(), qt() and rt()

```{r}
t_distribution <- rt(10000, 3)
histogram_t_distribution <- hist(t_distribution)
plot(histogram_t_distribution$mids,histogram_t_distribution$density)
```




*2) Chi square (χ2) distribution*

All positive and heavily skewed to the left.  
Used when data represents sum of squares of a normally distributed variables.

Use the following functions to visualize the chi sq distribution: 
dchisq(), pchisq(), qchisq(), rchisq().

```{r}
chisq_distribution <- rchisq(10000, 3, ncp = 0)
histogram_chisq_distribution <- hist(chisq_distribution)
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density)
```




3) F distribution

This one looka a bit like the chi square distribution. But this distribution comes into picture when 
one compares two chi sq distributions.

Use the following functions to visualize the chi sq distribution: 
df(), pf(), qf() and rf()

```{r}
f_distribution <- rf(10000, 3, 5)
histogram_f_distribution <- hist(f_distribution)
plot(histogram_f_distribution$mids,histogram_f_distribution$density)
```



The End

Reference - *Chapter 9, Navarro D.*

+
---
title: "Inferential Statistics: Probability & Distributions - 1"
output: html_notebook
---

So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.

#### Pre-requisite: Probability

##### Difference between probability and statistics**
Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?

In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]*

**A short note on Models**

A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]*
- - - -

But even though we know the models like `P(heads) = 0.5`, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?

We want to figure out which is the true model of the world. Is it *P(heads) = 0.5* or is it *P(heads) $\ne$ 0.5*?

##### What is probability really?

**The frequentist view**

![Frequentist_graph](Fig4.png)

According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.

 *Advantages*
 -  It is objective: the probability of an event is necessarily grounded in the world.
 -  It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.

But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?

**The Bayesian view**

Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this 'degree of belief'? 

One way is to use 'rational gambling'. So a “subjective probability” will be operationalized in terms of what bets you're willing to accept.

 *Advantage*
 - You don’t need to be limited to those events that are repeatable.
 
 *Disadvantage*
 - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an **intelligent agent** out there that believes in things. 


In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).

##### Definitions

Refer to the example described in *Section 9.3.1, Navarro D.* for the following content.

**Elementary event:** Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.

**Sample space:** The set of all possible events (e.g., the wardrobe)

**Probability:** Numbers between 0 and 1.

For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.

If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).

**Law of total probability:** The probabilities of the elementary events need to add up to 1

#### Distributions

Let's take a look at this and see what is a distribution. 

```{r}
pants <- data.frame(
   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
   label = c("X1", "X2", "X3", "X4", "X5"),
   probability = c(0.5,0.3,0.1,0,0.1))

pants
```
Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.

```{r}
#Try plotting a bar graph of all the probabilities above
```
Let's think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans" has occurred. 
Then what will be the probability of event E.

P(E) = P(X1) + P(X2) + P(X3)

If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:

![Probability_rules](Fig5.png)

##### Binomial Distribution

*Refer to section 9.4.1, Navarro D., for the detailed example*

Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`.

`X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N.

4 ~ Binomial(1/6, 20)

5 ~ Binomial(1/2, 10)

Let's generate a binomial distribution in R:

```{r}
dbinom( x = 1, size = 20, prob = 1/6)
```
The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.

What if the dice is replaced by a coin in the above example? How will the probability change? 

```{r}
#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
```
There are different functions in R for different distributions as well as different ones for finding different quantity of interest.

If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use `pbinom`.

```{r}
#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
pbinom( q= 3, size = 20, prob = 1/6)

#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
```
In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.

Now let’s say we want to calculate the 75th percentile of the binomial distribution.

```{r}
qbinom( p = 0.566, size = 20, prob = 1/6 )

#Practice - Find the 40th percentile
```

We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.

```{r}
z <- rbinom( n = 100, size = 20, prob = 1/6 )
z
#Let's also plot this and see how it looks
hist(z, col = 'steelblue')
```
#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.

All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. 


##### Normal Distribution

Most frequently encountered distribution.
Eg: heights of all students in the class, marks obtained in exams, etc

Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.


- `dnorm()` - For probability density
- `pnorm()` - For cumulative probability
- `qnorm()` - For quantile of
- `rnorm()` - For random number generation



```{r}
normal_distribution <- rnorm(10000, mean = 0, sd = 1) 
histogram_normal_distribution <- hist(normal_distribution)
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=3)


```

Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution

The notation for a normal distribution is: X ∼ Normal(μ,σ) 


dnorm tells you the probability of getting a particular outcome
```{r}
dnorm(x=0, mean=0, sd=1)

```
Cumulative normal distribution
```{r}
pnorm(2, mean = 0, sd = 1)
```

```{r}
qnorm(0.5 ,mean = , sd = 1)
```

*Checking for normality using the Shapiro-Wilk Test*
```{r}
norm <- rnorm(500, mean = 0, sd = 1) 
shapiro.test(norm)

binom <- rbinom(100, 20, 1/6)
shapiro.test(binom)

```





##### Other useful distributions

Some other distributions you may encounter include:
*1) t distribution*

Looks like the normal distribution but has heavier tails. 
Used when data looks like a normal distribution but the mean and SD are unknown.

Use the following functions to visualize the t distribution: 
dt(), pt(), qt() and rt()

```{r}
t_distribution <- rt(10000, 8)
histogram_t_distribution <- hist(t_distribution)
plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3)
```




*2) Chi square (χ2) distribution*

All positive and heavily skewed to the left.  
Used when data represents sum of squares of a normally distributed variables.

Use the following functions to visualize the chi sq distribution: 
dchisq(), pchisq(), qchisq(), rchisq().

```{r}
chisq_distribution <- rchisq(10000, 3, ncp = 0)
histogram_chisq_distribution <- hist(chisq_distribution)
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
```




3) F distribution

This one looka a bit like the chi square distribution. But this distribution comes into picture when 
one compares two chi sq distributions.

Use the following functions to visualize the chi sq distribution: 
df(), pf(), qf() and rf()

```{r}
f_distribution <- rf(10000, 5, 10)
histogram_f_distribution <- hist(f_distribution)
plot(histogram_f_distribution$mids,histogram_f_distribution$density, type="l", col="blue", lwd=3)
```



The End

Reference - *Chapter 9, Navarro D.*

From ea11e82c9157b366c2004f226cbac8ca7b620aa9 Mon Sep 17 00:00:00 2001 From: Arjun Date: Mon, 27 Sep 2021 19:13:59 +0000 Subject: [PATCH 14/55] Corrected some typos. Added assignment at the end for sample standard deviation. --- Module 3/Notebooks/Distributions.Rmd | 23 +++-- Module 3/Notebooks/Distributions.nb.html | 103 +++++++++++++++-------- 2 files changed, 81 insertions(+), 45 deletions(-) diff --git a/Module 3/Notebooks/Distributions.Rmd b/Module 3/Notebooks/Distributions.Rmd index fd8eca5c..503dc14f 100644 --- a/Module 3/Notebooks/Distributions.Rmd +++ b/Module 3/Notebooks/Distributions.Rmd @@ -139,7 +139,7 @@ qbinom( p = 0.566, size = 20, prob = 1/6 ) We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function. ```{r} -z <- rbinom( n = 100, size = 20, prob = 1/6 ) +z <- rbinom( n = 10000, size = 20, prob = 2/6 ) z #Let's also plot this and see how it looks hist(z, col = 'steelblue') @@ -163,7 +163,7 @@ Basically, whenever you have accumulation of data at the center, fewer extreme v - `rnorm()` - For random number generation - +mean = 0; sd = 1 -> standard normal distribution ```{r} normal_distribution <- rnorm(10000, mean = 0, sd = 1) histogram_normal_distribution <- hist(normal_distribution) @@ -179,21 +179,21 @@ The notation for a normal distribution is: X ∼ Normal(μ,σ) dnorm tells you the probability of getting a particular outcome ```{r} -dnorm(x=0, mean=0, sd=1) +dnorm(x=85, mean=80, sd=5) ``` Cumulative normal distribution ```{r} -pnorm(2, mean = 0, sd = 1) +pnorm(q = 80, mean = 80, sd = 5) ``` ```{r} -qnorm(0.5 ,mean = , sd = 1) +qnorm(0.25 ,mean = 0 , sd = 1) ``` *Checking for normality using the Shapiro-Wilk Test* ```{r} -norm <- rnorm(500, mean = 0, sd = 1) +norm <- rnorm(50, mean = 0, sd = 1) shapiro.test(norm) binom <- rbinom(100, 20, 1/6) @@ -217,7 +217,7 @@ Use the following functions to visualize the t distribution: dt(), pt(), qt() and rt() ```{r} -t_distribution <- rt(10000, 8) +t_distribution <- rt(10000, 3) histogram_t_distribution <- hist(t_distribution) plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3) ``` @@ -234,7 +234,12 @@ Use the following functions to visualize the chi sq distribution: dchisq(), pchisq(), qchisq(), rchisq(). ```{r} -chisq_distribution <- rchisq(10000, 3, ncp = 0) + +norm1 <- rnorm(100, mean = 10, sd = 5) +norm2 <- rnorm(100, mean = 20, sd = 7) +chisqdist <- norm1^2 + norm2^2 + +chisq_distribution <- rchisq(10000, 3) histogram_chisq_distribution <- hist(chisq_distribution) plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3) ``` @@ -244,7 +249,7 @@ plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, typ 3) F distribution -This one looka a bit like the chi square distribution. But this distribution comes into picture when +This one looks a bit like the chi square distribution. But this distribution comes into picture when one compares two chi sq distributions. Use the following functions to visualize the chi sq distribution: diff --git a/Module 3/Notebooks/Distributions.nb.html b/Module 3/Notebooks/Distributions.nb.html index 2f1b7b04..97a872b7 100644 --- a/Module 3/Notebooks/Distributions.nb.html +++ b/Module 3/Notebooks/Distributions.nb.html @@ -346,20 +346,50 @@
Binomial Distribution

We’ve found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the rbinom function.

- -
z <- rbinom( n = 100, size = 20, prob = 1/6 )
+
+
z <- rbinom( n = 10000, size = 20, prob = 2/6 )
 z

 
-
-
  [1] 2 5 3 1 6 4 2 5 4 4 5 3 3 4 0 3 3 3 4 3 3 1 3 2 2 6 3 5 2 8 2 5 4 4 2 2 1 6 2 3 4 2 3 3 4 4 3 1 2 3 2 1 2 1 4 4 6 3
- [59] 4 6 2 5 7 3 2 5 5 5 4 2 4 0 1 2 5 5 2 6 3 3 3 1 3 3 3 3 2 3 5 3 3 3 4 3 2 3 3 5 6 3

+
+
   [1]  6  7  4  8  2  6  6  8  2  9  6  5  6  7  8  5  4  4  5  7  9  7  8  6  8  8  8  4  7  8  6  6 10
+  [34]  9  3  8  6  7  9 10  5  7  2  3  6  6  5  9  9 12  8 10  6  7  6  4  7  8  6  4  7  8  8  5  6  3
+  [67]  5  7  6  6  8  7  3  8 14  3  5  4  4  5  7  8  4  7  8  3  8  7  7  7  5  5  9  4  8  4  6  7  8
+ [100]  3  6  6  5  7  6  8 12  1 12  9  7  5  9  9  8  6  5 12  5 10 10  7  8  8  6  8  9  6  7  8  3  9
+ [133]  4  5  4  2  8  6  9  6  7  9  6  7  7  7  5  8  5  2  6  9  3  9  6 11  6  4  4 10  4  7  8  6  8
+ [166]  4  3  6  6  7  8  1  5  7 10  6  6  9  7  4  8  7  4  4  6  6  9  7  7  4  8  4  6  6  5  6  5  7
+ [199]  3  9  2  9  5  8  6  8  6  2  9  8  5  4  4  7  7  4  7  6  5  6  4  8  6  8  6 11  9  9  6 10  7
+ [232]  7 10  8  4  7  7  6  3 10  4 11  4  7 10  6  4  3  5  5  6  6 11  9  5  8  7  7  6  5  7  6  7  4
+ [265]  6  6  4  2  1  7  6  7  5  7  7 11  8  7 10  4  7  8 10  9  4  5  9 10  7  9  9  9  7  5  6  4  7
+ [298]  8  9  4  9  6  7  7  7  7  5  5  5  9  5  6  7  5  6  7  6  8  6  5  8  8 10  9  5  6  3  8 10  6
+ [331]  7  7  4  6  5 10  5  6  5 10  9  8  7  8  7  7  5  8  6  7  7  7  6  4  9 11  6  6  5  7  6  7  8
+ [364]  5  8  6  8  8  5  7  6  7 11  6 10  4  7  7  2  3  7  6  3  5  8  9  7  7  4  4 13  4  8  8  7  6
+ [397]  6  7  6  6  7  7  8  6  8 11  6 12  5  4  6 11  6  9 11  7  8  7  7  9 10  9  6  3  6  9 10  5  7
+ [430]  7  5  3  9  9  6  9  7  8  5  8  4  9  7  3  6 11  7  4  9  5  7  7  4  6  7  5  7 12  7  7  5  9
+ [463]  4  8  9  9  4  8  3  5  8  6  7  4  9  6  8  5  5  3  6  9  7  5  9  7  6  7  7  6  8  9  9  8  7
+ [496]  7  7  8  5  3  7  5  9  5 10  8  8  7  6  1 12  9  7  4  8  7  4  7  6  3 11  5  5  9  3  8  3  8
+ [529]  6  6  6  6  7  5 10 11  7  6  5  7  4  2  5  7  6  8  9  6  6  5  4  6  6  8 10  8  5  8 10  6  9
+ [562]  6  8  5  7  7  8  9  6  1 11  3  5 10  8  7  6 11  7  4  5  1  6  5  9  7  7  7  9  9  4  8  9  9
+ [595] 10  6  7  7  6  8  7  6  6  3  4  7  5  8 12  6  4  9  6  6  9  8  5 13  6  6  6  7  7  3  4  4  7
+ [628]  3  7  4  5  7  8  7  3  7 10  7  5  4  6  6  7  5 10  7  2  5  8  7  6  8  8  5  6  7 10 10  8  8
+ [661]  8  7  8  9  8  4  6  4  5  8  7  7 10  9  9  9  3  9  6  8  7  6  4  5  7  6  9  3  3  8  1  5  8
+ [694] 10  3  7  9 11  8  7  5 12  4  2 10  3  6  5  9  7  4 11  8  8  5  8  6  8  6  7 10 10  5  3  6  4
+ [727]  6  7  6  9  5  7  7  3  8  9 10  3  9  7  4 10  6  7  4  6  7  8  6  3  7  8  7  6  7  6  7  7  6
+ [760] 10  7  6  3  6  7  7  3  6  7  5  7 10  4 11  9  4  9  5  6  4  6  6  7  7  4  7  9  6  8  4  9  8
+ [793]  6  8  5  7  7 10  9  4  9  6  6 10  7  5 12  7  5  6  4  3  8  6 10  4  6  5  8  4  4  6  7  9  6
+ [826]  8  6 10  7  8  9  5  8  5  7 10  5  6  8  8  2  7  5  3  5  3  9  7  5  8  8  6  5  7  8 10  5  7
+ [859]  9  4  7  7  8  9  8  6  2 10  7  6  5 10  6 10  3  6  5  3  7  8  2 11  4 10  4  8  9  4  7  6  3
+ [892]  6  6  9  5  4  6  5  3  5  6  8  3  7  6  6  8  4  5  4  7  7  8  7  5  8  2  8  4  2  8  9  5  8
+ [925]  7 11  5  4  8  4 10  9 11  7  4  8 10 12  8  7  7  7  5  5  5  4  5  9  9  9 10  8  7  5  5  8  8
+ [958]  5  6  5  4  4  5  4  8  7  4  8  4  5  4  6  5  6  7  6  8  4  8  3 10  5  9  6  7  4  9  5  7  6
+ [991]  7  6  5  4  8  3  4  8 10  6
+ [ reached getOption("max.print") -- omitted 9000 entries ]

 
 
 
#Let's also plot this and see how it looks
 hist(z, col = 'steelblue')

 
 
-

+

 
 
 
@@ -376,6 +406,7 @@ 
Normal Distribution

 
  • qnorm() - For quantile of
    • 
 
  • rnorm() - For random number generation
    • 
 
+
    mean = 0; sd = 1 -> standard normal distribution
    
 
 
 
@@ -383,13 +414,13 @@ 
    Normal Distribution
    
 histogram_normal_distribution <- hist(normal_distribution)
    -

    +

    plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=3)
    -

    +

    NA
@@ -402,59 +433,59 @@ 
    Normal Distribution
    
 
    dnorm tells you the probability of getting a particular outcome
    
 
 
-
-
    dnorm(x=0, mean=0, sd=1)
    
+
+
    dnorm(x=85, mean=80, sd=5)
    
 
-
-
    [1] 0.3989423
    
+
+
    [1] 0.04839414
    
 
 
 
 
    Cumulative normal distribution
    
 
 
-
-
    pnorm(2, mean = 0, sd = 1)
    
+
+
    pnorm(q = 80, mean = 80, sd = 5)
    
 
-
-
    [1] 0.9772499
    
+
+
    [1] 0.5
    
 
 
 
 
 
-
-
    qnorm(0.5 , mean = , sd = 1)
    
+
+
    qnorm(0.25 ,mean = 0 , sd = 1)
    
 
-
-
    [1] 0
    
+
+
    [1] -0.6744898
    
 
 
 
 
    Checking for normality using the Shapiro-Wilk Test
    
 
 
-
-
    norm <- rnorm(500, mean = 0, sd = 1) 
+
+
    norm <- rnorm(50, mean = 0, sd = 1) 
 shapiro.test(norm)
    
 
-
+
 
    
     Shapiro-Wilk normality test
 
 data:  norm
-W = 0.99593, p-value = 0.2251
    
+W = 0.9809, p-value = 0.5904
    
 
 
 
    binom <- rbinom(100, 20, 1/6)
 shapiro.test(binom)
    
 
-
+
 
    
     Shapiro-Wilk normality test
 
 data:  binom
-W = 0.95443, p-value = 0.001644
    
+W = 0.95971, p-value = 0.003833
    @@ -466,18 +497,18 @@
    Other useful distributions

    Use the following functions to visualize the t distribution: dt(), pt(), qt() and rt()

    - -
    t_distribution <- rt(10000, 8)
+
+
    t_distribution <- rt(10000, 3)
 histogram_t_distribution <- hist(t_distribution)
    
 
 
-
    
+
    
 
 
 
    plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3)
    
 
 
-
    
+
    
 
 
 
@@ -487,25 +518,25 @@ 
    Other useful distributions
    
 
    Use the following functions to visualize the chi sq distribution: dchisq(), pchisq(), qchisq(), rchisq().
    
 
 
-
-
    chisq_distribution <- rchisq(10000, 3, ncp = 0)
+
+
    chisq_distribution <- rchisq(10000, 3)
 histogram_chisq_distribution <- hist(chisq_distribution)
    
 
 
-
    
+
    
 
 
 
    plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
    
 
 
-
    
+
    
 
 
 
 
      
 
    1. F distribution
      2. 
 
    
-
    This one looka a bit like the chi square distribution. But this distribution comes into picture when one compares two chi sq distributions.
    
+
    This one looks a bit like the chi square distribution. But this distribution comes into picture when one compares two chi sq distributions.
    
 
    Use the following functions to visualize the chi sq distribution: df(), pf(), qf() and rf()
    
 
 
@@ -530,7 +561,7 @@ 
    Other useful distributions
    
 
 
 
-
    ---
title: "Inferential Statistics: Probability & Distributions - 1"
output: html_notebook
---

So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.

#### Pre-requisite: Probability

##### Difference between probability and statistics**
Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?

In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]*

**A short note on Models**

A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]*
- - - -

But even though we know the models like `P(heads) = 0.5`, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?

We want to figure out which is the true model of the world. Is it *P(heads) = 0.5* or is it *P(heads) $\ne$ 0.5*?

##### What is probability really?

**The frequentist view**

![Frequentist_graph](Fig4.png)

According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.

 *Advantages*
 -  It is objective: the probability of an event is necessarily grounded in the world.
 -  It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.

But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?

**The Bayesian view**

Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this 'degree of belief'? 

One way is to use 'rational gambling'. So a “subjective probability” will be operationalized in terms of what bets you're willing to accept.

 *Advantage*
 - You don’t need to be limited to those events that are repeatable.
 
 *Disadvantage*
 - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an **intelligent agent** out there that believes in things. 


In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).

##### Definitions

Refer to the example described in *Section 9.3.1, Navarro D.* for the following content.

**Elementary event:** Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.

**Sample space:** The set of all possible events (e.g., the wardrobe)

**Probability:** Numbers between 0 and 1.

For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.

If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).

**Law of total probability:** The probabilities of the elementary events need to add up to 1

#### Distributions

Let's take a look at this and see what is a distribution. 

```{r}
pants <- data.frame(
   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
   label = c("X1", "X2", "X3", "X4", "X5"),
   probability = c(0.5,0.3,0.1,0,0.1))

pants
```
Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.

```{r}
#Try plotting a bar graph of all the probabilities above
```
Let's think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans" has occurred. 
Then what will be the probability of event E.

P(E) = P(X1) + P(X2) + P(X3)

If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:

![Probability_rules](Fig5.png)

##### Binomial Distribution

*Refer to section 9.4.1, Navarro D., for the detailed example*

Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`.

`X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N.

4 ~ Binomial(1/6, 20)

5 ~ Binomial(1/2, 10)

Let's generate a binomial distribution in R:

```{r}
dbinom( x = 1, size = 20, prob = 1/6)
```
The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.

What if the dice is replaced by a coin in the above example? How will the probability change? 

```{r}
#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
```
There are different functions in R for different distributions as well as different ones for finding different quantity of interest.

If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use `pbinom`.

```{r}
#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
pbinom( q= 3, size = 20, prob = 1/6)

#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
```
In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.

Now let’s say we want to calculate the 75th percentile of the binomial distribution.

```{r}
qbinom( p = 0.566, size = 20, prob = 1/6 )

#Practice - Find the 40th percentile
```

We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.

```{r}
z <- rbinom( n = 100, size = 20, prob = 1/6 )
z
#Let's also plot this and see how it looks
hist(z, col = 'steelblue')
```
#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.

All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. 


##### Normal Distribution

Most frequently encountered distribution.
Eg: heights of all students in the class, marks obtained in exams, etc

Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.


- `dnorm()` - For probability density
- `pnorm()` - For cumulative probability
- `qnorm()` - For quantile of
- `rnorm()` - For random number generation



```{r}
normal_distribution <- rnorm(10000, mean = 0, sd = 1) 
histogram_normal_distribution <- hist(normal_distribution)
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=3)


```

Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution

The notation for a normal distribution is: X ∼ Normal(μ,σ) 


dnorm tells you the probability of getting a particular outcome
```{r}
dnorm(x=0, mean=0, sd=1)

```
Cumulative normal distribution
```{r}
pnorm(2, mean = 0, sd = 1)
```

```{r}
qnorm(0.5 ,mean = , sd = 1)
```

*Checking for normality using the Shapiro-Wilk Test*
```{r}
norm <- rnorm(500, mean = 0, sd = 1) 
shapiro.test(norm)

binom <- rbinom(100, 20, 1/6)
shapiro.test(binom)

```





##### Other useful distributions

Some other distributions you may encounter include:
*1) t distribution*

Looks like the normal distribution but has heavier tails. 
Used when data looks like a normal distribution but the mean and SD are unknown.

Use the following functions to visualize the t distribution: 
dt(), pt(), qt() and rt()

```{r}
t_distribution <- rt(10000, 8)
histogram_t_distribution <- hist(t_distribution)
plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3)
```




*2) Chi square (χ2) distribution*

All positive and heavily skewed to the left.  
Used when data represents sum of squares of a normally distributed variables.

Use the following functions to visualize the chi sq distribution: 
dchisq(), pchisq(), qchisq(), rchisq().

```{r}
chisq_distribution <- rchisq(10000, 3, ncp = 0)
histogram_chisq_distribution <- hist(chisq_distribution)
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
```




3) F distribution

This one looka a bit like the chi square distribution. But this distribution comes into picture when 
one compares two chi sq distributions.

Use the following functions to visualize the chi sq distribution: 
df(), pf(), qf() and rf()

```{r}
f_distribution <- rf(10000, 5, 10)
histogram_f_distribution <- hist(f_distribution)
plot(histogram_f_distribution$mids,histogram_f_distribution$density, type="l", col="blue", lwd=3)
```



The End

Reference - *Chapter 9, Navarro D.*

    
+
    ---
title: "Inferential Statistics: Probability & Distributions - 1"
output: html_notebook
---

So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.

#### Pre-requisite: Probability

##### Difference between probability and statistics**
Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?

In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]*

**A short note on Models**

A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]*
- - - -

But even though we know the models like `P(heads) = 0.5`, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?

We want to figure out which is the true model of the world. Is it *P(heads) = 0.5* or is it *P(heads) $\ne$ 0.5*?

##### What is probability really?

**The frequentist view**

![Frequentist_graph](Fig4.png)

According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.

 *Advantages*
 -  It is objective: the probability of an event is necessarily grounded in the world.
 -  It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.

But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?

**The Bayesian view**

Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this 'degree of belief'? 

One way is to use 'rational gambling'. So a “subjective probability” will be operationalized in terms of what bets you're willing to accept.

 *Advantage*
 - You don’t need to be limited to those events that are repeatable.
 
 *Disadvantage*
 - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an **intelligent agent** out there that believes in things. 


In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).

##### Definitions

Refer to the example described in *Section 9.3.1, Navarro D.* for the following content.

**Elementary event:** Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.

**Sample space:** The set of all possible events (e.g., the wardrobe)

**Probability:** Numbers between 0 and 1.

For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.

If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).

**Law of total probability:** The probabilities of the elementary events need to add up to 1

#### Distributions

Let's take a look at this and see what is a distribution. 

```{r}
pants <- data.frame(
   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
   label = c("X1", "X2", "X3", "X4", "X5"),
   probability = c(0.5,0.3,0.1,0,0.1))

pants
```
Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.

```{r}
#Try plotting a bar graph of all the probabilities above
```
Let's think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans" has occurred. 
Then what will be the probability of event E.

P(E) = P(X1) + P(X2) + P(X3)

If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:

![Probability_rules](Fig5.png)

##### Binomial Distribution

*Refer to section 9.4.1, Navarro D., for the detailed example*

Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`.

`X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N.

4 ~ Binomial(1/6, 20)

5 ~ Binomial(1/2, 10)

Let's generate a binomial distribution in R:

```{r}
dbinom( x = 1, size = 20, prob = 1/6)
```
The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.

What if the dice is replaced by a coin in the above example? How will the probability change? 

```{r}
#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
```
There are different functions in R for different distributions as well as different ones for finding different quantity of interest.

If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use `pbinom`.

```{r}
#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
pbinom( q= 3, size = 20, prob = 1/6)

#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
```
In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.

Now let’s say we want to calculate the 75th percentile of the binomial distribution.

```{r}
qbinom( p = 0.566, size = 20, prob = 1/6 )

#Practice - Find the 40th percentile
```

We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.

```{r}
z <- rbinom( n = 10000, size = 20, prob = 2/6 )
z
#Let's also plot this and see how it looks
hist(z, col = 'steelblue')
```
#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.

All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. 


##### Normal Distribution

Most frequently encountered distribution.
Eg: heights of all students in the class, marks obtained in exams, etc

Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.


- `dnorm()` - For probability density
- `pnorm()` - For cumulative probability
- `qnorm()` - For quantile of
- `rnorm()` - For random number generation


mean = 0; sd = 1 -> standard normal distribution
```{r}
normal_distribution <- rnorm(10000, mean = 0, sd = 1) 
histogram_normal_distribution <- hist(normal_distribution)
plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=3)


```

Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution

The notation for a normal distribution is: X ∼ Normal(μ,σ) 


dnorm tells you the probability of getting a particular outcome
```{r}
dnorm(x=85, mean=80, sd=5)

```
Cumulative normal distribution
```{r}
pnorm(q = 80, mean = 80, sd = 5)
```

```{r}
qnorm(0.25 ,mean = 0 , sd = 1)
```

*Checking for normality using the Shapiro-Wilk Test*
```{r}
norm <- rnorm(50, mean = 0, sd = 1) 
shapiro.test(norm)

binom <- rbinom(100, 20, 1/6)
shapiro.test(binom)

```





##### Other useful distributions

Some other distributions you may encounter include:
*1) t distribution*

Looks like the normal distribution but has heavier tails. 
Used when data looks like a normal distribution but the mean and SD are unknown.

Use the following functions to visualize the t distribution: 
dt(), pt(), qt() and rt()

```{r}
t_distribution <- rt(10000, 3)
histogram_t_distribution <- hist(t_distribution)
plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3)
```




*2) Chi square (χ2) distribution*

All positive and heavily skewed to the left.  
Used when data represents sum of squares of a normally distributed variables.

Use the following functions to visualize the chi sq distribution: 
dchisq(), pchisq(), qchisq(), rchisq().

```{r}

norm1 <- rnorm(100, mean = 10, sd = 5)
norm2 <- rnorm(100, mean = 20, sd = 7)
chisqdist <- norm1^2 + norm2^2

chisq_distribution <- rchisq(10000, 3)
histogram_chisq_distribution <- hist(chisq_distribution)
plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
```




3) F distribution

This one looks a bit like the chi square distribution. But this distribution comes into picture when 
one compares two chi sq distributions.

Use the following functions to visualize the chi sq distribution: 
df(), pf(), qf() and rf()

```{r}
f_distribution <- rf(10000, 5, 10)
histogram_f_distribution <- hist(f_distribution)
plot(histogram_f_distribution$mids,histogram_f_distribution$density, type="l", col="blue", lwd=3)
```



The End

Reference - *Chapter 9, Navarro D.*

    
 
 
 

From c2d800c073320491c1b6e488a7f8c6f7dcbfa9a3 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Mon, 25 Oct 2021 23:41:07 +0530
Subject: [PATCH 15/55] Sampling distribution and Power analysis

---
 Module 3/Notebooks/SamplingDistribution.Rmd | 84 +++++++++++++++++++++
 1 file changed, 84 insertions(+)
 create mode 100644 Module 3/Notebooks/SamplingDistribution.Rmd

diff --git a/Module 3/Notebooks/SamplingDistribution.Rmd b/Module 3/Notebooks/SamplingDistribution.Rmd
new file mode 100644
index 00000000..df89f310
--- /dev/null
+++ b/Module 3/Notebooks/SamplingDistribution.Rmd	
@@ -0,0 +1,84 @@
+---
+title: "R Notebook"
+output: html_notebook
+---
+
+This is an [R Markdown](http://rmarkdown.rstudio.com) Notebook. When you execute code within the notebook, the results appear beneath the code. 
+
+Try executing this chunk by clicking the *Run* button within the chunk or by placing your cursor inside it and pressing *Ctrl+Shift+Enter*. 
+
+ESP example:
+
+Out of 100, 62 individuals correctly predicted the card on their forehead. 
+
+Null hypothesis: p =  0.5
+n = 100
+
+
+
+*Null hypothesis*
+```{r}
+r <- rbinom(n = 10000, size = 100, prob = 0.5)
+hist(r)
+```
+
+
+
+How many people have scored above 60?
+```{r}
+pbinom(q = 63, size = 100,prob = 0.5,lower.tail = FALSE)
+```
+
+
+How many have scored below 40? 
+
+```{r}
+pbinom(q = 36, size = 100,prob = 0.5)
+
+```
+Overall including those below 40 and above 59, we have 0.028+0.028 = 0.056 or ~ 5.6% of the individuals.
+
+That means under the 5% type-1 error criterion, as long as the number of individuals is between 40 and 59, we still cannot reject the Null hypothesis.
+
+But at 62, we can! 
+
+
+How can we directly test this? 
+*Binomial Test in R*
+
+```{r}
+binom.test( x=62, n=100, p=.5 )
+
+
+
+```
+
+*Power Analysis in R*
+
+
+```{r}
+library(pwr)
+pwr.p.test(h = ES.h(p1 = 0.75, p2 = 0.5),
+           sig.level = 0.05,
+           power = 0.80,
+           alternative = "greater")
+```
+```{r}
+pwr.p.test(n = 100,
+           sig.level = 0.001,
+           power = 0.80,
+           alternative = "greater")
+```
+
+```{r}
+ES.h(p1 = 0.95, p2 = 0.5)
+```
+git config user.email "arjunr@iitk.ac.in"
+git config user.name "Arjun"
+
+
+Add a new chunk by clicking the *Insert Chunk* button on the toolbar or by pressing *Ctrl+Alt+I*.
+
+When you save the notebook, an HTML file containing the code and output will be saved alongside it (click the *Preview* button or press *Ctrl+Shift+K* to preview the HTML file).
+
+The preview shows you a rendered HTML copy of the contents of the editor. Consequently, unlike *Knit*, *Preview* does not run any R code chunks. Instead, the output of the chunk when it was last run in the editor is displayed.

From d4784ea17916223dffc4f85d9beb4e630c61c3e4 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 23 Aug 2022 08:16:00 +0530
Subject: [PATCH 16/55] ggplot related additions

Added some ggplots
---
 Module 3/Notebooks/Module3_Nb1.Rmd | 718 +++++++++++++++--------------
 1 file changed, 378 insertions(+), 340 deletions(-)

diff --git a/Module 3/Notebooks/Module3_Nb1.Rmd b/Module 3/Notebooks/Module3_Nb1.Rmd
index ea61e465..9f098df6 100644
--- a/Module 3/Notebooks/Module3_Nb1.Rmd	
+++ b/Module 3/Notebooks/Module3_Nb1.Rmd	
@@ -1,340 +1,378 @@
----
-title: "Descriptive Statistics: Central and Variability measures"
-output: html_notebook
----
-```{r}
-#Initial packages
-install.packages("lsr")
-```
-
-In this notebook, we'll take a look at how to explore a dataset.
-
-Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what **descriptive statistics** is all about.
-
-#### Describing data
-
-Imagine you've conducted an experiment involving measurements from 20 animals. If you wanted to report the outcome of your experiment to an audience, you wouldn’t want to talk through each and every data point. Instead, you report a summary, such as ‘The 20 animals had an average weight of 15 grams’, thus saving your audience valuable time and mental energy. This notebook focuses on such summaries of numerical information including distributions, measures of central tendency and measures of variability. 
-
-##### What exactly is a distribution?
-
-If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below.
-
-![Empirical and theoretical distributions](dice rolling probability.png)
-
-But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.)
-
-Apart from *looking* at how a data is distributed, the most important descriptive statistics for numerical data are those measuring the location of a frequency distribution and its spread. The location tells us something about the average or *typical* individual—where the observations are centered. The spread tells us how variable the measurements are from individual to individual—how widely scattered the observations are around the center. The proportion is the most important descriptive statistic for a categorical variable, measuring the fraction of observations in a given category. 
-
-##### But why is it needed?
-The importance of calculating some sort of a centre of a distribution seems obvious. How else do we address questions like “Which species is larger?” or “Which drug yielded the greatest response?” The importance of describing distribution spread is less obvious but no less crucial, at least in biology. In some fields of science, variability around a central value is instrument noise or measurement error, but in biology much of the variability signifies real differences among individuals. Different individuals respond differently to treatments, and this variability begs measurement. (Adapted from Chapter 3, Whitlock & Schluter, 2015)
-
-That's a lot of theory, let's dive into some data now.
-
-**Loading the Australian Football League Dataset**
-
-```{r}
-#Change the path according to your PC
-load("aflsmall.Rdata")
-library(lsr)
-who()
-```
-
-As you can see there are multiple variables of different class and size. 
-
-Let's take a look at afl.margins variable.
-
-```{r}
-print(afl.margins)
-```
-
-This output doesn’t make it easy to get a sense of what the data is actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data.
-
-Let's try to plot it.
-
-**Frequency distribution**
-
-```{r}
-hist (afl.margins)
-```
-
-As you can see, different margins in a sample will have different measurements. We can see this variability with a **frequency distribution**. The frequency of a specific measurement in a sample is the number of observations having a particular value of the measurement. The frequency distribution shows how often each value of the variable occurs in the sample. 
-
-Therefore, here we have plotted a histogram for the afl.margins variable which gives the frequency distribution of the different margin values.
-
-**Skewness**
-
-If you observe the graph, you will find that it is not entirely symmetrical. A measure of such asymmetry is called **Skewness**. If the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel), then we say that the data are _negatively skewed_. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are _positively skewed_.
-
-`psych` package contains a `skew()` function that you can use to calculate skewness. 
-
-Try finding the skewness for the above data for afl.margins using skew() function and also try to guess whether this data is positively or negatively skewed.
-
-```{r}
-library(psych)
-#Try finding skewness of afl.margins here
-```
-
-Although such a graphical representation gives a 'gist' of the data but it is useful to find some "summary" statistics as well.
-
-##### Measures of Central Tendency
-In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you’d like to know something about the “average” or “middle” of your data lies. The two most commonly used measures are the mean, median and mode.
-
-**Mean**
-
-As you've already seen in previous classes, the mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values.
-
-Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins
-```{r}
-mean(afl.margins)      # average margin
-mean(afl.margins[1:5]) # mean of the margin from the first 5 games
-
-```
-
-**Median**
-
-The second measure is the median. It is just the middle value of a set of observations. 
-*Try : Guess the median for 56, 31, 56, 8 and 32 *
-
-Probably you mentally arranged these numbers in ascending order first and then found the middle value. If there were a list of numbers like this `8, 14, 31, 32, 56, 56` . You will then find the average of middle 2 values. 
-
-Now try finding out the median for afl.margins.
-
-```{r}
-median(afl.margins)
-```
-
-**Difference between Mean and Median**
-
-Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data.
-
-![Difference between mean and median](pic2.png)
-*Fig 5.2 from Learning Statistics with R by D. Navarro*
-
-**Some key points**
-
-- If data is nominal scale, then it’s probably best to use the mode instead of mean or median.
-
-- If your data are ordinal scale, you’re more likely to want to use the median than the mean.
-
-- For interval and ratio scale data, either mean or median is generally acceptable. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values.
-
-*You can read more about this in Section 5.1.4, Learning Statistics with R by D. Navarro*
-
-Now let's take a look at some more data:
-
-` -100,2,3,4,5,6,7,8,9,10`
-
-If you observed such data in real life, you will probably think that -100 is an **_outlier_**, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely but you don’t always get such cut-and-dried examples. For instance, you might get this instead:
-
-` -15,2,3,4,5,6,7,8,9,12`
-
-The `-15` looks a bit suspicious, but not anywhere near as much as `-100` did. In this case, it’s a
-little trickier. It might be a legitimate observation, it might not. In such situations, the mean might give you an error as it is highly sensitive to one or two extreme values, and is thus not considered to be a robust measure.
-
-In such situations, one solution is to use the median or another is to use a **trimmed mean**. To calculate a trimmed mean, what you do is **discard** the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. This helps in taking the mean by excluding the outliers.
-
-Let's try trimming the mean for above data.
-
-```{r}
-dataset <- c(-15,2,3,4,5,6,7,8,9,12)
-mean(x = dataset, trim = .1)
-#Try calculating 5% trimmed mean for above dataset
-```
-
-**Mode**
-
-So far we've seen how to find the mean and median but what about mode. The **mode** of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the _lsr_ package has a function called modeOf() that does this. 
-
-Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins
-
-```{r}
-#afl.mode = 
-#afl.mode
-modeOf(afl.margins)
-maxFreq(afl.margins)
-```
-
-So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability.
-
-
-
-```{r}
-mean(afl.margins)
-```
-
-##### Measures of variability
-
-This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be?
-
-
-```{r}
-plot(afl.margins)
-```
-**Range**
-
-The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the `range()` function.
-
-```{r}
-#Find range of afl.margins here using the range function
-```
-
-But what about the earlier data we saw, ` -100,2,3,4,5,6,7,8,9,10`. Without removing the outlier, we'll get a range of 110 but without the outlier, we'll get a range of only 8.
-
-**Inter-quartile Range (IQR)**
-That is why there is something called the interquartile range (IQR) which is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. A 10% _quantile_ or _percentile_ of a data set is defined as the smallest number _x_ such that 10% of the data is less than _x_.
-
-Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.
-```{r}
-#Use the functions quantile(x = afl.margins, prob = 0.25) for 25% quantile and IQR() 
-quantile(x = afl.margins, prob = 0.2)
-
-```
-
-IQR can simply be thought as the range spanned by the “middle half” of the data.
-
-
-```{r}
-quantile( x = afl.margins, probs = c(.25,.75) )
-# try using IQR() here
-
-```
-
-
-**Variance**
-
-In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (X~i~ - $\overline{X}$). ($\overline{X}$ is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we'll get the **variance** of the data. 
-
-
-Try finding out the variance using `var()`.
-
-```{r}
-#Use var() for finding variance of afl.margins
-```
-
-_Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro_
-
-Also note that the division is by N-1 for variance for a sample! Why is that not N? 
-This we will discuss later. 
-
-**Standard Deviation**
-
-But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the _root mean square deviation_ for interpreting the spread of data points. This is called _Standard Deviation_ and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package.
-
-Try to find out the standard deviation of afl.margins.
-
-```{r}
-#Find out Std dev. here using sd()
-```
-
-##### Quick cheat sheet: When to use what?
-
-- Range: 
-  - Gives full spread of data. 
-  - Very vulnerable to outliers
-
-- Interquartile range: 
-  - Gives the “middle half” of data
-  - Robust, and complements the median nicely
-  
-- Variance:
-  - Average squared deviation from the mean
-  - It’s mathematically elegant but it’s completely uninterpretable
-
-- Standard deviation:
-  - Square root of the variance
-  - Fairly elegant mathematically, and can be interpreted pretty well
-  - Complements mean and is the most popular measure of variation
-  
-  
-##### Derivation for deviation based variance estimates 
-Mean absolute deviation; Variance; SD
- 
-Say, the sample is sample = [20,30,40]; 
-the mean of the sample is then 30
-
-The deviation of each sample data point from the mean is:
-deviation = [20-30, 30-30, 40-30];
-          = [-10, 0, 10]
-
-absolute deviation = [10,0,10] 
-mean of absolute deviation = (10+0+10)/3 
-
-square of the deviation = [-10^2, 0^2, 10^2];
-mean of the squared deviations = variance = (100+0+100)/3
-
-root of mean squared deviation = standard deviation = sqrt(variance)
-
-
-##### Bessel's correction
-While calculating variance and standard deviation of the sample, we are always
-trying to estimate the variance and standard deviation of the population. 
-
-Remember the heights of students example! 
-
-Now since the sample variance and standard deviation is biased and less than
-that of the population, we divide by N-1 instead of N to inflate the estimates.
-
-So the variance after Bessel's correction should be 
-mean of the squared deviations = variance = (100+0+100)/(3-1)
-
-  
-##### Summary function
-Now that we've learnt about the different methods of describing a data, it would've been awesome if R could summarize all of this for us together, right? 
-
-There's indeed a function called `summary()` in R.
-
-```{r}
-#Check out what summary() does for afl.margins
-summary(afl.margins)
-
-```
-
-Pretty cool, no?
-
-Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)`
-
-
-
-##### Summarizing dataframes 
-
-Let's try out summarizing a dataframe as well.
-
-```{r}
-load("clinicaltrial.Rdata")
-#Check the name of the variable in the environment which contains the dataframe and try summarizing it
-```
-
-The `psych` package also has a function called `describe()` for dataframes. Don't forget to check it out too!
-
-In fact, you can also describe these statistics group wise. 
-
-For instance, run `describeBy( x=clin.trial, group=clin.trial$therapy )`
-
-```{r}
-describeBy( x=clin.trial, group=clin.trial$therapy )
-```
-Notice that, the output displays asterisks for factor variables, in order to draw your attention to the fact that the descriptive statistics that it has calculated won’t be very meaningful for those variables.
-
-Another more general command for grouping is `by()` 
-
-Try running the following chunk and compare the results with the `describeBy()` command above.
-
-```{r}
-by(data=clin.trial, INDICES=clin.trial$therapy, FUN=describe)
-#Also try replacing describe in FUN above with summary
-```
-
-What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy.We can use `aggregate()` command.
-
-```{r}
-aggregate( formula = mood.gain ~ drug + therapy, 
-           data = clin.trial,
-           FUN = mean) 
-#1 mood.gain by drug/therapy combination
-#2 data is in the clin.trial data frame
-#3 print out group means
-
-#Try interchanging the positions of drug and therapy above
-```
-
-That's all for today!
+---
+title: "Descriptive Statistics: Central and Variability measures"
+output: html_notebook
+---
+```{r}
+#Initial packages
+install.packages("lsr")
+```
+
+In this notebook, we'll take a look at how to explore a dataset.
+
+Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what **descriptive statistics** is all about.
+
+#### Describing data
+
+Imagine you've conducted an experiment involving measurements from 20 animals. If you wanted to report the outcome of your experiment to an audience, you wouldn’t want to talk through each and every data point. Instead, you report a summary, such as ‘The 20 animals had an average weight of 15 grams’, thus saving your audience valuable time and mental energy. This notebook focuses on such summaries of numerical information including distributions, measures of central tendency and measures of variability. 
+
+##### What exactly is a distribution?
+
+If you throw a single dice 20 times in a row and note down how frequently each face occurs. The result of tallying all counts is a ‘frequency distribution’, which associates each possible outcome with a particular frequency value. Such a distribution is an empirically observed distribution because it is based on a set of 20 actual throws of a dice. Fig (a) below.
+
+![Empirical and theoretical distributions](dice rolling probability.png)
+
+But Fig (b) shows a theoretical distribution and represents probability rather than frequency. It depicts how probable is each outcome. In this case, all outcomes are equally probable and therefore it is a ‘uniform’ distribution because the probability is uniformly spread across all possible outcomes. It is furthermore a ‘discrete’ distribution because there are only six particular outcomes and no in-betweens. (Chapter 3, Winter B.)
+
+Apart from *looking* at how a data is distributed, the most important descriptive statistics for numerical data are those measuring the location of a frequency distribution and its spread. The location tells us something about the average or *typical* individual—where the observations are centered. The spread tells us how variable the measurements are from individual to individual—how widely scattered the observations are around the center. The proportion is the most important descriptive statistic for a categorical variable, measuring the fraction of observations in a given category. 
+
+##### But why is it needed?
+The importance of calculating some sort of a centre of a distribution seems obvious. How else do we address questions like “Which species is larger?” or “Which drug yielded the greatest response?” The importance of describing distribution spread is less obvious but no less crucial, at least in biology. In some fields of science, variability around a central value is instrument noise or measurement error, but in biology much of the variability signifies real differences among individuals. Different individuals respond differently to treatments, and this variability begs measurement. (Adapted from Chapter 3, Whitlock & Schluter, 2015)
+
+That's a lot of theory, let's dive into some data now.
+
+**Loading the Australian Football League Dataset**
+
+```{r}
+#Change the path according to your PC
+load("aflsmall.Rdata")
+library(lsr)
+who()
+```
+
+As you can see there are multiple variables of different class and size. 
+
+Let's take a look at afl.margins variable.
+
+```{r}
+print(afl.margins)
+```
+
+This output doesn’t make it easy to get a sense of what the data is actually saying. Just “looking at the data” isn’t a terribly effective way of understanding data.
+
+Let's try to plot it.
+
+**Frequency distribution**
+
+```{r}
+hist (afl.margins)
+```
+
+As you can see, different margins in a sample will have different measurements. We can see this variability with a **frequency distribution**. The frequency of a specific measurement in a sample is the number of observations having a particular value of the measurement. The frequency distribution shows how often each value of the variable occurs in the sample. 
+
+Therefore, here we have plotted a histogram for the afl.margins variable which gives the frequency distribution of the different margin values.
+
+Now let's try this using GGPLOT, shall we?!
+```{r}
+library(ggplot2)
+```
+
+```{r}
+df <- data.frame(afl.margins)
+```
+
+
+```{r}
+ggplot(df, aes(x = afl.margins)) + geom_histogram(bins=12)
+```
+
+```{r}
+# Basic histogram
+ggplot(df, aes(x = afl.margins)) + geom_histogram(bins=25,color="gray", fill="gray")
+```
+Note: if you want to learn more about plotting histograms check this out: http://www.sthda.com/english/wiki/ggplot2-histogram-plot-quick-start-guide-r-software-and-data-visualization
+
+
+**Skewness**
+
+If you observe the graph, you will find that it is not entirely symmetrical. A measure of such asymmetry is called **Skewness**. If the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel), then we say that the data are _negatively skewed_. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are _positively skewed_.
+
+`psych` package contains a `skew()` function that you can use to calculate skewness. 
+
+Try finding the skewness for the above data for afl.margins using skew() function and also try to guess whether this data is positively or negatively skewed.
+
+```{r}
+library(psych)
+#Try finding skewness of afl.margins here
+```
+
+Although such a graphical representation gives a 'gist' of the data but it is useful to find some "summary" statistics as well.
+
+##### Measures of Central Tendency
+In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you’d like to know something about the “average” or “middle” of your data lies. The two most commonly used measures are the mean, median and mode.
+
+**Mean**
+
+As you've already seen in previous classes, the mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values.
+
+Try finding the mean for the first 5 values from afl.margins and then for all the values of afl.margins
+```{r}
+mean(afl.margins)      # average margin
+mean(afl.margins[1:5]) # mean of the margin from the first 5 games
+
+```
+
+**Median**
+
+The second measure is the median. It is just the middle value of a set of observations. 
+*Try : Guess the median for 56, 31, 56, 8 and 32 *
+
+Probably you mentally arranged these numbers in ascending order first and then found the middle value. If there were a list of numbers like this `8, 14, 31, 32, 56, 56` . You will then find the average of middle 2 values. 
+
+Now try finding out the median for afl.margins.
+
+```{r}
+median(afl.margins)
+```
+
+**Difference between Mean and Median**
+
+Both of these are measures of central tendency but when to use which can be a bit confusing. In general, the mean is kind of like the “centre of gravity” of the data set, whereas the median is the “middle value” in the data.
+
+![Difference between mean and median](pic2.png)
+*Fig 5.2 from Learning Statistics with R by D. Navarro*
+
+**Some key points**
+
+- If data is nominal scale, then it’s probably best to use the mode instead of mean or median.
+
+- If your data are ordinal scale, you’re more likely to want to use the median than the mean.
+
+- For interval and ratio scale data, either mean or median is generally acceptable. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values.
+
+*You can read more about this in Section 5.1.4, Learning Statistics with R by D. Navarro*
+
+Now let's take a look at some more data:
+
+` -100,2,3,4,5,6,7,8,9,10`
+
+If you observed such data in real life, you will probably think that -100 is an **_outlier_**, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely but you don’t always get such cut-and-dried examples. For instance, you might get this instead:
+
+` -15,2,3,4,5,6,7,8,9,12`
+
+The `-15` looks a bit suspicious, but not anywhere near as much as `-100` did. In this case, it’s a
+little trickier. It might be a legitimate observation, it might not. In such situations, the mean might give you an error as it is highly sensitive to one or two extreme values, and is thus not considered to be a robust measure.
+
+In such situations, one solution is to use the median or another is to use a **trimmed mean**. To calculate a trimmed mean, what you do is **discard** the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. This helps in taking the mean by excluding the outliers.
+
+Let's try trimming the mean for above data.
+
+```{r}
+dataset <- c(-15,2,3,4,5,6,7,8,9,12)
+mean(x = dataset, trim = .1)
+#Try calculating 5% trimmed mean for above dataset
+```
+
+**Mode**
+
+So far we've seen how to find the mean and median but what about mode. The **mode** of a sample is very simple: it is the value that occurs most frequently. The core packages in R don’t have a function for calculating the mode. However, the _lsr_ package has a function called modeOf() that does this. 
+
+Say, you want to bet your money on the outcome of a match. You may want to find the most likely margin. This is when Mode is useful. Try to find out the mode for the variable afl.margins
+
+```{r}
+#afl.mode = 
+#afl.mode
+modeOf(afl.margins)
+maxFreq(afl.margins)
+```
+
+So far we've just seen the central measures of tendency, but we saw in the beginning that individual variability is quite important in biology. So, let's take a look at some of the measures of variability.
+
+
+
+```{r}
+mean(afl.margins)
+```
+
+##### Measures of variability
+
+This refers to how “spread out” are the data? How “far” away from the mean or median do the observed values tend to be?
+
+
+```{r}
+plot(afl.margins)
+```
+Again, do you what to try this using GGPLOT? Let's give it a shot!
+```{r}
+library(dplyr)
+df <- mutate(df, numgames= 1:length(afl.margins))
+```
+
+
+
+```{r}
+ggplot(df, aes(y = afl.margins, x = numgames)) + geom_point()
+```
+
+
+
+**Range**
+
+The range of a variable is very simple: it’s the biggest value minus the smallest value. Try to find out the range of afl.margins using the `range()` function.
+
+```{r}
+#Find range of afl.margins here using the range function
+range(afl.margins)
+```
+
+But what about the earlier data we saw, ` -100,2,3,4,5,6,7,8,9,10`. Without removing the outlier, we'll get a range of 110 but without the outlier, we'll get a range of only 8.
+
+**Inter-quartile Range (IQR)**
+That is why there is something called the interquartile range (IQR) which is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. A 10% _quantile_ or _percentile_ of a data set is defined as the smallest number _x_ such that 10% of the data is less than _x_.
+
+Try finding out 25%, 75% and 50% quantiles for afl.margins and also the Inter-quartile range.
+```{r}
+#Use the functions quantile(x = afl.margins, prob = 0.25) for 25% quantile
+quantile(x = afl.margins, prob = 0.25)
+
+```
+
+IQR can simply be thought as the range spanned by the “middle half” of the data.
+
+
+```{r}
+quantile( x = afl.margins, probs = c(.25,.75) )
+# try using IQR() here
+
+```
+
+
+**Variance**
+
+In order to find out the variance of data from the mean or median, we need to find the deviation such that abs (X~i~ - $\overline{X}$). ($\overline{X}$ is the mean of dataset). Mathematically, squared deviations are preferred over absolute deviations, and if we take the mean of all the squared deviations, we'll get the **variance** of the data. 
+
+
+Try finding out the variance using `var()`.
+
+```{r}
+#Use var() for finding variance of afl.margins
+var(afl.margins)
+```
+
+_Read more about var() function and absolute vs squared deviations in Section 5.2.4 from Learning Statistics with R by D. Navarro_
+
+Also note that the division is by N-1 for variance for a sample! Why is that not N? 
+This we will discuss later. 
+
+**Standard Deviation**
+
+But what does this variance signify? It is very difficult to interpret the squared value and therefore, we take the _root mean square deviation_ for interpreting the spread of data points. This is called _Standard Deviation_ and is calculated by taking the square root of variance mathematically, and using the sd() function in R base package.
+
+Try to find out the standard deviation of afl.margins.
+
+```{r}
+#Find out Std dev. here using sd()
+
+```
+
+##### Quick cheat sheet: When to use what?
+
+- Range: 
+  - Gives full spread of data. 
+  - Very vulnerable to outliers
+
+- Interquartile range: 
+  - Gives the “middle half” of data
+  - Robust, and complements the median nicely
+  
+- Variance:
+  - Average squared deviation from the mean
+  - It’s mathematically elegant but it’s completely uninterpretable
+
+- Standard deviation:
+  - Square root of the variance
+  - Fairly elegant mathematically, and can be interpreted pretty well
+  - Complements mean and is the most popular measure of variation
+  
+  
+##### Derivation for deviation based variance estimates 
+Mean absolute deviation; Variance; SD
+ 
+Say, the sample is sample = [20,30,40]; 
+the mean of the sample is then 30
+
+The deviation of each sample data point from the mean is:
+deviation = [20-30, 30-30, 40-30];
+          = [-10, 0, 10]
+
+absolute deviation = [10,0,10] 
+mean of absolute deviation = (10+0+10)/3 
+
+square of the deviation = [-10^2, 0^2, 10^2];
+mean of the squared deviations = variance = (100+0+100)/3
+
+root of mean squared deviation = standard deviation = sqrt(variance)
+
+
+##### Bessel's correction
+While calculating variance and standard deviation of the sample, we are always
+trying to estimate the variance and standard deviation of the population. 
+
+Remember the heights of students example! 
+
+Now since the sample variance and standard deviation is biased and less than
+that of the population, we divide by N-1 instead of N to inflate the estimates.
+
+So the variance after Bessel's correction should be 
+mean of the squared deviations = variance = (100+0+100)/(3-1)
+
+  
+##### Summary function
+Now that we've learnt about the different methods of describing a data, it would've been awesome if R could summarize all of this for us together, right? 
+
+There's indeed a function called `summary()` in R.
+
+```{r}
+#Check out what summary() does for afl.margins
+summary(afl.margins)
+
+```
+
+Pretty cool, no?
+
+Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)`
+
+
+
+##### Summarizing dataframes 
+
+Let's try out summarizing a dataframe as well.
+
+```{r}
+load("clinicaltrial.Rdata")
+#Check the name of the variable in the environment which contains the dataframe and try summarizing it
+```
+
+The `psych` package also has a function called `describe()` for dataframes. Don't forget to check it out too!
+
+In fact, you can also describe these statistics group wise. 
+
+For instance, run `describeBy( x=clin.trial, group=clin.trial$therapy )`
+
+```{r}
+describeBy( x=clin.trial, group=clin.trial$therapy )
+```
+Notice that, the output displays asterisks for factor variables, in order to draw your attention to the fact that the descriptive statistics that it has calculated won’t be very meaningful for those variables.
+
+Another more general command for grouping is `by()` 
+
+Try running the following chunk and compare the results with the `describeBy()` command above.
+
+```{r}
+by(data=clin.trial, INDICES=clin.trial$therapy, FUN=describe)
+#Also try replacing describe in FUN above with summary
+```
+
+What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy.We can use `aggregate()` command.
+
+```{r}
+aggregate( formula = mood.gain ~ drug + therapy, 
+           data = clin.trial,
+           FUN = mean) 
+#1 mood.gain by drug/therapy combination
+#2 data is in the clin.trial data frame
+#3 print out group means
+
+#Try interchanging the positions of drug and therapy above
+```
+
+That's all for today!

From 80e85fcc412bd72c7c5e340fd8a9b653874500e4 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 30 Aug 2022 09:49:39 +0530
Subject: [PATCH 17/55] Via upload. Summary with different objects.

---
 Module 3/Notebooks/Module3_Nb1.Rmd | 32 +++++++++++++++++++++++++-----
 1 file changed, 27 insertions(+), 5 deletions(-)

diff --git a/Module 3/Notebooks/Module3_Nb1.Rmd b/Module 3/Notebooks/Module3_Nb1.Rmd
index 9f098df6..fa7d0e9f 100644
--- a/Module 3/Notebooks/Module3_Nb1.Rmd	
+++ b/Module 3/Notebooks/Module3_Nb1.Rmd	
@@ -331,17 +331,41 @@ Pretty cool, no?
 
 Also try it out for other kinds of variables like `afl.finalists` or `as.character(afl.finalists)`
 
+```{r}
+ggplot(df, aes(x=afl.margins, )) + 
+  geom_boxplot()+ coord_flip()
+```
 
 
-##### Summarizing dataframes 
 
-Let's try out summarizing a dataframe as well.
+##### Summary function
+
+If the Object is a numeric
+```{r}
+summary(object = afl.margins)
+```
+
+```{r}
+blowouts <-  afl.margins > 50
+blowouts
+```
+```{r}
+summary (object = blowouts)
+```
+
+
+Now let's try out summarizing a dataframe as well.
+
 
 ```{r}
 load("clinicaltrial.Rdata")
 #Check the name of the variable in the environment which contains the dataframe and try summarizing it
 ```
 
+```{r}
+summary(clin.trial)
+```
+
 The `psych` package also has a function called `describe()` for dataframes. Don't forget to check it out too!
 
 In fact, you can also describe these statistics group wise. 
@@ -365,9 +389,7 @@ by(data=clin.trial, INDICES=clin.trial$therapy, FUN=describe)
 What if you have multiple grouping variables? Suppose, for example, you would like to look at the average mood gain separately for all possible combinations of drug and therapy.We can use `aggregate()` command.
 
 ```{r}
-aggregate( formula = mood.gain ~ drug + therapy, 
-           data = clin.trial,
-           FUN = mean) 
+aggregate( mood.gain ~ drug + therapy, data = clin.trial,FUN = mean) 
 #1 mood.gain by drug/therapy combination
 #2 data is in the clin.trial data frame
 #3 print out group means

From 64cc6290a878b5a0e8c6688ee4451da0eeb2f045 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 6 Sep 2022 07:35:57 +0530
Subject: [PATCH 18/55] Included ggcorrplot, correlate functions

---
 Module 3/Notebooks/Module3_Nb2.Rmd | 656 ++++++++++++++++-------------
 1 file changed, 360 insertions(+), 296 deletions(-)

diff --git a/Module 3/Notebooks/Module3_Nb2.Rmd b/Module 3/Notebooks/Module3_Nb2.Rmd
index 879883c4..9d13312f 100644
--- a/Module 3/Notebooks/Module3_Nb2.Rmd	
+++ b/Module 3/Notebooks/Module3_Nb2.Rmd	
@@ -1,296 +1,360 @@
----
-title: "Descriptive Statistics: Scaling and Correlations"
-output: html_notebook
----
-
-After taking a first look at our data in the last notebook, now we want to start looking at it more closely as per our needs and requirements. 
-
-#### Scaling
-
-In simple terms, scaling refers to changing size of an object without affecting its shape.
-
-##### Linear Transformation:
-
-A linear transformation involves addition, subtraction, multiplication, or division with a constant value. For example, if you add 1 to the numbers 2, 4, and 6, the resulting numbers (3, 5, and 7) are a linear transformation of the original numbers. 
-Linear transformations are useful, because they allow you to represent your data in a metric that is suitable to you and your audience.
-
-**Centering:**
-
-‘Centering’ is a particularly common linear transformation. This linear transfor- mation is frequently applied to continuous predictor variables. 
-To center a predictor variable, subtract the mean of that predictor variable from each data point. As a result, each data point is expressed in terms of how much it is above the mean (positive score) or below the mean (negative score). Thus, subtracting the mean out of the variable expresses each data point as a mean-deviation score. The value zero now has a new meaning for this variable: it is at the ‘center’ of the variable’s distribution, namely, the mean.
-
-**Standardizing:**
-
-A second common linear transformation is ‘standardizing’ or ‘z–scoring’. For standardizing, the centered variable is divided by the standard deviation of the sample.
-
-Let's look at an example: 
-
-The following are response durations from a psycholinguistic experiment:
-
-`460ms 480ms 500ms 520ms 540ms`
-
-The mean of these five numbers is `500ms`. 
-
-Centering these numbers results in the following:
-
-`− 40ms − 20ms 0ms +20ms + 40ms`
-
-The standard deviation (learnt in last notebook) for these numbers is `~32ms`. 
-
-To ‘standardize’, we have to divide the centered data by the standard deviation. For example, the first point, `–40ms`, divided by `32ms`, yields `–1.3`. Since each data point is divided by the same number, this change qualifies as a linear transformation.
-
-As a result of standardization, you get the following numbers (rounded to one digit):
-
-`−1.3z − 0.6z 0z + 0.6z +1.3z`
-
-The raw response duration `460ms` is `–40ms` (after centering), which corresponds to being `1.3` standard deviations below the mean. Thus, standardization involves re-expressing the data in terms of **how many standard deviations they are away from the mean**.
-
-##### But why this extra effort?
-
-Standardizing is a way of getting rid of a variable’s metric. In a situation with multiple variables, each variable may have a different standard deviation, but by dividing each variable by the respective standard deviation, it is possible to convert all variables into a scale of **standard units**. This sometimes may help in making variables comparable, for example, when assessing the relative impact of multiple predictors. For example, if you can imagine we have two questionnaires - one for extraversion where you scored 2 out of 10 and the other for grumpiness where you scored 35 out of 50, then it doesn’t make a lot of sense to try to compare your raw score of 2 on the extraversion questionnaire to your raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are “about” fundamentally different things, so this would be like comparing apples to oranges. But if you standardize them, they will still become comparable in some sense.
-
-Let's also examine the score of 35 out of 50 for grumpiness. Would this mean that you're 70% grumpy? Instead of interpreting raw data this way, it would make more sense if we describe your grumpiness in terms of the overall distribution of the grumpiness of humans which is possible through  standardisation i.e. where do you lie on the grumpiness spectrum of the all humans? ;)
-
-```{r}
-#Try it out yourself
-#Define a vector with Grumpiness scores of you and your friends and find the z score for your self
-X =                        
-z = (X - mean(X)) / sd(X)
-```
-
-Using scale() to center and normalize
-```{r}
-load("aflsmall.Rdata")
-afl.margins_c <- scale(afl.margins, scale = FALSE)
-afl.margins_z <- scale(afl.margins)
-```
-
-Plotting the histogram
-```{r}
-hist(afl.margins)
-hist(afl.margins_c)
-hist(afl.margins_z)
-```
-
-
-
-_Reference: Chapter 5, Winter B._
-
-#### Correlation
-
-So far we have focused entirely on how to construct descriptive statistics for a single variable. We haven’t talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables.
-
-```{r}
-#Let's load some data
-load( "parenthood.Rdata" )
-who(TRUE)
-```
-
-```{r}
-#Try describe() for the above dataframe
-```
-
-
-```{r}
-#Let's also take a graphical look at the data 
-hist(parenthood$dan.sleep)
-
-#Try plotting for the other 2 variables
-
-```
-
-But we now want to take a look at the relationship between two variables. n order to visualize that, it is better to plot a **scatter plot.** (Plotting graphs will be covered in detail a separate notebook).
-
-_Brief note on Scatterplots:_
-
-In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don’t really have a clear opinion about what the causal relationship is (e.g., does A cause B, or does B cause A, or does some other variable C controls both A and B). If that’s the case, it doesn’t really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it’s conventional to plot the **cause** variable on the **x-axis**, and the **effect** variable on the **y-axis**. 
-
-Suppose our goal is to draw a scatterplot displaying the relationship between the amount of sleep that Dan gets (dan.sleep) and how grumpy she is the next day (dan.grump). _Do you suspect a causal relationship here?_
-
-A simple way to plot these scatter plots is to use the scatterplot() function in the car package. 
-
-Let's load the package and get started.
-
-```{r}
-install.packages("car")
-install.packages("Rcpp")
-```
-
-
-```{r}
-library(car)
-scatterplot( dan.grump ~ dan.sleep, data = parenthood, regLine = FALSE, smooth = FALSE)
-scatterplot
-```
-
-```{r}
-#Plot a scatter plot for baby.sleep and dan.grump variables
-```
-
-
-Just by plain observation and comparison, you can see that the relationship is qualitatively the same in both cases: more sleep equals less grump! However, it’s also pretty obvious that the relationship between dan.sleep and dan.grump is stronger than the relationship between baby.sleep and dan.grump. 
-
-But what about the plot between baby.sleep and dan.sleep?
-
-```{r}
-#Plot baby sleep and dan sleep here
-```
-
-Is the direction of this plot same as the earlier plots? What about strength?
-
-##### Correlation coefficient
-
-In order to to quantitatively represent the relationships of strength and direction we discussed above, we can use correlation coefficient.
-
-The correlation coefficient (or Pearson's correlation coefficient) between two variables X and Y (sometimes denoted _r~XY~_ ) is a measure that varies from -1 to 1. When _r_ = -1 it means that we have a perfect negative relationship, and when _r_ = 1 it means we have a perfect positive relationship. When _r_ = 0, there’s no relationship at all.
-
-Look at the plots for different _r_ values:
-
-![Correlation plots](fig 4.png)
-
-##### Covariance
-
-The covariance between two variables X and Y is a generalisation of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables:
-
- \begin{align*}
- 
- Cov (X, Y) = \frac{1}{N-1}\sum_{i=1}^{N} (X- \overline{X} ) (Y- \overline{Y} )  \\
- 
- \end{align*}
- 
-Covariance can be understood as an “average cross product” between X and Y . The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If it is positive, then the covariance is also positive; and if the relationship is negative then the covariance is also negative. But as it has weird units (try seeing for yourself), it si difficult to interpret and therefore we standardise the covariance, the exact same way that the z-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations. 
-
-This is what we call as the correlation coefficent, _r_:
-
-\begin{align*}
-
- r~XY~ = \frac{Cov(X,Y)}{\sigma_{X} \sigma_{Y}}
-
-\end{align*}
-
-This way, covariance properties are retained and it also becomes interpretable.
-
-Now let's check out how to code this using cor().
-
-```{r}
-cor(x = parenthood$dan.sleep, y = parenthood$dan.grump)
-
-#Try giving the entire dataframe 'parenthood' as input in cor()
-```
-
-What did you find?
-
-##### What does r = 0.4 mean?
-
-It really depends on what you want to use the data for, and on how strong the correlations in your field tend to be.
-
-![Correlation coefficient interpretation table](fig 5.png)
- 
-Now let's take a look at this data called "Anscombe's Quartet"
- 
-```{r}
-load( "anscombesquartet.Rdata" )
-cor( X1, Y1 )
-cor( X2, Y2 )
-cor (X3, Y3)
-cor (X4, Y4)
-```
-
-Were the correlation coefficients same?
-
-Now try plotting them.
-
-```{r}
-scatterplot(x = X1, y = Y1,regLine = FALSE, smooth = FALSE)
-scatterplot(x = X2, y = Y2,regLine = FALSE, smooth = FALSE)
-scatterplot(x = X3, y = Y3,regLine = FALSE, smooth = FALSE)
-scatterplot(x = X4, y = Y4,regLine = FALSE, smooth = FALSE)
-
-
-```
-
-Therefore, remember to always look at the scatterplot before attaching any interpretation to the data!
-
-If we have to properly define the role of Pearson's coefficient, we can say that it actually measures the strength of the linear relationship between two variables. In other words, it gives a measure of the extent to which the data all tend to fall on a single, perfectly straight line.
-
-##### Spearman's Rank Order Correlation Coefficient
- 
-But let's take a look at another dataset and find correlation between its variables.
-
-```{r}
-load( "effort.Rdata" )
-effort
-cor( effort$hours, effort$grade )
-```
-
-If you plot this - 
-
-```{r}
-scatterplot(effort$hours, effort$grade, regLine = TRUE, smooth = FALSE)
-```
-
-The correlation _r_ = 0.91 we get above doe snot represent the actual relationship the plot is depicting. What we’re looking for is something that captures the fact that there is a perfect **ordinal relationship** here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade.
-
-If we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, let's rank all 10 of the students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2).
-
-```{r}
-hours.rank <- rank( effort$hours )   # rank students by hours worked
-grade.rank <- rank( effort$grade )   # rank students by grade received
-
-#Now try cor() function for these
-cor( hours.rank, grade.rank )
-```
-
-Now the correlation coefficient we get is different from the Perason's correlation coefficient _r_ we got earlier. This new correlation coefficient that we got is called '**Spearman's Correlation Coefficient**', denoted by $\rho$.
-
-```{r}
-#Execute this and compare with the correlation coefficient we got above
-cor( effort$hours, effort$grade, method = "spearman")
-```
-
-##### the correlate() function
-Try using this function to find the relationship between several variables in a dataframe at once.
-
-
-##### Handling missing values
-
-We've seen in earlier lectures that there could be missing values in data which are represented by `NA` in R. One easy way to remove them is using `na.rm = TRUE` as argument in many functions.
-
-But what if we have missing values in a dataframe where we have to find correlations across variables.
-
-Let's look at such a dataset.
-
-```{r}
-load( "parenthood2.Rdata" )
-print( parenthood2 )
-describe( parenthood2 ) 
-#Check how many missing values are there for each variable - compare the values in 'n' with the number of days.
-```
-
-Now, let's try finding correlations for this dataframe.
-
-```{r}
-cor(parenthood2)
-```
-
-In order top overcome this problem, we can use `use` as an argument in the cor() function. Try out the following.
-
-```{r}
-cor(parenthood2, use = "complete.obs")
-cor(parenthood2, use = "pairwise.complete.obs")
-```
-
-When we choose `use = "complete.obs"`, R will completely ignore all cases (i.e., all rows in our parenthood2 data frame) that have any missing values at all. For eg., if you choose use = "complete.obs" R will ignore that row completely: that is, even when it’s trying to calculate the correlation between dan.sleep and dan.grump, observation 1 will be ignored, because the value of baby.sleep is missing for that observation.
-
-Whereas when we set `use = "pairwise.complete.obs"` R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of parenthood2 is for baby.sleep R will only drop observation 1 when baby.sleep is one of the variables involved: and so R keeps observation 1 when trying to correlate dan.sleep and dan.grump.
-
-The above operation can also be performed by another function called `correlate()` in `lsr` package.
-
-Try it out.
-```{r}
-#Try correlate() for parenthood2 here
-```
-
-_Reference : Chapter 5, D. Navarro_
-
-That's all folks!
+---
+title: "Descriptive Statistics: Scaling and Correlations"
+output: html_notebook
+---
+
+After taking a first look at our data in the last notebook, now we want to start looking at it more closely as per our needs and requirements. 
+
+#### Scaling
+
+In simple terms, scaling refers to changing size of an object without affecting its shape.
+
+##### Linear Transformation:
+
+A linear transformation involves addition, subtraction, multiplication, or division with a constant value. For example, if you add 1 to the numbers 2, 4, and 6, the resulting numbers (3, 5, and 7) are a linear transformation of the original numbers. 
+Linear transformations are useful, because they allow you to represent your data in a metric that is suitable to you and your audience.
+
+**Centering:**
+
+‘Centering’ is a particularly common linear transformation. This linear transfor- mation is frequently applied to continuous predictor variables. 
+To center a predictor variable, subtract the mean of that predictor variable from each data point. As a result, each data point is expressed in terms of how much it is above the mean (positive score) or below the mean (negative score). Thus, subtracting the mean out of the variable expresses each data point as a mean-deviation score. The value zero now has a new meaning for this variable: it is at the ‘center’ of the variable’s distribution, namely, the mean.
+
+**Standardizing:**
+
+A second common linear transformation is ‘standardizing’ or ‘z–scoring’. For standardizing, the centered variable is divided by the standard deviation of the sample.
+
+Let's look at an example: 
+
+The following are response durations from a psycholinguistic experiment:
+
+`460ms 480ms 500ms 520ms 540ms`
+
+The mean of these five numbers is `500ms`. 
+
+Centering these numbers results in the following:
+
+`− 40ms − 20ms 0ms +20ms + 40ms`
+
+The standard deviation (learnt in last notebook) for these numbers is `~32ms`. 
+
+To ‘standardize’, we have to divide the centered data by the standard deviation. For example, the first point, `–40ms`, divided by `32ms`, yields `–1.3`. Since each data point is divided by the same number, this change qualifies as a linear transformation.
+
+As a result of standardization, you get the following numbers (rounded to one digit):
+
+`−1.3z − 0.6z 0z + 0.6z +1.3z`
+
+The raw response duration `460ms` is `–40ms` (after centering), which corresponds to being `1.3` standard deviations below the mean. Thus, standardization involves re-expressing the data in terms of **how many standard deviations they are away from the mean**.
+
+##### But why this extra effort?
+
+Standardizing is a way of getting rid of a variable’s metric. In a situation with multiple variables, each variable may have a different standard deviation, but by dividing each variable by the respective standard deviation, it is possible to convert all variables into a scale of **standard units**. This sometimes may help in making variables comparable, for example, when assessing the relative impact of multiple predictors. For example, if you can imagine we have two questionnaires - one for extraversion where you scored 2 out of 10 and the other for grumpiness where you scored 35 out of 50, then it doesn’t make a lot of sense to try to compare your raw score of 2 on the extraversion questionnaire to your raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are “about” fundamentally different things, so this would be like comparing apples to oranges. But if you standardize them, they will still become comparable in some sense.
+
+Let's also examine the score of 35 out of 50 for grumpiness. Would this mean that you're 70% grumpy? Instead of interpreting raw data this way, it would make more sense if we describe your grumpiness in terms of the overall distribution of the grumpiness of humans which is possible through  standardisation i.e. where do you lie on the grumpiness spectrum of the all humans? ;)
+
+```{r}
+#Try it out yourself
+#Define a vector with Grumpiness scores of you and your friends and find the z score for your self
+X =                        
+z = (X - mean(X)) / sd(X)
+```
+
+Using scale() to center and normalize
+```{r}
+load("aflsmall.Rdata")
+afl.margins_c <- scale(afl.margins, scale = FALSE)
+afl.margins_z <- scale(afl.margins)
+```
+
+Plotting the histogram
+```{r}
+hist(afl.margins)
+hist(afl.margins_c)
+hist(afl.margins_z)
+```
+
+
+
+_Reference: Chapter 5, Winter B._
+
+#### Correlation
+
+So far we have focused entirely on how to construct descriptive statistics for a single variable. We haven’t talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables.
+
+```{r}
+#Let's load some data
+load( "parenthood.Rdata" )
+who(TRUE)
+```
+
+```{r}
+#Try describe() for the above dataframe
+```
+
+
+```{r}
+#Let's also take a graphical look at the data 
+hist(parenthood$dan.sleep)
+
+#Try plotting for the other 2 variables
+
+```
+
+But we now want to take a look at the relationship between two variables. n order to visualize that, it is better to plot a **scatter plot.** (Plotting graphs will be covered in detail a separate notebook).
+
+_Brief note on Scatterplots:_
+
+In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don’t really have a clear opinion about what the causal relationship is (e.g., does A cause B, or does B cause A, or does some other variable C controls both A and B). If that’s the case, it doesn’t really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it’s conventional to plot the **cause** variable on the **x-axis**, and the **effect** variable on the **y-axis**. 
+
+Suppose our goal is to draw a scatterplot displaying the relationship between the amount of sleep that Dan gets (dan.sleep) and how grumpy she is the next day (dan.grump). _Do you suspect a causal relationship here?_
+
+A simple way to plot these scatter plots is to use the scatterplot() function in the car package. 
+
+Let's load the package and get started.
+
+```{r}
+install.packages("car")
+install.packages("Rcpp")
+```
+
+
+```{r}
+library(car)
+scatterplot( dan.grump ~ dan.sleep, data = parenthood, regLine = FALSE, smooth = FALSE)
+scatterplot
+```
+
+
+```{r}
+library(ggplot2)
+# Basic scatter plot
+ggplot(parenthood, aes(x = dan.sleep, y = dan.grump)) + geom_point() + geom_point() + geom_smooth()
+```
+
+
+
+
+
+```{r}
+#Plot a scatter plot for baby.sleep and dan.grump variables
+ggplot(parenthood, aes(x = baby.sleep, y = dan.grump)) + geom_point()
+
+```
+
+
+Just by plain observation and comparison, you can see that the relationship is qualitatively the same in both cases: more sleep equals less grump! However, it’s also pretty obvious that the relationship between dan.sleep and dan.grump is stronger than the relationship between baby.sleep and dan.grump. 
+
+But what about the plot between baby.sleep and dan.sleep?
+
+```{r}
+#Plot baby sleep and dan sleep here
+```
+
+Is the direction of this plot same as the earlier plots? What about strength?
+
+##### Correlation coefficient
+
+In order to to quantitatively represent the relationships of strength and direction we discussed above, we can use correlation coefficient.
+
+The correlation coefficient (or Pearson's correlation coefficient) between two variables X and Y (sometimes denoted _r~XY~_ ) is a measure that varies from -1 to 1. When _r_ = -1 it means that we have a perfect negative relationship, and when _r_ = 1 it means we have a perfect positive relationship. When _r_ = 0, there’s no relationship at all.
+
+Look at the plots for different _r_ values:
+
+![Correlation plots](fig 4.png)
+
+##### Covariance
+
+The covariance between two variables X and Y is a generalisation of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables:
+
+ \begin{align*}
+ 
+ Cov (X, Y) = \frac{1}{N-1}\sum_{i=1}^{N} (X- \overline{X} ) (Y- \overline{Y} )  \\
+ 
+ \end{align*}
+ 
+Covariance can be understood as an “average cross product” between X and Y . The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If it is positive, then the covariance is also positive; and if the relationship is negative then the covariance is also negative. But as it has weird units (try seeing for yourself), it si difficult to interpret and therefore we standardise the covariance, the exact same way that the z-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations. 
+
+This is what we call as the correlation coefficent, _r_:
+
+\begin{align*}
+
+ r~XY~ = \frac{Cov(X,Y)}{\sigma_{X} \sigma_{Y}}
+
+\end{align*}
+
+This way, covariance properties are retained and it also becomes interpretable.
+
+Now let's check out how to code this using cor().
+
+```{r}
+cor(x = parenthood$dan.sleep, y = parenthood$dan.grump)
+
+#Try giving the entire dataframe 'parenthood' as input in cor()
+```
+
+What did you find?
+
+##### What does r = 0.4 mean?
+
+It really depends on what you want to use the data for, and on how strong the correlations in your field tend to be.
+
+![Correlation coefficient interpretation table](fig 5.png)
+
+Let's make some correlation plots using ggcorrplot library 
+```{r}
+install.packages("ggcorrplot")
+```
+
+
+```{r}
+library(ggcorrplot)
+```
+
+```{r}
+corr <- round(cor(parenthood), 1)
+corr
+```
+```{r}
+p.mat <- cor_pmat(parenthood)
+p.mat
+```
+
+```{r}
+ggcorrplot(corr)
+```
+
+```{r}
+ggcorrplot(corr, method = "circle")
+```
+
+
+```{r}
+ggcorrplot(corr, hc.order = TRUE, type = "lower",
+   lab = TRUE)
+```
+For more such options refer:
+http://www.sthda.com/english/wiki/ggcorrplot-visualization-of-a-correlation-matrix-using-ggplot2
+
+
+Now let's take a look at this data called "Anscombe's Quartet"
+ 
+```{r}
+load( "anscombesquartet.Rdata" )
+cor( X1, Y1 )
+cor( X2, Y2 )
+cor (X3, Y3)
+cor (X4, Y4)
+```
+
+Were the correlation coefficients same?
+
+Now try plotting them.
+
+```{r}
+scatterplot(x = X1, y = Y1,regLine = FALSE, smooth = FALSE)
+scatterplot(x = X2, y = Y2,regLine = FALSE, smooth = FALSE)
+scatterplot(x = X3, y = Y3,regLine = FALSE, smooth = FALSE)
+scatterplot(x = X4, y = Y4,regLine = FALSE, smooth = FALSE)
+
+
+```
+
+Therefore, remember to always look at the scatterplot before attaching any interpretation to the data!
+
+If we have to properly define the role of Pearson's coefficient, we can say that it actually measures the strength of the linear relationship between two variables. In other words, it gives a measure of the extent to which the data all tend to fall on a single, perfectly straight line.
+
+##### Spearman's Rank Order Correlation Coefficient
+ 
+But let's take a look at another dataset and find correlation between its variables.
+
+```{r}
+load( "effort.Rdata" )
+effort
+cor( effort$hours, effort$grade )
+```
+
+If you plot this - 
+
+```{r}
+scatterplot(effort$hours, effort$grade, regLine = TRUE, smooth = FALSE)
+```
+
+The correlation _r_ = 0.91 we get above does not represent the actual relationship the plot is depicting. What we’re looking for is something that captures the fact that there is a perfect **ordinal relationship** here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade.
+
+If we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, let's rank all 10 of the students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2).
+
+```{r}
+hours.rank <- rank( effort$hours )   # rank students by hours worked
+grade.rank <- rank( effort$grade )   # rank students by grade received
+
+#Now try cor() function for these
+cor( hours.rank, grade.rank )
+```
+
+Now the correlation coefficient we get is different from the Perason's correlation coefficient _r_ we got earlier. This new correlation coefficient that we got is called '**Spearman's Correlation Coefficient**', denoted by $\rho$.
+
+```{r}
+#Execute this and compare with the correlation coefficient we got above
+cor( effort$hours, effort$grade, method = "spearman")
+```
+
+##### the correlate() function
+Try using this function to find the relationship between several variables in a dataframe at once even if some of them are not numeric!
+
+```{r}
+load("work.Rdata")
+head(work)
+```
+```{r}
+cor(work)
+```
+```{r}
+correlate(work)
+```
+Or if you want to use the Spearman method
+```{r}
+correlate( work, corr.method="spearman" )
+```
+
+
+##### Handling missing values
+
+We've seen in earlier lectures that there could be missing values in data which are represented by `NA` in R. One easy way to remove them is using `na.rm = TRUE` as argument in many functions.
+
+But what if we have missing values in a dataframe where we have to find correlations across variables.
+
+Let's look at such a dataset.
+
+```{r}
+load( "parenthood2.Rdata" )
+print( parenthood2 )
+head(parenthood2)
+#Check how many missing values are there for each variable - compare the values in 'n' with the number of days.
+```
+
+Now, let's try finding correlations for this dataframe.
+
+```{r}
+cor(parenthood2)
+```
+
+In order top overcome this problem, we can use `use` as an argument in the cor() function. Try out the following.
+
+```{r}
+cor(parenthood2, use = "complete.obs")
+cor(parenthood2, use = "pairwise.complete.obs")
+```
+
+When we choose `use = "complete.obs"`, R will completely ignore all cases (i.e., all rows in our parenthood2 data frame) that have any missing values at all. For eg., if you choose use = "complete.obs" R will ignore that row completely: that is, even when it’s trying to calculate the correlation between dan.sleep and dan.grump, observation 1 will be ignored, because the value of baby.sleep is missing for that observation.
+
+Whereas when we set `use = "pairwise.complete.obs"` R only looks at the variables that it’s trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of parenthood2 is for baby.sleep R will only drop observation 1 when baby.sleep is one of the variables involved: and so R keeps observation 1 when trying to correlate dan.sleep and dan.grump.
+
+The above operation can also be performed by another function called `correlate()` in `lsr` package.
+
+Try it out.
+```{r}
+#Try correlate() for parenthood2 here
+```
+
+_Reference : Chapter 5, D. Navarro_
+
+That's all folks!

From d07b479596257a1068e7cdcaa210c7e0bdc867ac Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Mon, 12 Sep 2022 16:37:17 +0530
Subject: [PATCH 19/55] Binomial plots, ggplot histogram, density plots

---
 Module 3/Notebooks/Distributions.Rmd | 556 ++++++++++++++-------------
 1 file changed, 288 insertions(+), 268 deletions(-)

diff --git a/Module 3/Notebooks/Distributions.Rmd b/Module 3/Notebooks/Distributions.Rmd
index 503dc14f..e2d99e02 100644
--- a/Module 3/Notebooks/Distributions.Rmd	
+++ b/Module 3/Notebooks/Distributions.Rmd	
@@ -1,268 +1,288 @@
----
-title: "Inferential Statistics: Probability & Distributions - 1"
-output: html_notebook
----
-
-So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.
-
-#### Pre-requisite: Probability
-
-##### Difference between probability and statistics**
-Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?
-
-In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]*
-
-**A short note on Models**
-
-A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]*
-- - - -
-
-But even though we know the models like `P(heads) = 0.5`, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?
-
-We want to figure out which is the true model of the world. Is it *P(heads) = 0.5* or is it *P(heads) $\ne$ 0.5*?
-
-##### What is probability really?
-
-**The frequentist view**
-
-![Frequentist_graph](Fig4.png)
-
-According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.
-
- *Advantages*
- -  It is objective: the probability of an event is necessarily grounded in the world.
- -  It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.
-
-But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?
-
-**The Bayesian view**
-
-Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this 'degree of belief'? 
-
-One way is to use 'rational gambling'. So a “subjective probability” will be operationalized in terms of what bets you're willing to accept.
-
- *Advantage*
- - You don’t need to be limited to those events that are repeatable.
- 
- *Disadvantage*
- - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an **intelligent agent** out there that believes in things. 
-
-
-In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).
-
-##### Definitions
-
-Refer to the example described in *Section 9.3.1, Navarro D.* for the following content.
-
-**Elementary event:** Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.
-
-**Sample space:** The set of all possible events (e.g., the wardrobe)
-
-**Probability:** Numbers between 0 and 1.
-
-For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.
-
-If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).
-
-**Law of total probability:** The probabilities of the elementary events need to add up to 1
-
-#### Distributions
-
-Let's take a look at this and see what is a distribution. 
-
-```{r}
-pants <- data.frame(
-   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
-   label = c("X1", "X2", "X3", "X4", "X5"),
-   probability = c(0.5,0.3,0.1,0,0.1))
-
-pants
-```
-Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.
-
-```{r}
-#Try plotting a bar graph of all the probabilities above
-```
-Let's think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans" has occurred. 
-Then what will be the probability of event E.
-
-P(E) = P(X1) + P(X2) + P(X3)
-
-If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:
-
-![Probability_rules](Fig5.png)
-
-##### Binomial Distribution
-
-*Refer to section 9.4.1, Navarro D., for the detailed example*
-
-Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`.
-
-`X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N.
-
-4 ~ Binomial(1/6, 20)
-
-5 ~ Binomial(1/2, 10)
-
-Let's generate a binomial distribution in R:
-
-```{r}
-dbinom( x = 1, size = 20, prob = 1/6)
-```
-The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.
-
-What if the dice is replaced by a coin in the above example? How will the probability change? 
-
-```{r}
-#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
-```
-There are different functions in R for different distributions as well as different ones for finding different quantity of interest.
-
-If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use `pbinom`.
-
-```{r}
-#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
-pbinom( q= 3, size = 20, prob = 1/6)
-
-#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
-```
-In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.
-
-Now let’s say we want to calculate the 75th percentile of the binomial distribution.
-
-```{r}
-qbinom( p = 0.566, size = 20, prob = 1/6 )
-
-#Practice - Find the 40th percentile
-```
-
-We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.
-
-```{r}
-z <- rbinom( n = 10000, size = 20, prob = 2/6 )
-z
-#Let's also plot this and see how it looks
-hist(z, col = 'steelblue')
-```
-#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.
-
-All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. 
-
-
-##### Normal Distribution
-
-Most frequently encountered distribution.
-Eg: heights of all students in the class, marks obtained in exams, etc
-
-Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.
-
-
-- `dnorm()` - For probability density
-- `pnorm()` - For cumulative probability
-- `qnorm()` - For quantile of
-- `rnorm()` - For random number generation
-
-
-mean = 0; sd = 1 -> standard normal distribution
-```{r}
-normal_distribution <- rnorm(10000, mean = 0, sd = 1) 
-histogram_normal_distribution <- hist(normal_distribution)
-plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=3)
-
-
-```
-
-Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution
-
-The notation for a normal distribution is: X ∼ Normal(μ,σ) 
-
-
-dnorm tells you the probability of getting a particular outcome
-```{r}
-dnorm(x=85, mean=80, sd=5)
-
-```
-Cumulative normal distribution
-```{r}
-pnorm(q = 80, mean = 80, sd = 5)
-```
-
-```{r}
-qnorm(0.25 ,mean = 0 , sd = 1)
-```
-
-*Checking for normality using the Shapiro-Wilk Test*
-```{r}
-norm <- rnorm(50, mean = 0, sd = 1) 
-shapiro.test(norm)
-
-binom <- rbinom(100, 20, 1/6)
-shapiro.test(binom)
-
-```
-
-
-
-
-
-##### Other useful distributions
-
-Some other distributions you may encounter include:
-*1) t distribution*
-
-Looks like the normal distribution but has heavier tails. 
-Used when data looks like a normal distribution but the mean and SD are unknown.
-
-Use the following functions to visualize the t distribution: 
-dt(), pt(), qt() and rt()
-
-```{r}
-t_distribution <- rt(10000, 3)
-histogram_t_distribution <- hist(t_distribution)
-plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3)
-```
-
-
-
-
-*2) Chi square (χ2) distribution*
-
-All positive and heavily skewed to the left.  
-Used when data represents sum of squares of a normally distributed variables.
-
-Use the following functions to visualize the chi sq distribution: 
-dchisq(), pchisq(), qchisq(), rchisq().
-
-```{r}
-
-norm1 <- rnorm(100, mean = 10, sd = 5)
-norm2 <- rnorm(100, mean = 20, sd = 7)
-chisqdist <- norm1^2 + norm2^2
-
-chisq_distribution <- rchisq(10000, 3)
-histogram_chisq_distribution <- hist(chisq_distribution)
-plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
-```
-
-
-
-
-3) F distribution
-
-This one looks a bit like the chi square distribution. But this distribution comes into picture when 
-one compares two chi sq distributions.
-
-Use the following functions to visualize the chi sq distribution: 
-df(), pf(), qf() and rf()
-
-```{r}
-f_distribution <- rf(10000, 5, 10)
-histogram_f_distribution <- hist(f_distribution)
-plot(histogram_f_distribution$mids,histogram_f_distribution$density, type="l", col="blue", lwd=3)
-```
-
-
-
-The End
-
-Reference - *Chapter 9, Navarro D.*
+---
+title: "Inferential Statistics: Probability & Distributions - 1"
+output: html_notebook
+---
+
+So far we have discussed about descriptive statistics - summarizing data and plotting it. But in order gain the power of making inferences, we will be strating with inferential statistics.
+
+#### Pre-requisite: Probability
+
+##### Difference between probability and statistics**
+Probability theory is a branch of mathematics that tells you how often different kinds of events will happen. For eg. What are the chances of a fair coin coming up heads 10 times in a row? or What are the chances that I’ll win the lottery?
+
+In each case the “truth of the world” is known. We know that the coin is fair, so there’s a 50% chance that any individual coin flip will come up heads. We know that the lottery follows specific rules. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations. *[Chapter 9, Navarro D.]*
+
+**A short note on Models**
+
+A model is a simplified representation of a system. For example, the map of a city represents a city in a simplified fashion. A map providing as much detail as the original city would not only be impossible to construct, it would also be pointless. Humans build models, such as maps and statistical models, to make their lives simpler. *[Chapter 3, Winter B.]*
+- - - -
+
+But even though we know the models like `P(heads) = 0.5`, we do not know the data (Whetehr heads will come 10 times or 3 times). However, for statistics, it is the opposite. We have the data and we want to infer the truth about the world. For eg., If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me? or If the lottery commissioner’s spouse wins the lottery, how likely is it that the lottery was rigged?
+
+We want to figure out which is the true model of the world. Is it *P(heads) = 0.5* or is it *P(heads) $\ne$ 0.5*?
+
+##### What is probability really?
+
+**The frequentist view**
+
+![Frequentist_graph](Fig4.png)
+
+According to the frequentist view, flip a fair coin over and over again, and as N grows large (approaches infinity, denoted N Ñ 8), the proportion of heads will converge to 50%.
+
+ *Advantages*
+ -  It is objective: the probability of an event is necessarily grounded in the world.
+ -  It is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer.
+
+But it all depends on infinite flips of coin. Do infinities really exist in the physical universe? What about the probability for a single non-repeatable event like the chances of rain on 21 September 2021?
+
+**The Bayesian view**
+
+Bayesian view is subjectivist view. The most common way of thinking about subjective probability is to define the probability of an event as the degree of belief that an intelligent and rational agent assigns to that truth of that event. But how to operationalize this 'degree of belief'? 
+
+One way is to use 'rational gambling'. So a “subjective probability” will be operationalized in terms of what bets you're willing to accept.
+
+ *Advantage*
+ - You don’t need to be limited to those events that are repeatable.
+ 
+ *Disadvantage*
+ - Can’t be purely objective – specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an **intelligent agent** out there that believes in things. 
+
+
+In short, frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to) while the Bayesian view is sometimes thought to be too broad (allows too many differences between observers).
+
+##### Definitions
+
+Refer to the example described in *Section 9.3.1, Navarro D.* for the following content.
+
+**Elementary event:** Every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events.
+
+**Sample space:** The set of all possible events (e.g., the wardrobe)
+
+**Probability:** Numbers between 0 and 1.
+
+For an event X, the probability of that event P(X) is a number that lies between 0 and 1. The bigger the value of P(X), the more likely the event is to occur.
+
+If P(X) = 0, it means the event X is impossible (i.e., I never wear those pants). On the other hand, if P(X)= 1 it means that event X is certain to occur (i.e., I always wear those pants).
+
+**Law of total probability:** The probabilities of the elementary events need to add up to 1
+
+#### Distributions
+
+Let's take a look at this and see what is a distribution. 
+
+```{r}
+pants <- data.frame(
+   type = c("Blue jeans","Grey jeans","Black jeans","Black suit","Blue tracksuit"),
+   label = c("X1", "X2", "X3", "X4", "X5"),
+   probability = c(0.5,0.3,0.1,0,0.1))
+
+pants
+```
+Probability distribution is simply the probabilities of these different events above. Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1.
+
+```{r}
+#Try plotting a bar graph of all the probabilities above
+```
+Let's think about what happens in case of non-elementary events. E.g. An event E where either “blue jeans” or “black jeans” or “grey jeans" has occurred. 
+Then what will be the probability of event E.
+
+P(E) = P(X1) + P(X2) + P(X3)
+
+If any of these elementary events occurs, then E is also said to have occurred. Similarly, there are other rules satisfying probabilities:
+
+![Probability_rules](Fig5.png)
+
+##### Binomial Distribution
+
+*Refer to section 9.4.1, Navarro D., for the detailed example*
+
+Some basic terminology - We’ll let `N` denote the number of dice rolls in our experiment; which is often referred to as the `size parameter` of our binomial distribution. We’ll also use `θ` to refer to the the probability that a single die comes up skull, a quantity that is usually called the `success probability` of the binomial. Finally, we’ll use `X` to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of X is due to chance, we refer to it as a `random variable`.
+
+`X ~ Binomial(θ, N)` denotes X is generated randomly from a binomial distribution with parameters θ and N.
+
+4 ~ Binomial(1/6, 20)
+
+5 ~ Binomial(1/2, 10)
+
+Let's generate a binomial distribution in R:
+
+```{r}
+dbinom( x = 1, size = 20, prob = 1/6)
+```
+The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.
+
+What if the dice is replaced by a coin in the above example? How will the probability change? 
+
+```{r}
+#Try finding the probability for N = 20 and N=100 trials for a fair coin flip.
+```
+There are different functions in R for different distributions as well as different ones for finding different quantity of interest.
+
+If we want to find the probability of obtaining an outcome smaller than or equal to quantile q, then we can directly use `pbinom`.
+
+```{r}
+#Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
+pbinom( q= 3, size = 20, prob = 1/6)
+
+#Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
+```
+In other words, value of 4 is actually the 76.9th percentile of this binomial distribution.
+
+Now let’s say we want to calculate the 75th percentile of the binomial distribution.
+
+```{r}
+qbinom( p = 0.566, size = 20, prob = 1/6 )
+
+#Practice - Find the 40th percentile
+```
+
+We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.
+
+```{r}
+z <- rbinom( n = 1000, size = 20, prob = 1/2 )
+#Let's also plot this and see how it looks
+hist(z, breaks=15, col = 'steelblue')
+```
+#Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.
+```{r}
+library(ggplot2)
+df<- data.frame(z)
+# Basic histogram with custom binwidth
+ggplot(df, aes(x=z)) + geom_histogram(aes(y = ..density..), binwidth=1, color="darkblue", fill="lightblue")
+```
+
+```{r}
+dbinom(x = 10, size = 20, prob = 1/2)
+```
+```{r}
+pbinom(q = 10, size = 20, prob = 1/2)
+```
+```{r}
+qbinom(p = 0.95, size = 20, prob = 1/2)
+```
+
+
+
+
+
+
+All these different functions *d, p, q, n* are also applicable to other distributions. E.g. *dnorm, pnorm, qnorm, rnorm* for Normal distribution. 
+
+
+##### Normal Distribution
+
+Most frequently encountered distribution.
+Eg: heights of all students in the class, marks obtained in exams, etc
+
+Basically, whenever you have accumulation of data at the center, fewer extreme values and a near symmetric spread, you should recall the normal distribution.
+
+
+- `dnorm()` - For probability density
+- `pnorm()` - For cumulative probability
+- `qnorm()` - For quantile of
+- `rnorm()` - For random number generation
+
+
+mean = 0; sd = 1 -> standard normal distribution
+```{r}
+normal_distribution <- rnorm(10000, mean = 10, sd = 5) 
+histogram_normal_distribution <- hist(normal_distribution)
+plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=1)
+
+
+```
+
+Note: Normal distribution is sometimes referred to as the bell curve or Gaussian distribution
+
+The notation for a normal distribution is: X ∼ Normal(μ,σ) 
+
+
+dnorm tells you the probability of getting a particular outcome
+```{r}
+dnorm(x=85, mean=80, sd=5)
+
+```
+Cumulative normal distribution
+```{r}
+pnorm(q = 80, mean = 80, sd = 5)
+```
+
+```{r}
+qnorm(0.25 ,mean = 0 , sd = 1)
+```
+
+*Checking for normality using the Shapiro-Wilk Test*
+```{r}
+norm <- rnorm(50, mean = 0, sd = 1) 
+shapiro.test(norm)
+
+binom <- rbinom(100, 20, 1/6)
+shapiro.test(binom)
+
+```
+
+
+
+
+
+##### Other useful distributions
+
+Some other distributions you may encounter include:
+*1) t distribution*
+
+Looks like the normal distribution but has heavier tails. 
+Used when data looks like a normal distribution but the mean and SD are unknown.
+
+Use the following functions to visualize the t distribution: 
+dt(), pt(), qt() and rt()
+
+```{r}
+t_distribution <- rt(10000, 15)
+histogram_t_distribution <- hist(t_distribution)
+plot(histogram_t_distribution$mids,histogram_t_distribution$density, type="l", col="blue", lwd=3)
+```
+
+
+
+
+*2) Chi square (χ2) distribution*
+
+All positive and heavily skewed to the left.  
+Used when data represents sum of squares of a normally distributed variables.
+
+Use the following functions to visualize the chi sq distribution: 
+dchisq(), pchisq(), qchisq(), rchisq().
+
+```{r}
+
+norm1 <- rnorm(100, mean = 10, sd = 5)
+norm2 <- rnorm(100, mean = 20, sd = 7)
+chisqdist <- norm1^2 + norm2^2
+
+chisq_distribution <- rchisq(10000, 3)
+histogram_chisq_distribution <- hist(chisq_distribution)
+plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
+```
+
+
+
+
+3) F distribution
+
+This one looks a bit like the chi square distribution. But this distribution comes into picture when 
+one compares two chi sq distributions.
+
+Use the following functions to visualize the chi sq distribution: 
+df(), pf(), qf() and rf()
+
+```{r}
+f_distribution <- rf(10000, 5, 10)
+histogram_f_distribution <- hist(f_distribution)
+plot(histogram_f_distribution$mids,histogram_f_distribution$density, type="l", col="blue", lwd=3)
+```
+
+
+
+The End
+
+Reference - *Chapter 9, Navarro D.*

From 8bc3e9cfc8c88dde0c5a27ce98ed647302a2c127 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Sat, 1 Oct 2022 09:28:45 +0530
Subject: [PATCH 20/55] power calculation + null vs alternative comparison

---
 Module 3/Notebooks/Distributions.Rmd | 61 ++++++++++++++++++++++++----
 1 file changed, 54 insertions(+), 7 deletions(-)

diff --git a/Module 3/Notebooks/Distributions.Rmd b/Module 3/Notebooks/Distributions.Rmd
index e2d99e02..f38d5a7e 100644
--- a/Module 3/Notebooks/Distributions.Rmd	
+++ b/Module 3/Notebooks/Distributions.Rmd	
@@ -139,7 +139,7 @@ qbinom( p = 0.566, size = 20, prob = 1/6 )
 We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.
 
 ```{r}
-z <- rbinom( n = 1000, size = 20, prob = 1/2 )
+z <- rbinom( n = 1000000, size = 100, prob = 1/2 )
 #Let's also plot this and see how it looks
 hist(z, breaks=15, col = 'steelblue')
 ```
@@ -152,14 +152,61 @@ ggplot(df, aes(x=z)) + geom_histogram(aes(y = ..density..), binwidth=1, color="d
 ```
 
 ```{r}
-dbinom(x = 10, size = 20, prob = 1/2)
+dbinom(x = 43, size = 100, prob = 1/2)
 ```
 ```{r}
-pbinom(q = 10, size = 20, prob = 1/2)
+pbinom(q = 17, size = 20, prob = 1/2)
 ```
 ```{r}
-qbinom(p = 0.95, size = 20, prob = 1/2)
+qbinom(p = 0.975, size = 10, prob = 1/2)
 ```
+```{r}
+qbinom(p = 0.025, size = 10, prob = 1/2)
+```
+
+```{r}
+binom.test( x=62, n=100, p=.5 )
+```
+
+
+```{r}
+z <- rbinom( n = 1000000, size = 100, prob = 0.9 )
+
+library(ggplot2)
+df<- data.frame(z)
+# Basic histogram with custom binwidth
+ggplot(df, aes(x=z)) + geom_histogram(aes(y = ..density..), binwidth=1, color="darkblue", fill="lightblue")
+```
+```{r}
+library(ggplot2)
+library("ggpubr")
+```
+
+
+```{r}
+num <- 10000
+z_null <- rbinom( n = num, size = 100, prob = 0.5 )
+z_alt <- rbinom( n = num, size = 100, prob = 0.6 )
+df1 <- data.frame(z_null,z_alt)
+# Basic histogram with custom binwidth
+p1 <- ggplot(df1, aes(x=z_null)) + geom_histogram(aes(y = ..density..), binwidth=1, color="lightblue", fill="lightblue")
+p2 <- ggplot(df1, aes(x=z_alt)) + geom_histogram(aes(y = ..density..), binwidth=1, color="lightgreen", fill="lightgreen")
+
+figure <- ggarrange(p1, p2,
+                    labels = c("Null", "Alt"))
+figure
+```
+
+
+```{r}
+library(pwr)
+pwr.p.test(h  = ES.h(p1 = 0.7, p2 = 0.5), sig.level = 0.05, power = 0.8)
+```
+```{r}
+effsize <- ES.h(p1 = 0.7, p2 = 0.5)
+effsize
+```
+
 
 
 
@@ -199,12 +246,12 @@ The notation for a normal distribution is: X ∼ Normal(μ,σ)
 
 dnorm tells you the probability of getting a particular outcome
 ```{r}
-dnorm(x=85, mean=80, sd=5)
+dnorm(x=10, mean=10, sd=5)
 
 ```
 Cumulative normal distribution
 ```{r}
-pnorm(q = 80, mean = 80, sd = 5)
+pnorm(q = 19.8, mean = 10, sd = 5)
 ```
 
 ```{r}
@@ -259,7 +306,7 @@ norm1 <- rnorm(100, mean = 10, sd = 5)
 norm2 <- rnorm(100, mean = 20, sd = 7)
 chisqdist <- norm1^2 + norm2^2
 
-chisq_distribution <- rchisq(10000, 3)
+chisq_distribution <- rchisq(10000, 2)
 histogram_chisq_distribution <- hist(chisq_distribution)
 plot(histogram_chisq_distribution$mids,histogram_chisq_distribution$density, type="l", col="blue", lwd=3)
 ```

From f93b8e20ab3bc31e9cfe817ac6de00d7eaae8fc2 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Mon, 17 Oct 2022 13:28:59 +0530
Subject: [PATCH 21/55] Added Normality Tests

---
 .gitignore             |   4 ++
 BSE658.Rproj           |  13 ++++
 Module 5/Ttest.Rmd     |  22 ++++++-
 Module 5/Ttest.nb.html | 146 +++++++++++++++++++++++++++++++++--------
 4 files changed, 156 insertions(+), 29 deletions(-)
 create mode 100644 .gitignore
 create mode 100644 BSE658.Rproj

diff --git a/.gitignore b/.gitignore
new file mode 100644
index 00000000..5b6a0652
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,4 @@
+.Rproj.user
+.Rhistory
+.RData
+.Ruserdata
diff --git a/BSE658.Rproj b/BSE658.Rproj
new file mode 100644
index 00000000..8e3c2ebc
--- /dev/null
+++ b/BSE658.Rproj
@@ -0,0 +1,13 @@
+Version: 1.0
+
+RestoreWorkspace: Default
+SaveWorkspace: Default
+AlwaysSaveHistory: Default
+
+EnableCodeIndexing: Yes
+UseSpacesForTab: Yes
+NumSpacesForTab: 2
+Encoding: UTF-8
+
+RnwWeave: Sweave
+LaTeX: pdfLaTeX
diff --git a/Module 5/Ttest.Rmd b/Module 5/Ttest.Rmd
index 7279e0c7..cb5d2209 100644
--- a/Module 5/Ttest.Rmd	
+++ b/Module 5/Ttest.Rmd	
@@ -81,6 +81,26 @@ head( chico )
 oneSampleTTest( chico$improvement, mu=0 )
 ```
 ```{r}
-wilcox.test( x = chico$improvement, mu=0)
+qqnorm( y = chico$improvement )        # draw the QQ plot
+```
+
+```{r}
+shapiro.test( x = chico$improvement )
+```
+
+```{r}
+improvement2 <- rchisq(100, 2) 
+hist(x = improvement2)
+```
+```{r}
+qqnorm( y = improvement2 ) 
+```
+```{r}
+shapiro.test( x = improvement2 )
+```
+
+
+```{r}
+wilcox.test( x = improvement2, mu=0)
 ```
 
diff --git a/Module 5/Ttest.nb.html b/Module 5/Ttest.nb.html
index ca4e9724..b1f72e0e 100644
--- a/Module 5/Ttest.nb.html	
+++ b/Module 5/Ttest.nb.html	
@@ -1774,14 +1774,34 @@
 
    setwd(dirname(rstudioapi::getActiveDocumentContext()$path)) # set curretn path as workpath
 install.packages('psych')
    
 
-
-
    Error in install.packages : Updating loaded packages
    
+
+
    Installing package into ‘C:/Users/Arjun/AppData/Local/R/win-library/4.2’
+(as ‘lib’ is unspecified)
+trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/psych_2.2.9.zip'
+Content type 'application/zip' length 3821660 bytes (3.6 MB)
+downloaded 3.6 MB
    
+
+
+
    package ‘psych’ successfully unpacked and MD5 sums checked
+
+The downloaded binary packages are in
+    C:\Users\Arjun\AppData\Local\Temp\RtmpgT0xwo\downloaded_packages
    
 
 
 
    install.packages('lsr')
    
 
-
-
    Error in install.packages : Updating loaded packages
    
+
+
    Installing package into ‘C:/Users/Arjun/AppData/Local/R/win-library/4.2’
+(as ‘lib’ is unspecified)
+trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/lsr_0.5.2.zip'
+Content type 'application/zip' length 209262 bytes (204 KB)
+downloaded 204 KB
    
+
+
+
    package ‘lsr’ successfully unpacked and MD5 sums checked
+
+The downloaded binary packages are in
+    C:\Users\Arjun\AppData\Local\Temp\RtmpgT0xwo\downloaded_packages
    
 
 
 
@@ -1838,22 +1858,40 @@
 
 
 
-
-
    head( harpo )
    
-
-
-
    
-
-
    
-
-
-
-
-
-
-
    independentSamplesTTest( 
+
+
    ```r
+head( harpo )
    
+
    
+<!-- rnb-source-end -->
+
+<!-- rnb-frame-begin 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 -->
+
+<div data-pagedtable="false">
+  <script data-pagedtable-source type="application/json">
+{"columns":[{"label":[""],"name":["_rn_"],"type":[""],"align":["left"]},{"label":["grade"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["tutor"],"name":[2],"type":["fctr"],"align":["left"]}],"data":[{"1":"65","2":"Anastasia","_rn_":"1"},{"1":"72","2":"Bernadette","_rn_":"2"},{"1":"66","2":"Bernadette","_rn_":"3"},{"1":"74","2":"Anastasia","_rn_":"4"},{"1":"73","2":"Anastasia","_rn_":"5"},{"1":"71","2":"Bernadette","_rn_":"6"}],"options":{"columns":{"min":{},"max":[10],"total":[2]},"rows":{"min":[10],"max":[10],"total":[6]},"pages":{}}}
+  </script>
+</div>
+
+<!-- rnb-frame-end -->
+
+<!-- rnb-chunk-end -->
+
+
+<!-- rnb-text-begin -->
+
+
+
+
+<!-- rnb-text-end -->
+
+
+<!-- rnb-chunk-begin -->
+
+
+<!-- rnb-source-begin eyJkYXRhIjoiYGBgclxuaW5kZXBlbmRlbnRTYW1wbGVzVFRlc3QoIFxuICAgICAgZm9ybXVsYSA9IGdyYWRlIH4gdHV0b3IsICAjIGZvcm11bGEgc3BlY2lmeWluZyBvdXRjb21lIGFuZCBncm91cCB2YXJpYWJsZXNcbiAgICAgIGRhdGEgPSBoYXJwbywgICAgICAgICAgICAgIyBkYXRhIGZyYW1lIHRoYXQgY29udGFpbnMgdGhlIHZhcmlhYmxlc1xuICAgICAgdmFyLmVxdWFsID0gVFJVRSAgICAgICAgICAjIGFzc3VtZSB0aGF0IHRoZSB0d28gZ3JvdXBzIGhhdmUgdGhlIHNhbWUgdmFyaWFuY2VcbiAgKVxuYGBgIn0= -->
+
+```r
+independentSamplesTTest( 
       formula = grade ~ tutor,  # formula specifying outcome and group variables
       data = harpo,             # data frame that contains the variables
       var.equal = TRUE          # assume that the two groups have the same variance
@@ -2012,25 +2050,77 @@
 
 
 
-
-
    wilcox.test( x = chico$improvement, mu=0)
    
+
+
    qqnorm( y = chico$improvement )        # draw the QQ plot
    
+
+
+
    
+
+
+
+
+
+
+
    shapiro.test( x = chico$improvement )
    
 
-
-
    Warning: cannot compute exact p-value with tiesWarning: cannot compute exact p-value with zeroes
    
+
+
    
+    Shapiro-Wilk normality test
+
+data:  chico$improvement
+W = 0.9664, p-value = 0.6778
    
 
-
+
+
+
+
+
+
    improvement2 <- rchisq(100, 2) 
+hist(x = improvement2)
    
+
+
+
    
+
+
+
+
+
    qqnorm( y = improvement2 ) 
    
+
+
+
    
+
+
+
+
+
    shapiro.test( x = improvement2 )
    
+
+
+
    
+    Shapiro-Wilk normality test
+
+data:  improvement2
+W = 0.80117, p-value = 2.689e-10
    
+
+
+
+
+
+
+
    wilcox.test( x = improvement2, mu=0)
    
+
+
 
    
     Wilcoxon signed rank test with continuity correction
 
-data:  chico$improvement
-V = 188, p-value = 0.0001965
+data:  improvement2
+V = 5050, p-value < 2.2e-16
 alternative hypothesis: true location is not equal to 0
    
 
 
 
 
 
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+
    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
    
 
 
 

From d91a361dde1a084c9ab262615a8a38e5321fc6c7 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Mon, 17 Oct 2022 13:37:05 +0530
Subject: [PATCH 22/55] Added Levene Test

---
 Module 5/Ttest.Rmd     |  9 +++++++++
 Module 5/Ttest.nb.html | 30 +++++++++++++++++++++++++++++-
 2 files changed, 38 insertions(+), 1 deletion(-)

diff --git a/Module 5/Ttest.Rmd b/Module 5/Ttest.Rmd
index cb5d2209..716072e0 100644
--- a/Module 5/Ttest.Rmd	
+++ b/Module 5/Ttest.Rmd	
@@ -43,12 +43,21 @@ independentSamplesTTest(
       var.equal = TRUE          # assume that the two groups have the same variance
   )
 ```
+```{r}
+library(car)
+```
+
+
 ```{r}
 independentSamplesTTest( 
       formula = grade ~ tutor,  # formula specifying outcome and group variables
       data = harpo              # data frame that contains the variables
   )
 ```
+```{r}
+leveneTest(grade ~ tutor,  # formula specifying outcome and group variables
+      data = harpo)              # data frame that contains the variables)
+```
 
 
 ```{r}
diff --git a/Module 5/Ttest.nb.html b/Module 5/Ttest.nb.html
index b1f72e0e..0f648d7a 100644
--- a/Module 5/Ttest.nb.html	
+++ b/Module 5/Ttest.nb.html	
@@ -1924,6 +1924,22 @@
 
 
 
+
+
    library(car)
    
+
+
+
    Loading required package: carData
+
+Attaching package: ‘car’
+
+The following object is masked from ‘package:psych’:
+
+    logit
    
+
+
+
+
+
 
 
    independentSamplesTTest( 
       formula = grade ~ tutor,  # formula specifying outcome and group variables
@@ -1956,6 +1972,18 @@
    estimated effect size (Cohen's d):  0.724 
    
 
 
+
+
+
    leveneTest(grade ~ tutor,  # formula specifying outcome and group variables
+      data = harpo)              # data frame that contains the variables)
    
+
+
+
    Levene's Test for Homogeneity of Variance (center = median)
+      Df F value Pr(>F)
+group  1  2.1287 0.1546
+      31               
    
+
+
 
 
 
@@ -2120,7 +2148,7 @@
 
 
 
-
    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
    
+
    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
    
 
 
 

From bdcbfebb952a2ebb91b1396e2af0e836cd392f97 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 18 Oct 2022 09:20:52 +0530
Subject: [PATCH 23/55] Added Chisq.test

---
 Module 5/Chi_sq.Rmd     |  21 +++++++
 Module 5/Chi_sq.nb.html | 126 ++++++++++++++++++++++++++++++++--------
 2 files changed, 124 insertions(+), 23 deletions(-)

diff --git a/Module 5/Chi_sq.Rmd b/Module 5/Chi_sq.Rmd
index 466071dd..79944c10 100644
--- a/Module 5/Chi_sq.Rmd	
+++ b/Module 5/Chi_sq.Rmd	
@@ -58,6 +58,16 @@ pchisq( q = 8.44, df = 3, lower.tail = FALSE )
 goodnessOfFitTest( cards$choice_1 )
 ```
 
+Doing the same Goodness of Fit test using chisq.test. 
+Note: observed frequencies are provided instead of choices.
+```{r}
+chisq.test( x = observed  )
+```
+
+
+
+
+
 ```{r}
 nullProbs <- c(clubs = .2, diamonds = .3, hearts = .3, spades = .2)
 nullProbs
@@ -67,6 +77,14 @@ nullProbs
 goodnessOfFitTest( x = cards$choice_1, p = nullProbs )
 ```
 
+```{r}
+chisq.test( x = observed, p = c(.2, .3, .3, .2) )
+```
+
+
+
+
+
 Chi Sq Association test With Chapek9
 
 ```{r}
@@ -96,4 +114,7 @@ associationTest( formula = ~choice+species, data = chapek9 )
 ```{r}
 cramersV( chapekFrequencies )
 ```
+```{r}
+chisq.test( chapekFrequencies )
+```
 
diff --git a/Module 5/Chi_sq.nb.html b/Module 5/Chi_sq.nb.html
index 7d12ea0c..3a525551 100644
--- a/Module 5/Chi_sq.nb.html	
+++ b/Module 5/Chi_sq.nb.html	
@@ -1877,28 +1877,66 @@ 
    R Notebook
    
 
 
 
-
-
    pchisq( q = 8.44, df = 3, lower.tail = FALSE )
    
-
-
-
    [1] 0.03774185
    
-
-
-
-
-
-
-
    1-pchisq( q = 8.44, df = 3 )
    
-
-
-
    [1] 0.03774185
    
-
-
-
-
-
-
-
    goodnessOfFitTest( cards$choice_1 )
    
+
+
    ```r
+pchisq( q = 8.44, df = 3, lower.tail = FALSE )
    
+
    
+<!-- rnb-source-end -->
+
+<!-- rnb-output-begin eyJkYXRhIjoiWzFdIDAuMDM3NzQxODVcbiJ9 -->
+
    
+
    [1] 0.03774185
    
+
    
+
+
+<!-- rnb-output-end -->
+
+<!-- rnb-chunk-end -->
+
+
+<!-- rnb-text-begin -->
+
+
+
+<!-- rnb-text-end -->
+
+
+<!-- rnb-chunk-begin -->
+
+
+<!-- rnb-source-begin eyJkYXRhIjoiYGBgclxuYGBgclxuMS1wY2hpc3EoIHEgPSA4LjQ0LCBkZiA9IDMgKVxuYGBgXG5gYGAifQ== -->
+
+```r
+```r
+1-pchisq( q = 8.44, df = 3 )
    
+
    
+<!-- rnb-source-end -->
+
+<!-- rnb-output-begin eyJkYXRhIjoiWzFdIDAuMDM3NzQxODVcbiJ9 -->
+
    
+
    [1] 0.03774185
    
+
    
+
+
+<!-- rnb-output-end -->
+
+<!-- rnb-chunk-end -->
+
+
+<!-- rnb-text-begin -->
+
+
+
+<!-- rnb-text-end -->
+
+
+<!-- rnb-chunk-begin -->
+
+
+<!-- rnb-source-begin eyJkYXRhIjoiYGBgclxuZ29vZG5lc3NPZkZpdFRlc3QoIGNhcmRzJGNob2ljZV8xIClcbmBgYCJ9 -->
+
+```r
+goodnessOfFitTest( cards$choice_1 )
    
 
 
 
    
@@ -1924,6 +1962,22 @@ 
    R Notebook
    
 
 
 
+
    Doing the same Goodness of Fit test using chisq.test. Note: observed
+frequencies are provided instead of choices.
    
+
+
+
+
    chisq.test( x = observed  )
    
+
+
+
    
+    Chi-squared test for given probabilities
+
+data:  observed
+X-squared = 8.44, df = 3, p-value = 0.03774
    
+
+
+
 
 
 
@@ -1965,6 +2019,20 @@ 
    R Notebook
    
 
 
 
+
+
+
+
    chisq.test( x = observed, p = c(.2, .3, .3, .2) )
    
+
+
+
    
+    Chi-squared test for given probabilities
+
+data:  observed
+X-squared = 4.7417, df = 3, p-value = 0.1917
    
+
+
+
 
    Chi Sq Association test With Chapek9
    
 
 
@@ -2069,10 +2137,22 @@ 
    R Notebook
    
 
    [1] 0.244058
    
 
 
+
+
+
    chisq.test( chapekFrequencies )
    
+
+
+
    
+    Pearson's Chi-squared test
+
+data:  chapekFrequencies
+X-squared = 10.722, df = 2, p-value = 0.004697
    
+
+
 
 
 
-
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+
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From afe28c77a5b0b07f0ba20942bf8206f7cc359e0c Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 18 Oct 2022 09:23:13 +0530
Subject: [PATCH 24/55] Added Salem.Rdata

---
 Module 5/salem.Rdata | Bin 0 -> 155 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 Module 5/salem.Rdata

diff --git a/Module 5/salem.Rdata b/Module 5/salem.Rdata
new file mode 100644
index 0000000000000000000000000000000000000000..7e591e801cc300bb8a8c49d1de432c781d3a0b44
GIT binary patch
literal 155
zcmV;M0A&9kiwFP!000000}FDAFye~fVqjokVqoHBWMEAnzXt0|2n8
Jc^p{*004o>H~9bn

literal 0
HcmV?d00001


From 2f9a17ebc4179c3b37235a7e9b1f9d760572a906 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 18 Oct 2022 09:28:28 +0530
Subject: [PATCH 25/55] Added Fischer Exact Test

---
 Module 5/Chi_sq.Rmd     | 27 +++++++++++++++++
 Module 5/Chi_sq.nb.html | 64 ++++++++++++++++++++++++++++++++++++++++-
 2 files changed, 90 insertions(+), 1 deletion(-)

diff --git a/Module 5/Chi_sq.Rmd b/Module 5/Chi_sq.Rmd
index 79944c10..612631bd 100644
--- a/Module 5/Chi_sq.Rmd	
+++ b/Module 5/Chi_sq.Rmd	
@@ -113,8 +113,35 @@ associationTest( formula = ~choice+species, data = chapek9 )
 
 ```{r}
 cramersV( chapekFrequencies )
+
 ```
+
+
+
+
 ```{r}
 chisq.test( chapekFrequencies )
 ```
 
+
+Fisher Exact test: when observed frequencies are less than 5 
+80% of the time in larger datasets
+
+```{r}
+load( file.path( "salem.Rdata" ))
+```
+
+
+```{r}
+salem.tabs <- table( trial )
+print( salem.tabs )
+```
+
+```{r}
+chisq.test( salem.tabs )
+```
+
+```{r}
+fisher.test( salem.tabs )
+```
+
diff --git a/Module 5/Chi_sq.nb.html b/Module 5/Chi_sq.nb.html
index 3a525551..6e1e2f2c 100644
--- a/Module 5/Chi_sq.nb.html	
+++ b/Module 5/Chi_sq.nb.html	
@@ -2137,6 +2137,8 @@ 
    R Notebook
    
 
    [1] 0.244058
    
 
 
+
+
 
 
 
    chisq.test( chapekFrequencies )
    
@@ -2150,9 +2152,69 @@ 
    R Notebook
    
 
 
 
+
    Fisher Exact test: when observed frequencies are less than 5 80% of
+the time in larger datasets
    
+
+
+
+
    load( file.path( "salem.Rdata" ))
    
+
+
+
+
+
+
+
    salem.tabs <- table( trial )
+print( salem.tabs )
    
+
+
+
           on.fire
+happy   FALSE TRUE
+  FALSE     3    3
+  TRUE     10    0
    
+
+
+
+
+
+
+
    chisq.test( salem.tabs )
    
+
+
+
    Warning: Chi-squared approximation may be incorrect
    
+
+
+
    
+    Pearson's Chi-squared test with Yates' continuity correction
+
+data:  salem.tabs
+X-squared = 3.3094, df = 1, p-value = 0.06888
    
+
+
+
+
+
+
+
    fisher.test( salem.tabs )
    
+
+
+
    
+    Fisher's Exact Test for Count Data
+
+data:  salem.tabs
+p-value = 0.03571
+alternative hypothesis: true odds ratio is not equal to 1
+95 percent confidence interval:
+ 0.000000 1.202913
+sample estimates:
+odds ratio 
+         0 
    
+
+
+
 
 
-
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From 491ae81bb8d351283ffef29141b9b20b2efed424 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 18 Oct 2022 09:34:51 +0530
Subject: [PATCH 26/55] Added agpp data

---
 Module 5/agpp.Rdata | Bin 0 -> 677 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 Module 5/agpp.Rdata

diff --git a/Module 5/agpp.Rdata b/Module 5/agpp.Rdata
new file mode 100644
index 0000000000000000000000000000000000000000..a856b2627c439eaccaaeb473896d7d55b977d4ed
GIT binary patch
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From 601f6dc74bc23d5e7f584191b0e3bc410cdf2f4f Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 18 Oct 2022 09:37:50 +0530
Subject: [PATCH 27/55] Added McNemar Test

---
 Module 5/Chi_sq.Rmd     | 26 +++++++++++++
 Module 5/Chi_sq.nb.html | 83 ++++++++++++++++++++++++++++++++++++++++-
 2 files changed, 108 insertions(+), 1 deletion(-)

diff --git a/Module 5/Chi_sq.Rmd b/Module 5/Chi_sq.Rmd
index 612631bd..62019904 100644
--- a/Module 5/Chi_sq.Rmd	
+++ b/Module 5/Chi_sq.Rmd	
@@ -144,4 +144,30 @@ chisq.test( salem.tabs )
 ```{r}
 fisher.test( salem.tabs )
 ```
+McNemar Test (remember paired samples t test? This is the equivalent test for nominal variables) 
+
+```{r}
+load( file.path( "agpp.Rdata" ))
+```
+
+```{r}
+str(agpp)
+```
+
+```{r}
+head(agpp)
+```
+
+```{r}
+summary(agpp)    
+```
+
+```{r}
+right.table <- xtabs( ~ response_before + response_after, data = agpp)
+print( right.table )
+```
+
+```{r}
+mcnemar.test( right.table )
+```
 
diff --git a/Module 5/Chi_sq.nb.html b/Module 5/Chi_sq.nb.html
index 6e1e2f2c..fb3b061c 100644
--- a/Module 5/Chi_sq.nb.html	
+++ b/Module 5/Chi_sq.nb.html	
@@ -2212,9 +2212,90 @@ 
    R Notebook
    
 
 
 
+
    McNemar Test (remember paired samples t test? This is the equivalent
+test for nominal variables)
    
+
+
+
+
    load( file.path( "agpp.Rdata" ))
    
+
+
+
+
+
+
+
    str(agpp)
    
+
+
+
    'data.frame':   100 obs. of  3 variables:
+ $ id             : Factor w/ 100 levels "subj.1","subj.10",..: 1 13 24 35 46 57 68 79 90 2 ...
+ $ response_before: Factor w/ 2 levels "no","yes": 1 2 2 2 1 1 1 1 1 1 ...
+ $ response_after : Factor w/ 2 levels "no","yes": 2 1 1 1 1 1 1 2 1 1 ...
    
+
+
+
+
+
+
+
    head(agpp)
    
+
+
+
    
+
+
    
+
+
+
+
+
+
+
    summary(agpp)    
    
+
+
+
            id     response_before response_after
+ subj.1  : 1   no :70          no :90        
+ subj.10 : 1   yes:30          yes:10        
+ subj.100: 1                                 
+ subj.11 : 1                                 
+ subj.12 : 1                                 
+ subj.13 : 1                                 
+ (Other) :94                                 
    
+
+
+
+
+
+
+
    right.table <- xtabs( ~ response_before + response_after, data = agpp)
+print( right.table )
    
+
+
+
                   response_after
+response_before no yes
+            no  65   5
+            yes 25   5
    
+
+
+
+
+
+
+
    mcnemar.test( right.table )
    
+
+
+
    
+    McNemar's Chi-squared test with continuity correction
+
+data:  right.table
+McNemar's chi-squared = 12.033, df = 1, p-value = 0.0005226
    
+
+
+
 
 
-
    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
    
+
    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
    
 
 
 

From f43a5353c75b142f62a61c3f1e94cf9fa070aaf1 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 25 Oct 2022 09:10:04 +0530
Subject: [PATCH 28/55] Added Post hoc testing

---
 Module 5/Chi_sq.Rmd     |  7 ++--
 Module 5/Chi_sq.nb.html | 82 ++++++++---------------------------------
 2 files changed, 19 insertions(+), 70 deletions(-)

diff --git a/Module 5/Chi_sq.Rmd b/Module 5/Chi_sq.Rmd
index 62019904..0df200ef 100644
--- a/Module 5/Chi_sq.Rmd	
+++ b/Module 5/Chi_sq.Rmd	
@@ -82,9 +82,6 @@ chisq.test( x = observed, p = c(.2, .3, .3, .2) )
 ```
 
 
-
-
-
 Chi Sq Association test With Chapek9
 
 ```{r}
@@ -122,6 +119,10 @@ cramersV( chapekFrequencies )
 ```{r}
 chisq.test( chapekFrequencies )
 ```
+```{r}
+library(chisq.posthoc.test)
+chisq.posthoc.test(chapekFrequencies)
+```
 
 
 Fisher Exact test: when observed frequencies are less than 5 
diff --git a/Module 5/Chi_sq.nb.html b/Module 5/Chi_sq.nb.html
index fb3b061c..f0c07ba5 100644
--- a/Module 5/Chi_sq.nb.html	
+++ b/Module 5/Chi_sq.nb.html	
@@ -2052,7 +2052,7 @@ 
    R Notebook
    
 
 
    head(chapek9)
    
 
-
+
 
    
 
+
    
+
+
 
 
    Fisher Exact test: when observed frequencies are less than 5 80% of
 the time in larger datasets
    
@@ -2167,12 +2180,6 @@ 
    R Notebook
    
 
    salem.tabs <- table( trial )
 print( salem.tabs )
    
 
-
-
           on.fire
-happy   FALSE TRUE
-  FALSE     3    3
-  TRUE     10    0
    
-
 
 
 
@@ -2180,16 +2187,6 @@ 
    R Notebook
    
 
 
    chisq.test( salem.tabs )
    
 
-
-
    Warning: Chi-squared approximation may be incorrect
    
-
-
-
    
-    Pearson's Chi-squared test with Yates' continuity correction
-
-data:  salem.tabs
-X-squared = 3.3094, df = 1, p-value = 0.06888
    
-
 
 
 
@@ -2197,19 +2194,6 @@ 
    R Notebook
    
 
 
    fisher.test( salem.tabs )
    
 
-
-
    
-    Fisher's Exact Test for Count Data
-
-data:  salem.tabs
-p-value = 0.03571
-alternative hypothesis: true odds ratio is not equal to 1
-95 percent confidence interval:
- 0.000000 1.202913
-sample estimates:
-odds ratio 
-         0 
    
-
 
 
 
    McNemar Test (remember paired samples t test? This is the equivalent
@@ -2226,12 +2210,6 @@ 
    R Notebook
    
 
 
    str(agpp)
    
 
-
-
    'data.frame':   100 obs. of  3 variables:
- $ id             : Factor w/ 100 levels "subj.1","subj.10",..: 1 13 24 35 46 57 68 79 90 2 ...
- $ response_before: Factor w/ 2 levels "no","yes": 1 2 2 2 1 1 1 1 1 1 ...
- $ response_after : Factor w/ 2 levels "no","yes": 2 1 1 1 1 1 1 2 1 1 ...
    
-
 
 
 
@@ -2239,13 +2217,6 @@ 
    R Notebook
    
 
 
    head(agpp)
    
 
-
-
    
-
-
    
-
 
 
 
@@ -2253,16 +2224,6 @@ 
    R Notebook
    
 
 
    summary(agpp)    
    
 
-
-
            id     response_before response_after
- subj.1  : 1   no :70          no :90        
- subj.10 : 1   yes:30          yes:10        
- subj.100: 1                                 
- subj.11 : 1                                 
- subj.12 : 1                                 
- subj.13 : 1                                 
- (Other) :94                                 
    
-
 
 
 
@@ -2271,12 +2232,6 @@ 
    R Notebook
    
 
    right.table <- xtabs( ~ response_before + response_after, data = agpp)
 print( right.table )
    
 
-
-
                   response_after
-response_before no yes
-            no  65   5
-            yes 25   5
    
-
 
 
 
@@ -2284,18 +2239,11 @@ 
    R Notebook
    
 
 
    mcnemar.test( right.table )
    
 
-
-
    
-    McNemar's Chi-squared test with continuity correction
-
-data:  right.table
-McNemar's chi-squared = 12.033, df = 1, p-value = 0.0005226
    
-
 
 
 
 
-
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+
    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
    
 
 
 

From 307d090449d617faab7c8d1924ac9980b9a14121 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 1 Nov 2022 09:13:09 +0530
Subject: [PATCH 29/55] Module 6 ANOVA

---
 Module 6                     |   1 -
 Module 6/ANOVA.Rmd           | 299 +++++++++++++++++++++++++++++++++++
 Module 6/clinicaltrial.Rdata | Bin 0 -> 326 bytes
 3 files changed, 299 insertions(+), 1 deletion(-)
 delete mode 100644 Module 6
 create mode 100644 Module 6/ANOVA.Rmd
 create mode 100644 Module 6/clinicaltrial.Rdata

diff --git a/Module 6 b/Module 6
deleted file mode 100644
index 8b137891..00000000
--- a/Module 6	
+++ /dev/null
@@ -1 +0,0 @@
-
diff --git a/Module 6/ANOVA.Rmd b/Module 6/ANOVA.Rmd
new file mode 100644
index 00000000..bfd040c2
--- /dev/null
+++ b/Module 6/ANOVA.Rmd	
@@ -0,0 +1,299 @@
+---
+title: "ANOVA"
+output: html_document
+date: "2022-10-31"
+---
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+```
+## R Markdown
+Suppose you are testing a new antidepressant drug called Joyzepam. To test of the drug’s effectiveness, the study involves three separate drugs to be administered.
+One is a placebo, and the other is an existing antidepressant / anti-anxiety drug called Anxifree.
+18 participants with moderate to severe depression are recruited for your initial testing
+the drugs are sometimes administered in conjunction with psychological therapy, your study includes 9 people undergoing cognitive behavioural therapy (CBT) and 9 who are not.
+Participants are randomly assigned (doubly blinded, of course) a treatment, such that there are 3 CBT people and 3 no-therapy people assigned to each of the 3 drugs. A psychologist assesses the mood of each person after a 3 month run with each drug: and the overall improvement in each person’s mood is assessed on a scale ranging from  −5 to +5.
+let’s now look at what we’ve got in the data file:
+```{r ANOVA}
+projecthome = "D:/Stats class"; #enter folder name where the data is downloaded
+load(file.path(projecthome, "clinicaltrial.Rdata")) # load data
+str(clin.trial)
+print( clin.trial )
+```
+Lets see how many people we have in each group:
+
+```{r Opening Data}
+xtabs( ~drug, clin.trial )
+```
+calculate means and standard deviations
+
+```{r Opening Data}
+aggregate( mood.gain ~ drug, clin.trial, mean )
+```
+
+produce a pretty picture plots
+You might want to install it by using (install.packages("gplots"))
+observe the graph carefully 
+```{r Opening Data}
+library(gplots)
+plotmeans(  formula = mood.gain ~ drug,  # plot mood.gain by drug
+            data = clin.trial,           # the data frame
+            xlab = "Drug Administered",  # x-axis label
+            ylab = "Mood Gain",          # y-axis label
+            n.label = FALSE              # don't display sample size
+)
+```
+The question that we want to answer is: are these difference “real”, or are they just due to chance? To answer the question posed by our clinical trial data, we’re going to run a one-way ANOVA. We’re interested in comparing the average mood change for the three different drugs.
+
+let μP denote the population mean for the mood change induced by the placebo, and let μA and μJ denote the corresponding means for our two drugs, Anxifree and Joyzepam.
+
+null hypothesis = H0: it is true that μP=μA=μJ (all three equal)
+alternative hypothesis = H1:it is *not* true that μP=μA=μJ
+
+We’ll start out by playing around with variances, and it will turn out that this gives us a useful tool for investigating means.
+
+There are Two formulas for the variance, Can you recall the formula ?
+sum of squares = same as variance but not divided by N i.e. instead of averaging the squared deviations, which is what we do when calculating the variance, we just add them up
+
+This can be done within grp as well as between grps (students can try and plot within grp vs between grp sum of squares.)
+
+Qualitative idea behind ANOVA is to compare the two sums of squares values
+
+SSb and SSw to each other: if the between-group variation is SSb is large relative to the within-group variation, SSw then we have reason to suspect that the population means for the different groups aren’t identical to each other
+
+What we do to calculate our test statistic – which is called an F ratio
+
+convert our SS values into an  F-ratio
+“the variation due to the differences in the sample means for the different groups” (SSb) plus “all the rest of the variation” (SSw)
+
+calculate the degrees of freedom associated with the SSb and SSw values. The degrees of freedom corresponds to the number of unique “data points” that contribute to a particular calculation, minus the number of “constraints” that they need to satisfy.
+
+Within-groups variability, what we’re calculating is the variation of the individual observations (Ndata points) around the group means for the between groups variability, we’re interested in the variation of the group means (G data points) around the grand mean (1 constraint)
+
+The intuition behind the  F statistic is straightforward: bigger values of F means that the between-groups variation is large, relative to the within-groups variation , larger the value of F, the more evidence we have against the null hypothesis.
+
+lets see an example, recall the the means of the three groups that were administered different drugs
+
+```{r Opening Data}
+outcome <- clin.trial$mood.gain
+group <- clin.trial$drug
+gp.means <- tapply(outcome,group,mean)
+gp.means <- gp.means[group]
+dev.from.gp.means <- outcome - gp.means
+squared.devs <- dev.from.gp.means ^2
+#putting variables in the dataframe 
+Y <- data.frame( group, outcome, gp.means,
+                 dev.from.gp.means, squared.devs )
+print(Y, digits = 2)
+```
+Calculations of the within-group sum of squares 
+
+```{r Opening Data}
+SSw <- sum( squared.devs )
+print( SSw )
+```
+Now that we’ve calculated the within groups variation, SSw, it’s time to turn our attention to the between-group sum of squares, SSb.
+We calculate the differences between the group means and the grand mean.
+
+However, for the between group calculations we need to multiply each of these squared deviations by Nk, the number of observations in the group (guess why?)
+
+For between group calculations
+```{r Opening Data}
+gp.means <- tapply(outcome,group,mean)
+grand.mean <- mean(outcome)
+dev.from.grand.mean <- gp.means - grand.mean
+squared.devs <- dev.from.grand.mean ^2
+gp.sizes <- tapply(outcome,group,length)
+wt.squared.devs <- gp.sizes * squared.devs
+```
+dump all our variables into a data frame
+
+```{r Opening Data}
+Y <- data.frame( gp.means, grand.mean, dev.from.grand.mean, 
+                 squared.devs, gp.sizes, wt.squared.devs )
+print(Y, digits = 2)
+```
+rounded all my numbers to 2 decimal places ;)
+
+```{r Opening Data}
+SSb <- sum( wt.squared.devs )
+print( SSb )
+```
+We’ve calculated our sums of squares values, SSb and SSw
+
+The next step is to calculate the degrees of freedom. Since we have G=3 groups and  N=18 observations in total, our degrees of freedom can be calculated by simple subtraction:
+  dfb = G−1 = 2
+  dfw = N−G = 15
+
+since we’ve now calculated the values for the sums of squares and the degrees of freedom, for both the within-groups variability and the between-groups variability, we can obtain the mean square values by dividing one by the other:
+  MSb = SSb/dfb = 3.45/2 = 1.73
+  MSw = SSw/dfw = 1.39/15= 0.09
+
+We calculate F-values by dividing the between-groups MS value by the and within-groups MS value.
+  F = MSb/MSw = 1.73/0.09 = 18.6
+
+It is easier to directly calculate the p-value. 
+
+reject the null hypothesis for very large F-values
+
+
+```{r Opening Data}
+pf( 18.6, df1 = 2, df2 = 15, lower.tail = FALSE)
+```
+You get a p-value, we’re pretty much guaranteed to reject the null hypothesis.
+
+A pretty standard way of reporting this result would be to write something like this:
+One-way ANOVA showed a significant effect of drug on mood gain (F (2,15) = 18.6, p<.001).
+
+Using the aov() function to specify your ANOVA: type '?aov' and have a look at the help documentation
+
+```{r Opening Data}
+my.anova <-  aov( formula = mood.gain ~ drug, data = clin.trial ) 
+print( my.anova )
+```
+R doesn’t use the names “between-group” and “within-group”
+instead : between groups variance corresponds to the effect that the drug has on the outcome variable; and the within groups variance is corresponds to the “leftover” variability, so it calls that the residuals.
+
+but wait  Where’s the  F-value? The  p-value? These are the most important numbers in our hypothesis test
+
+ask for a summary() 
+
+
+
+```{r Opening Data}
+summary( my.anova )
+```
+
+Effect size
+most commonly used measures to calculate effect size are  η2 (eta squared) and partial  η2
+  η2 = SSb/SStot
+
+interpretation of  η2 is equally straightforward: it refers to the proportion of the variability in the outcome variable (mood.gain) that can be explained in terms of the predictor (drug). A value of  
+  η2 = 0, means that there is no relationship at all between the two, whereas  
+  η2 = 1, means that the relationship is perfect.
+
+you can derive pearson correlation from η2 by taking an underoot of it i.e. η
+
+core packages in R don’t include any functions for calculating  η2
+mannually we can do this
+```{r Opening Data}
+SStot <- SSb + SSw          # total sums of squares
+eta.squared <- SSb / SStot  # eta-squared value
+print( eta.squared )
+```
+or use function directly
+load libraray(lsr) 
+```{r Opening Data}
+etaSquared( x = my.anova ) 
+```
+Multiple comparisons and post hoc tests
+
+Running “pairwise”  t-tests ask your professor why to do a t-test
+There’s a couple of ways that we could do this. One method would be to construct new variables corresponding the groups you want to compare (e.g., anxifree, placebo and joyzepam), and then run a  t-test on these new variables:
+
+```{r Opening Data}
+anxifree <- with(clin.trial, mood.gain[drug == "anxifree"])  # mood change due to anxifree
+placebo <- with(clin.trial, mood.gain[drug == "placebo"])    # mood change due to placebo 
+
+t.test( anxifree, placebo, var.equal = TRUE )   # Student t-test
+```
+or, you could use 
+
+```{r Opening Data}
+t.test( formula = mood.gain ~ drug, 
+        data = clin.trial, 
+        subset = drug %in% c("placebo","anxifree"), 
+        var.equal = TRUE )
+```
+function called pairwise.t.test() that automatically runs all of the  t-tests for you.
+
+```{r Opening Data}
+pairwise.t.test( x = clin.trial$mood.gain,   # outcome variable
+                 g = clin.trial$drug,        # grouping variable
+                 p.adjust.method = "none" )   # which correction to use?
+```
+Corrections for multiple testing
+#each individual  
+
+t-test is designed to have a 5% Type I error rate (i.e.,α = 0.05), imagine if you have more than 10 groups !
+correction for multiple comparisons, though it is sometimes referred to as “simultaneous inference”
+
+Bonferroni corrections
+
+post hoc analysis consists of  m separate tests, and I want to ensure that the total probability of making any Type I errors at all is at most α.
+the Bonferroni correction just says “multiply all your raw  p-values by m”
+```{r Opening Data}
+pairwise.t.test( x = clin.trial$mood.gain,   # outcome variable
+                 g = clin.trial$drug,        # grouping variable
+                 p.adjust.method = "bonferroni" )   # set p.adjust.method = "bonferroni"
+```
+Holm corrections
+
+Holm correction is to pretend that you’re doing the tests sequentially; starting with the smallest (raw) p-value and moving onto the largest one.
+First, you sort all of your p-values in order, from smallest to largest. For the smallest p-value all you do is multiply it by m when you move to the second smallest p value, you first multiply it by  m−1.If this produces a number that is bigger than the adjusted p-value that you got last time, then you keep it. But if it’s smaller than the last one, then you copy the last p-value. 
+
+To run the Holm correction in R, you could specify p.adjust.method = "Holm" in the above equation  or 
+```{r Opening Data}
+posthocPairwiseT( my.anova ) #takes Holm's correction by default.
+```
+Assumptions of one-way ANOVA
+
+There are three key assumptions that you need to be aware of: normality, homogeneity of variance and independence
+
+Checking the homogeneity of variance assumption
+Levene test involve checking the assumptions of an ANOVA
+
+```{r Opening Data}
+leveneTest(y = mood.gain ~ drug, data = clin.trial)   # y is a formula in this case
+leveneTest(y = clin.trial$mood.gain, group = clin.trial$drug)   # y is the outcome
+```
+Is your levene test significant ? What do you observe ?
+
+Removing the homogeneity of variance assumption
+```{r Opening Data}
+oneway.test(mood.gain ~ drug, data = clin.trial) #  Welch one-way ANOVA
+```
+Originally our ANOVA gave us the result F(2,15) = 18.6, oneway.test(mood.gain ~ drug, data = clin.trial, var.equal = TRUE)
+whereas the Welch one-way test gave us F(2,9.49)=26.32.
+In other words, the Welch test has reduced the within-groups degrees of freedom from 15 to 9.49, and the F-value has increased from 18.6 to 26.32.
+
+Checking the normality assumption
+we need to know how to pull out the residuals (i.e., the ϵik values) so that we can draw our QQ plot and run our Shapiro-Wilk test. 
+
+First, let’s extract the residuals (Can you recall what are residuals ?)
+
+```{r Opening Data}
+my.anova.residuals <- residuals( object = my.anova )   # extract the residuals
+hist( x = my.anova.residuals )           # plot a histogram (similar to Figure @ref{fig:normalityanova}a)
+qqnorm( y = my.anova.residuals )         # draw a QQ plot (similar to Figure @ref{fig:normalityanova}b)
+shapiro.test( x = my.anova.residuals )   # run Shapiro-Wilk test
+```
+
+Removing the normality assumption (what we can do to address violations of normality)
+
+switch to a non-parametric test (i.e., one that doesn’t rely on any particular assumption about the kind of distribution involved)
+Wilcoxon test provides the non-parametric alternative for two groups
+
+But, What if I got three or more groups? 
+you can use the Kruskal-Wallis rank sum test
+
+```{r Opening Data}
+kruskal.test(mood.gain ~ drug, data = clin.trial)
+```
+Relationship between ANOVA and the Student  t-test
+
+```{r Opening Data}
+summary( aov( mood.gain ~ therapy, data = clin.trial ))
+```
+looks like there’s no significant effect here at all 
+
+```{r Opening Data}
+t.test( mood.gain ~ therapy, data = clin.trial, var.equal = TRUE )
+```
+p-values are identical
+what about the test statistic? Having run a  t-test instead of an ANOVA, we get a somewhat different answer, namely t(16) = −1.3068.
+
+there is a fairly straightforward relationship here. If we square the t-statistic we get the  F-statistic from before.
+
+```{r Opening Data}
+1.3068 ^ 2
\ No newline at end of file
diff --git a/Module 6/clinicaltrial.Rdata b/Module 6/clinicaltrial.Rdata
new file mode 100644
index 0000000000000000000000000000000000000000..82e9762dc54eff9f67c9859da60f1b137c6bbcf6
GIT binary patch
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z@^3=Q7=_Xn)(JXHTVsJR*F
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From ddd83f1ba9cb1444a601a77ab0a6fcf25c340595 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 1 Nov 2022 09:13:23 +0530
Subject: [PATCH 30/55] minor stuff chi sq html

---
 Module 5/Chi_sq.nb.html | 58 +++++++++++++++++++++++++++++++++++++++++
 1 file changed, 58 insertions(+)

diff --git a/Module 5/Chi_sq.nb.html b/Module 5/Chi_sq.nb.html
index f0c07ba5..45367b3a 100644
--- a/Module 5/Chi_sq.nb.html	
+++ b/Module 5/Chi_sq.nb.html	
@@ -2180,6 +2180,12 @@ 
    R Notebook
    
 
    salem.tabs <- table( trial )
 print( salem.tabs )
    
 
+
+
           on.fire
+happy   FALSE TRUE
+  FALSE     3    3
+  TRUE     10    0
    
+
 
 
 
@@ -2187,6 +2193,16 @@ 
    R Notebook
    
 
 
    chisq.test( salem.tabs )
    
 
+
+
    Warning: Chi-squared approximation may be incorrect
    
+
+
+
    
+    Pearson's Chi-squared test with Yates' continuity correction
+
+data:  salem.tabs
+X-squared = 3.3094, df = 1, p-value = 0.06888
    
+
 
 
 
@@ -2194,6 +2210,19 @@ 
    R Notebook
    
 
 
    fisher.test( salem.tabs )
    
 
+
+
    
+    Fisher's Exact Test for Count Data
+
+data:  salem.tabs
+p-value = 0.03571
+alternative hypothesis: true odds ratio is not equal to 1
+95 percent confidence interval:
+ 0.000000 1.202913
+sample estimates:
+odds ratio 
+         0 
    
+
 
 
 
    McNemar Test (remember paired samples t test? This is the equivalent
@@ -2210,6 +2239,12 @@ 
    R Notebook
    
 
 
    str(agpp)
    
 
+
+
    'data.frame':   100 obs. of  3 variables:
+ $ id             : Factor w/ 100 levels "subj.1","subj.10",..: 1 13 24 35 46 57 68 79 90 2 ...
+ $ response_before: Factor w/ 2 levels "no","yes": 1 2 2 2 1 1 1 1 1 1 ...
+ $ response_after : Factor w/ 2 levels "no","yes": 2 1 1 1 1 1 1 2 1 1 ...
    
+
 
 
 
@@ -2217,6 +2252,13 @@ 
    R Notebook
    
 
 
    head(agpp)
    
 
+
+
    
+
+
    
+
 
 
 
@@ -2224,6 +2266,16 @@ 
    R Notebook
    
 
 
    summary(agpp)    
    
 
+
+
            id     response_before response_after
+ subj.1  : 1   no :70          no :90        
+ subj.10 : 1   yes:30          yes:10        
+ subj.100: 1                                 
+ subj.11 : 1                                 
+ subj.12 : 1                                 
+ subj.13 : 1                                 
+ (Other) :94                                 
    
+
 
 
 
@@ -2232,6 +2284,12 @@ 
    R Notebook
    
 
    right.table <- xtabs( ~ response_before + response_after, data = agpp)
 print( right.table )
    
 
+
+
                   response_after
+response_before no yes
+            no  65   5
+            yes 25   5
    
+
 
 
 

From 78e96166fe95086bb18b5430f4a2f2fd12eae72c Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 1 Nov 2022 09:35:32 +0530
Subject: [PATCH 31/55] Minor corrections.

---
 Module 6/ANOVA.Rmd | 6 ++----
 1 file changed, 2 insertions(+), 4 deletions(-)

diff --git a/Module 6/ANOVA.Rmd b/Module 6/ANOVA.Rmd
index bfd040c2..89ef849f 100644
--- a/Module 6/ANOVA.Rmd	
+++ b/Module 6/ANOVA.Rmd	
@@ -15,8 +15,7 @@ the drugs are sometimes administered in conjunction with psychological therapy,
 Participants are randomly assigned (doubly blinded, of course) a treatment, such that there are 3 CBT people and 3 no-therapy people assigned to each of the 3 drugs. A psychologist assesses the mood of each person after a 3 month run with each drug: and the overall improvement in each person’s mood is assessed on a scale ranging from  −5 to +5.
 let’s now look at what we’ve got in the data file:
 ```{r ANOVA}
-projecthome = "D:/Stats class"; #enter folder name where the data is downloaded
-load(file.path(projecthome, "clinicaltrial.Rdata")) # load data
+load( file.path("clinicaltrial.Rdata" ))
 str(clin.trial)
 print( clin.trial )
 ```
@@ -182,7 +181,6 @@ eta.squared <- SSb / SStot  # eta-squared value
 print( eta.squared )
 ```
 or use function directly
-load libraray(lsr) 
 ```{r Opening Data}
 etaSquared( x = my.anova ) 
 ```
@@ -213,9 +211,9 @@ pairwise.t.test( x = clin.trial$mood.gain,   # outcome variable
                  p.adjust.method = "none" )   # which correction to use?
 ```
 Corrections for multiple testing
-#each individual  
 
 t-test is designed to have a 5% Type I error rate (i.e.,α = 0.05), imagine if you have more than 10 groups !
+
 correction for multiple comparisons, though it is sometimes referred to as “simultaneous inference”
 
 Bonferroni corrections

From cdd32dafacb76854cc288658f9ba00b8eab0e331 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Mon, 7 Nov 2022 10:19:45 +0530
Subject: [PATCH 32/55] Update ANOVA.Rmd

---
 Module 6/ANOVA.Rmd | 49 ++++++++++++++++++++++++++++++++++------------
 1 file changed, 37 insertions(+), 12 deletions(-)

diff --git a/Module 6/ANOVA.Rmd b/Module 6/ANOVA.Rmd
index 89ef849f..1640599e 100644
--- a/Module 6/ANOVA.Rmd	
+++ b/Module 6/ANOVA.Rmd	
@@ -7,6 +7,13 @@ date: "2022-10-31"
 ```{r setup, include=FALSE}
 knitr::opts_chunk$set(echo = TRUE)
 ```
+
+```{r}
+library(lsr)
+```
+
+
+
 ## R Markdown
 Suppose you are testing a new antidepressant drug called Joyzepam. To test of the drug’s effectiveness, the study involves three separate drugs to be administered.
 One is a placebo, and the other is an existing antidepressant / anti-anxiety drug called Anxifree.
@@ -20,6 +27,9 @@ str(clin.trial)
 print( clin.trial )
 ```
 Lets see how many people we have in each group:
+```{r}
+
+```
 
 ```{r Opening Data}
 xtabs( ~drug, clin.trial )
@@ -214,7 +224,7 @@ Corrections for multiple testing
 
 t-test is designed to have a 5% Type I error rate (i.e.,α = 0.05), imagine if you have more than 10 groups !
 
-correction for multiple comparisons, though it is sometimes referred to as “simultaneous inference”
+c   orrection for multiple comparisons, though it is sometimes referred to as “simultaneous inference”
 
 Bonferroni corrections
 
@@ -278,20 +288,35 @@ you can use the Kruskal-Wallis rank sum test
 ```{r Opening Data}
 kruskal.test(mood.gain ~ drug, data = clin.trial)
 ```
-Relationship between ANOVA and the Student  t-test
 
-```{r Opening Data}
-summary( aov( mood.gain ~ therapy, data = clin.trial ))
+Two-factor ANOVA:
+What if you want to look at the effect of Drug and Therapy?
+
+```{r One factor ANOVA}
+model.1 <- aov( mood.gain ~ drug, clin.trial )  
+summary( model.1 )  
 ```
-looks like there’s no significant effect here at all 
 
-```{r Opening Data}
-t.test( mood.gain ~ therapy, data = clin.trial, var.equal = TRUE )
+```{r Two factor ANOVA}
+model.2 <- aov( mood.gain ~ drug + therapy, clin.trial )  
+summary(model.2)
+```
+Computing Effect Size
+```{r}
+etaSquared( model.2 )
+```
+
+
+
+
+
+Now, how do you determine whether the drug and therapy interact? 
+```{r Two factor ANOVA with interaction effects}
+model.3 <- aov( mood.gain ~ drug + therapy + drug:therapy, clin.trial )  
+summary(model.3)
+```
+```{r}
+etaSquared( model.3 )
 ```
-p-values are identical
-what about the test statistic? Having run a  t-test instead of an ANOVA, we get a somewhat different answer, namely t(16) = −1.3068.
 
-there is a fairly straightforward relationship here. If we square the t-statistic we get the  F-statistic from before.
 
-```{r Opening Data}
-1.3068 ^ 2
\ No newline at end of file

From 11b71ab321a4509aa9a48016818227e266ac1232 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Mon, 7 Nov 2022 10:22:06 +0530
Subject: [PATCH 33/55] Regression Initial Commit

---
 Module 6/Regression.Rmd   | 171 ++++++++++++++++++++++++++++++++++++++
 Module 6/parenthood.Rdata | Bin 0 -> 1150 bytes
 2 files changed, 171 insertions(+)
 create mode 100644 Module 6/Regression.Rmd
 create mode 100644 Module 6/parenthood.Rdata

diff --git a/Module 6/Regression.Rmd b/Module 6/Regression.Rmd
new file mode 100644
index 00000000..95ec5879
--- /dev/null
+++ b/Module 6/Regression.Rmd	
@@ -0,0 +1,171 @@
+---
+title: "Regression"
+output: html_document
+date: "2022-11-06"
+---
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+setwd(dirname(rstudioapi::getActiveDocumentContext()$path)) # set curretn path as workpath
+```
+
+
+
+```{r}
+load( file.path("parenthood.Rdata" ))
+colnames(parenthood) <- c('dan.sleep','baby.sleep','dan.grump','day')
+```
+
+linear regression:
+dan.grump ~ dan.sleep
+```{r}
+regression.1 <- lm( formula = dan.grump ~ dan.sleep,  
+                    data = parenthood )
+print( regression.1 )
+```
+##Multiple linear regression
+dan.grump ~ dan.sleep + baby.sleep
+
+```{r}
+regression.2 <- lm( formula = dan.grump ~ dan.sleep + baby.sleep,  
+                     data = parenthood )
+print( regression.2 )
+```
+
+
+The  R-squared value Calculation 
+
+
+#Calculating y-predicted
+```{r}
+X <- parenthood$dan.sleep  # the predictor
+Y <- parenthood$dan.grump  # the outcome
+Y.pred <-  -8.95025 * X  +  125.96557
+
+```
+
+#calculating residue
+```{r}
+SS.resid <- sum( (Y - Y.pred)^2 )
+print( SS.resid )
+```
+
+
+```{r}
+SS.tot <- sum( (Y - mean(Y))^2 )
+print( SS.tot )
+```
+
+
+```{r}
+R.squared <- 1 - (SS.resid / SS.tot)
+print( R.squared )
+```
+
+```{r}
+r <- cor(X, Y)  # calculate the correlation
+print( r^2 )  
+```
+
+```{r}
+print( regression.2 ) 
+```
+
+```{r}
+summary( regression.2 )
+```
+```{r}
+summary( regression.1 )
+```
+
+```{r}
+cor.test( x = parenthood$dan.sleep, y = parenthood$dan.grump )
+```
+
+```{r}
+library(lsr)
+correlate(parenthood, test=TRUE)
+```
+
+```{r}
+confint( object = regression.2,
+        level = .99)
+```
+
+```{r}
+standardCoefs( regression.2 )
+
+```
+
+```{r}
+residuals( object = regression.2 )
+```
+
+```{r}
+rstandard( model = regression.2 )
+```
+
+```{r}
+rstudent( model = regression.2 )
+```
+
+```{r}
+hatvalues( model = regression.2 )
+```
+
+```{r}
+ckd <- cooks.distance( model = regression.2 )
+plot(regression.2,which=4)
+```
+```{r}
+lm( formula = dan.grump ~ dan.sleep + baby.sleep,  # same formula
+     data = parenthood,       # same data frame...
+     subset = -64             # ...but observation 64 is deleted
+ )
+```
+```{r}
+ hist( x = residuals( regression.2 ),   # data are the residuals
+       xlab = "Value of residual",      # x-axis label
+       main = "",                       # no title 
+       breaks = 20                      # lots of breaks
+ )
+```
+```{r}
+plot( x = regression.2, which = 2 )
+```
+
+Backward elimination
+
+```{r}
+full.model <- lm( formula = dan.grump ~ dan.sleep + baby.sleep + day,  
+                   data = parenthood  
+ )
+```
+
+```{r}
+ step( object = full.model,     # start at the full model
+       direction = "backward"   # allow it remove predictors but not add them
+ )
+```
+Forward selection
+```{r}
+ null.model <- lm( dan.grump ~ 1, parenthood )   # intercept only.
+ step( object = null.model,     # start with null.model
+       direction = "forward",   # only consider "addition" moves
+       scope =  dan.grump ~ dan.sleep + baby.sleep + day  # largest model allowed
+ )
+```
+ Comparing two regression models
+```{r}
+M0 <- lm( dan.grump ~ dan.sleep + day, parenthood )
+M1 <- lm( dan.grump ~ dan.sleep + day + baby.sleep, parenthood )
+```
+
+```{r}
+AIC( M0, M1 )
+```
+
+```{r}
+anova( M0, M1 )
+```
+
diff --git a/Module 6/parenthood.Rdata b/Module 6/parenthood.Rdata
new file mode 100644
index 0000000000000000000000000000000000000000..b571c3317c0267477ba5563148bae9bef6cb1a0e
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Date: Wed, 9 Nov 2022 14:14:04 +0530
Subject: [PATCH 34/55] Added car package and InteractionPlot

---
 Module 6/ANOVA.Rmd      | 11 +++----
 Module 6/Regression.Rmd | 65 ++++++++++++-----------------------------
 2 files changed, 24 insertions(+), 52 deletions(-)

diff --git a/Module 6/ANOVA.Rmd b/Module 6/ANOVA.Rmd
index 1640599e..3289038a 100644
--- a/Module 6/ANOVA.Rmd	
+++ b/Module 6/ANOVA.Rmd	
@@ -25,10 +25,6 @@ let’s now look at what we’ve got in the data file:
 load( file.path("clinicaltrial.Rdata" ))
 str(clin.trial)
 print( clin.trial )
-```
-Lets see how many people we have in each group:
-```{r}
-
 ```
 
 ```{r Opening Data}
@@ -224,7 +220,7 @@ Corrections for multiple testing
 
 t-test is designed to have a 5% Type I error rate (i.e.,α = 0.05), imagine if you have more than 10 groups !
 
-c   orrection for multiple comparisons, though it is sometimes referred to as “simultaneous inference”
+correction for multiple comparisons, though it is sometimes referred to as “simultaneous inference”
 
 Bonferroni corrections
 
@@ -252,6 +248,7 @@ Checking the homogeneity of variance assumption
 Levene test involve checking the assumptions of an ANOVA
 
 ```{r Opening Data}
+library(car)
 leveneTest(y = mood.gain ~ drug, data = clin.trial)   # y is a formula in this case
 leveneTest(y = clin.trial$mood.gain, group = clin.trial$drug)   # y is the outcome
 ```
@@ -307,6 +304,10 @@ etaSquared( model.2 )
 ```
 
 
+```{r}
+interaction.plot(x.factor = clin.trial$drug, response = clin.trial$mood.gain, 
+                 trace.factor = clin.trial$therapy)
+```
 
 
 
diff --git a/Module 6/Regression.Rmd b/Module 6/Regression.Rmd
index 95ec5879..ba0210c3 100644
--- a/Module 6/Regression.Rmd	
+++ b/Module 6/Regression.Rmd	
@@ -6,7 +6,7 @@ date: "2022-11-06"
 
 ```{r setup, include=FALSE}
 knitr::opts_chunk$set(echo = TRUE)
-setwd(dirname(rstudioapi::getActiveDocumentContext()$path)) # set curretn path as workpath
+setwd(dirname(rstudioapi::getActiveDocumentContext()$path)) # set current path as work path
 ```
 
 
@@ -16,70 +16,38 @@ load( file.path("parenthood.Rdata" ))
 colnames(parenthood) <- c('dan.sleep','baby.sleep','dan.grump','day')
 ```
 
-linear regression:
-dan.grump ~ dan.sleep
-```{r}
-regression.1 <- lm( formula = dan.grump ~ dan.sleep,  
-                    data = parenthood )
-print( regression.1 )
-```
-##Multiple linear regression
-dan.grump ~ dan.sleep + baby.sleep
 
 ```{r}
-regression.2 <- lm( formula = dan.grump ~ dan.sleep + baby.sleep,  
-                     data = parenthood )
-print( regression.2 )
+head(parenthood)
 ```
 
-
-The  R-squared value Calculation 
-
-
-#Calculating y-predicted
-```{r}
-X <- parenthood$dan.sleep  # the predictor
-Y <- parenthood$dan.grump  # the outcome
-Y.pred <-  -8.95025 * X  +  125.96557
-
-```
-
-#calculating residue
+## Simple linear regression:
+dan.grump ~ dan.sleep (Is Dan's sleep deprivation leading to grumpiness the next day?)
 ```{r}
-SS.resid <- sum( (Y - Y.pred)^2 )
-print( SS.resid )
+regression.1 <- lm( formula = dan.grump ~ dan.sleep,  
+                    data = parenthood )
+print( regression.1 )
 ```
 
-
 ```{r}
-SS.tot <- sum( (Y - mean(Y))^2 )
-print( SS.tot )
+summary(regression.1)
 ```
 
-
 ```{r}
-R.squared <- 1 - (SS.resid / SS.tot)
-print( R.squared )
+cor.test( x = parenthood$dan.sleep, y = parenthood$dan.grump )
 ```
 
-```{r}
-r <- cor(X, Y)  # calculate the correlation
-print( r^2 )  
-```
 
-```{r}
-print( regression.2 ) 
-```
+## Multiple linear regression
+dan.grump ~ dan.sleep + baby.sleep (Is Dan's sleep and the baby's sleep both together leading to grumpiness the next day?)
 
 ```{r}
-summary( regression.2 )
-```
-```{r}
-summary( regression.1 )
+regression.2 <- lm( formula = dan.grump ~ dan.sleep + baby.sleep,  
+                     data = parenthood )
+print( regression.2 )
 ```
-
 ```{r}
-cor.test( x = parenthood$dan.sleep, y = parenthood$dan.grump )
+summary(regression.2)
 ```
 
 ```{r}
@@ -97,6 +65,9 @@ standardCoefs( regression.2 )
 
 ```
 
+## Checking Model Assumptions
+First, we need to generate the residuals!
+
 ```{r}
 residuals( object = regression.2 )
 ```

From 0509c92f1692cb34464f8977e1a1c965cadb486b Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Wed, 9 Nov 2022 14:22:29 +0530
Subject: [PATCH 35/55] Added tukeyHSD instead of pairwise t test

---
 Module 6/ANOVA.Rmd | 9 +++++++++
 1 file changed, 9 insertions(+)

diff --git a/Module 6/ANOVA.Rmd b/Module 6/ANOVA.Rmd
index 3289038a..0c1b8478 100644
--- a/Module 6/ANOVA.Rmd	
+++ b/Module 6/ANOVA.Rmd	
@@ -302,6 +302,15 @@ Computing Effect Size
 ```{r}
 etaSquared( model.2 )
 ```
+Running a posthoc test on 2-way ANOVA
+Using tukey HSD because it can run on more complex models.
+```{r}
+TukeyHSD( model.2 )
+```
+
+
+
+
 
 
 ```{r}

From 309adcd15ed9b1b6061888d8bb9610180f8ff4d1 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Wed, 9 Nov 2022 14:26:49 +0530
Subject: [PATCH 36/55] updated Cook's distance and plot of residuals

---
 Module 6/Regression.Rmd | 29 +++++++++++++++++++++++++++++
 1 file changed, 29 insertions(+)

diff --git a/Module 6/Regression.Rmd b/Module 6/Regression.Rmd
index ba0210c3..3cb12ca1 100644
--- a/Module 6/Regression.Rmd	
+++ b/Module 6/Regression.Rmd	
@@ -86,14 +86,27 @@ hatvalues( model = regression.2 )
 
 ```{r}
 ckd <- cooks.distance( model = regression.2 )
+ckd
+```
+
+Directly plotting Cook's distance using plot from car package
+```{r}
 plot(regression.2,which=4)
 ```
+
+
+
+
 ```{r}
 lm( formula = dan.grump ~ dan.sleep + baby.sleep,  # same formula
      data = parenthood,       # same data frame...
      subset = -64             # ...but observation 64 is deleted
  )
 ```
+
+
+
+
 ```{r}
  hist( x = residuals( regression.2 ),   # data are the residuals
        xlab = "Value of residual",      # x-axis label
@@ -101,10 +114,26 @@ lm( formula = dan.grump ~ dan.sleep + baby.sleep,  # same formula
        breaks = 20                      # lots of breaks
  )
 ```
+
+
+Directly make the QQ plot of residuals
+
 ```{r}
 plot( x = regression.2, which = 2 )
 ```
 
+
+
+
+
+
+
+
+
+
+
+
+
 Backward elimination
 
 ```{r}

From b50de3a360cd3dd09f8a9a25e94007302aee5fd3 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Sat, 12 Nov 2022 16:21:20 +0530
Subject: [PATCH 37/55] Added diagnostics

---
 Module 6/Regression.Rmd | 32 ++++++++++++++++++++++++++++++++
 1 file changed, 32 insertions(+)

diff --git a/Module 6/Regression.Rmd b/Module 6/Regression.Rmd
index 3cb12ca1..0a9c0d3d 100644
--- a/Module 6/Regression.Rmd	
+++ b/Module 6/Regression.Rmd	
@@ -122,14 +122,46 @@ Directly make the QQ plot of residuals
 plot( x = regression.2, which = 2 )
 ```
 
+Checking linearity of relationship
+```{r}
+yhat.2 <- fitted.values( object = regression.2 )
+ plot( x = yhat.2, 
+       y = parenthood$dan.grump,
+       xlab = "Fitted Values",
+       ylab = "Observed Values" 
+ )
+```
 
 
+Checking this using residuals -- using plot
+```{r}
+plot(x = regression.2, which = 1)
 
+```
 
 
+```{r}
+residualPlots( model = regression.2 ) 
+```
 
+If the curvature is significant, then you might want to transform the predictor using Box Cox Transformation. Or use the powerTransform() in the car package.
 
 
+Checking homogeneity of variance
+```{r}
+plot(x = regression.2, which = 3)
+
+```
+```{r}
+ncvTest( regression.2 )
+
+```
+
+Lastly, we assess the variance inflation factor -- to diagnose for collinearity
+
+```{r}
+vif( mod = regression.2 )
+```
 
 
 

From d84a5dad0d6e10b3cc807fef49f826b27793d36b Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Sat, 12 Nov 2022 16:22:02 +0530
Subject: [PATCH 38/55] Added Bootstrapping from MKinfer

---
 Module 5/Ttest.Rmd     |  29 +++++++--
 Module 5/Ttest.nb.html | 137 ++++++++++++++++++++++++++++++++++++-----
 2 files changed, 146 insertions(+), 20 deletions(-)

diff --git a/Module 5/Ttest.Rmd b/Module 5/Ttest.Rmd
index 716072e0..65b456aa 100644
--- a/Module 5/Ttest.Rmd	
+++ b/Module 5/Ttest.Rmd	
@@ -13,18 +13,29 @@ install.packages('lsr')
 ```
 
 ```{r}
+library(lsr)
+load( file.path("zeppo.Rdata" )) 
+oneSampleTTest( x=grades, mu=67.5 ) #add the mu value here
 
 ```
-
+```{r}
+install.packages('MKinfer')
+```
 
 ```{r}
-library(lsr)
-load( file.path("zeppo.Rdata" )) 
-oneSampleTTest( x=grades, mu=67.5 ) #add the mu value here
+library(MKinfer)
+```
+
 
+```{r}
+boot.t.test( x=grades, mu=67.5 )
 ```
 
 
+
+
+
+
 ```{r}
 
 load (file.path("harpo.Rdata" ))
@@ -43,6 +54,16 @@ independentSamplesTTest(
       var.equal = TRUE          # assume that the two groups have the same variance
   )
 ```
+
+```{r}
+boot.t.test( 
+      formula = grade ~ tutor,  # formula specifying outcome and group variables
+      data = harpo            # data frame that contains the variables
+  )
+```
+
+
+
 ```{r}
 library(car)
 ```
diff --git a/Module 5/Ttest.nb.html b/Module 5/Ttest.nb.html
index 0f648d7a..8c144833 100644
--- a/Module 5/Ttest.nb.html	
+++ b/Module 5/Ttest.nb.html	
@@ -1774,46 +1774,47 @@
 
    setwd(dirname(rstudioapi::getActiveDocumentContext()$path)) # set curretn path as workpath
 install.packages('psych')
    
 
-
+
 
    Installing package into ‘C:/Users/Arjun/AppData/Local/R/win-library/4.2’
 (as ‘lib’ is unspecified)
 trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/psych_2.2.9.zip'
-Content type 'application/zip' length 3821660 bytes (3.6 MB)
+Content type 'application/zip' length 3821161 bytes (3.6 MB)
 downloaded 3.6 MB
    
 
-
+
 
    package ‘psych’ successfully unpacked and MD5 sums checked
 
 The downloaded binary packages are in
-    C:\Users\Arjun\AppData\Local\Temp\RtmpgT0xwo\downloaded_packages
    
+    C:\Users\Arjun\AppData\Local\Temp\RtmpSmUVDl\downloaded_packages
    
 
 
 
    install.packages('lsr')
    
 
-
+
 
    Installing package into ‘C:/Users/Arjun/AppData/Local/R/win-library/4.2’
 (as ‘lib’ is unspecified)
 trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/lsr_0.5.2.zip'
-Content type 'application/zip' length 209262 bytes (204 KB)
+Content type 'application/zip' length 209226 bytes (204 KB)
 downloaded 204 KB
    
 
-
+
 
    package ‘lsr’ successfully unpacked and MD5 sums checked
 
 The downloaded binary packages are in
-    C:\Users\Arjun\AppData\Local\Temp\RtmpgT0xwo\downloaded_packages
    
+    C:\Users\Arjun\AppData\Local\Temp\RtmpSmUVDl\downloaded_packages
    
 
 
 
 
 
-
-
-
-
-
-
    library(lsr)
-load( file.path("zeppo.Rdata" )) 
+
+
    library(lsr)
    
+
+
+
    Warning: package ‘lsr’ was built under R version 4.2.2
    
+
+
+
    load( file.path("zeppo.Rdata" )) 
 oneSampleTTest( x=grades, mu=67.5 ) #add the mu value here
    
 
 
@@ -1841,6 +1842,79 @@
    estimated effect size (Cohen's d):  0.504 
    
 
 
+
+
+
    install.packages('MKinfer')
    
+
+
+
    Installing package into ‘C:/Users/Arjun/AppData/Local/R/win-library/4.2’
+(as ‘lib’ is unspecified)
+also installing the dependencies ‘gmp’, ‘MKdescr’, ‘arrangements’, ‘exactRankTests’
+
+trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/gmp_0.6-8.zip'
+Content type 'application/zip' length 737069 bytes (719 KB)
+downloaded 719 KB
+
+trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/MKdescr_0.8.zip'
+Content type 'application/zip' length 383849 bytes (374 KB)
+downloaded 374 KB
+
+trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/arrangements_1.1.9.zip'
+Content type 'application/zip' length 379010 bytes (370 KB)
+downloaded 370 KB
+
+trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/exactRankTests_0.8-35.zip'
+Content type 'application/zip' length 157132 bytes (153 KB)
+downloaded 153 KB
+
+trying URL 'https://cran.rstudio.com/bin/windows/contrib/4.2/MKinfer_0.8.zip'
+Content type 'application/zip' length 288495 bytes (281 KB)
+downloaded 281 KB
    
+
+
+
    package ‘gmp’ successfully unpacked and MD5 sums checked
+package ‘MKdescr’ successfully unpacked and MD5 sums checked
+package ‘arrangements’ successfully unpacked and MD5 sums checked
+package ‘exactRankTests’ successfully unpacked and MD5 sums checked
+package ‘MKinfer’ successfully unpacked and MD5 sums checked
+
+The downloaded binary packages are in
+    C:\Users\Arjun\AppData\Local\Temp\RtmpSmUVDl\downloaded_packages
    
+
+
+
+
+
+
+
    library(MKinfer)
    
+
+
+
+
+
+
+
    boot.t.test( x=grades, mu=67.5 )
    
+
+
+
    
+    Bootstrap One Sample t-test
+
+data:  grades
+bootstrap p-value = 0.05361 
+bootstrap mean of x (SE) = 72.27616 (2.052497) 
+95 percent bootstrap percentile confidence interval:
+ 68.15 76.25
+
+Results without bootstrap:
+t = 2.2547, df = 19, p-value = 0.03615
+alternative hypothesis: true mean is not equal to 67.5
+95 percent confidence interval:
+ 67.84422 76.75578
+sample estimates:
+mean of x 
+     72.3 
    
+
+
 
 
 
@@ -1923,6 +1997,37 @@
    estimated effect size (Cohen's d):  0.74 
    
 
 
+
+
+
+
+
    boot.t.test( 
+      formula = grade ~ tutor,  # formula specifying outcome and group variables
+      data = harpo            # data frame that contains the variables
+  )
    
+
+
+
    
+    Bootstrap Welch Two Sample t-test
+
+data:  grade by tutor
+bootstrap p-value = 0.05841 
+bootstrap difference of means (SE) = 5.464636 (2.582662) 
+95 percent bootstrap percentile confidence interval:
+  0.4222222 10.4444444
+
+Results without bootstrap:
+t = 2.0342, df = 23.025, p-value = 0.05361
+alternative hypothesis: true difference in means is not equal to 0
+95 percent confidence interval:
+ -0.09249349 11.04804904
+sample estimates:
+ mean in group Anastasia mean in group Bernadette 
+                74.53333                 69.05556 
    
+
+
+
+
 
 
 
    library(car)
    
@@ -2148,7 +2253,7 @@
 
 
 
-
    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
    
+
    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
    
 
 
 

From 90d05c0b3dcce05ed76f6b6c85350d2e98547ac0 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Tue, 15 Nov 2022 09:15:33 +0530
Subject: [PATCH 39/55] Linear Mixed Models first update

---
 Module 6/linearMixedModels.Rmd     |  154 ++
 Module 6/linearMixedModels.nb.html | 2256 ++++++++++++++++++++++++++++
 2 files changed, 2410 insertions(+)
 create mode 100644 Module 6/linearMixedModels.Rmd
 create mode 100644 Module 6/linearMixedModels.nb.html

diff --git a/Module 6/linearMixedModels.Rmd b/Module 6/linearMixedModels.Rmd
new file mode 100644
index 00000000..e401d899
--- /dev/null
+++ b/Module 6/linearMixedModels.Rmd	
@@ -0,0 +1,154 @@
+---
+title: "R Notebook"
+output: html_notebook
+---
+
+Linear Mixed Models
+Adapted from: https://chenzixu.rbind.io/slides/lme/lmer.html
+
+
+```{r}
+library(lme4)
+library(magrittr)
+library(tidyverse)
+library(ggplot2)
+library(lmerTest)
+```
+
+```{r}
+data("sleepstudy")
+head(sleepstudy)
+```
+
+Plotting subjectwise data
+
+```{r}
+sleep2 <- sleepstudy %>%
+  filter(Days >= 2) %>%
+  mutate(days_deprived = Days - 2)
+ggplot(sleep2, aes(x = days_deprived, 
+                   y = Reaction)) +
+  geom_point() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
+```
+
+
+Complete Pooling model
+
+
+```{r}
+cp_model <- lm(Reaction ~ days_deprived, sleep2)
+summary(cp_model)
+ggplot(sleep2, aes(x = days_deprived, y = Reaction)) +
+  geom_abline(intercept = coef(cp_model)[1],
+              slope = coef(cp_model)[2],
+              color = '#f4cae2', size = 1.5) +
+  geom_point() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject, nrow = 3) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
+```
+
+No Pooling Model
+
+```{r}
+sleep2 %>% pull(Subject) %>% is.factor()
+np_model <- lm(Reaction ~ days_deprived + Subject + days_deprived:Subject,
+               data = sleep2)
+  summary(np_model)
+all_intercepts <- c(coef(np_model)["(Intercept)"],
+                    coef(np_model)[3:19] + coef(np_model)["(Intercept)"])
+all_slopes  <- c(coef(np_model)["days_deprived"],
+                 coef(np_model)[20:36] + coef(np_model)["days_deprived"])
+ids <- sleep2 %>% pull(Subject) %>% levels() %>% factor()
+np_coef <- tibble(Subject = ids,
+                  intercept = all_intercepts,
+                  slope = all_slopes)
+```
+
+```{r}
+ggplot(sleep2, aes(x = days_deprived, y = Reaction)) +
+  geom_abline(data = np_coef,
+              mapping = aes(intercept = intercept,
+                            slope = slope),
+              color = '#f4cae2', size = 1.5) +
+  geom_point() + theme_bw() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject, nrow=3) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
+```
+
+Partial Pooling Model
+
+```{r}
+pp_mod <- lmer(Reaction ~ days_deprived + (days_deprived | Subject), sleep2)
+summary(pp_mod)
+newdata <- crossing(
+  Subject = sleep2 %>% pull(Subject) %>% levels() %>% factor(),
+  days_deprived = 0:7)
+newdata2 <- newdata %>%
+  mutate(Reaction = predict(pp_mod, newdata))
+```
+
+
+```{r}
+ggplot(sleep2, aes(x = days_deprived, y = Reaction)) +
+  geom_line(data = newdata2,
+            color = '#f4cae2', size = 1.5) +
+  geom_point() + theme_bw() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject, nrow = 3) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
+```
+
+##Research Question
+We're interested in the relationship between pitch and politeness (Winter & Grawunder, 2012).
+
+Politeness: formal/polite and informal register (categorical factor)
+multiple measures per subject (inter-dependent!)
+
+```{r}
+data = read.csv("http://www.bodowinter.com/tutorial/politeness_data.csv")
+head(data)
+```
+
+Convert attitude, gender, subject into factors.
+```{r}
+data = data %>% mutate(attitude=as.factor(attitude), gender=as.factor(gender), subject=as.factor(subject))
+
+```
+
+
+Random Intercept Models
+
+```{r}
+politeness.model0 = lmer(frequency ~ attitude + (1|subject) + (1|scenario), data=data)
+summary(politeness.model0)
+```
+
+```{r}
+politeness.model = lmer(frequency ~ attitude + gender + (1|subject) + (1|scenario), data=data)
+summary(politeness.model)
+```
+Likelihood Ratio Test
+
+```{r}
+politeness.null = lmer(frequency ~ gender + (1|subject) + (1|scenario), data=data, REML=FALSE)
+politeness.full = lmer(frequency ~ attitude + gender + (1|subject) + (1|scenario), data=data, REML=FALSE)
+anova(politeness.null, politeness.full)
+```
+
+Random Slope Model
+
+```{r}
+politeness.model1 = lmer(frequency~attitude + gender + (1+attitude|subject) + (1+attitude|scenario), data = data)
+coef(politeness.model1)
+```
+
+
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+R Notebook
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+
    Linear Mixed Models Adapted from: https://chenzixu.rbind.io/slides/lme/lmer.html
    
+
+
+
+
    library(lmerTest)
+
    
+
+
+
    Warning: package ‘lmerTest’ was built under R version 4.2.2
+Attaching package: ‘lmerTest’
+
+The following object is masked from ‘package:lme4’:
+
+    lmer
+
+The following object is masked from ‘package:stats’:
+
+    step
    
+
+
+
+
+
+
+
    data("sleepstudy")
+head(sleepstudy)
    
+
+
+
    
+
+
    
+
+
+
+
    Plotting subjectwise data
    
+
+
+
+
    sleep2 <- sleepstudy %>%
+  filter(Days >= 2) %>%
+  mutate(days_deprived = Days - 2)
+ggplot(sleep2, aes(x = days_deprived, 
+                   y = Reaction)) +
+  geom_point() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
    
+
+
+
+
    Complete Pooling model
    
+
+
+
+
    cp_model <- lm(Reaction ~ days_deprived, sleep2)
+summary(cp_model)
    
+
+
+
    
+Call:
+lm(formula = Reaction ~ days_deprived, data = sleep2)
+
+Residuals:
+     Min       1Q   Median       3Q      Max 
+-112.284  -26.732    2.143   27.734  140.453 
+
+Coefficients:
+              Estimate Std. Error t value Pr(>|t|)    
+(Intercept)    267.967      7.737  34.633  < 2e-16 ***
+days_deprived   11.435      1.850   6.183 6.32e-09 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Residual standard error: 50.85 on 142 degrees of freedom
+Multiple R-squared:  0.2121,    Adjusted R-squared:  0.2066 
+F-statistic: 38.23 on 1 and 142 DF,  p-value: 6.316e-09
    
+
+
+
    ggplot(sleep2, aes(x = days_deprived, y = Reaction)) +
+  geom_abline(intercept = coef(cp_model)[1],
+              slope = coef(cp_model)[2],
+              color = '#f4cae2', size = 1.5) +
+  geom_point() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject, nrow = 3) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
    
+
+
+
    
+
+
+
+
    No Pooling Model
    
+
+
+
+
    sleep2 %>% pull(Subject) %>% is.factor()
    
+
+
+
    [1] TRUE
    
+
+
+
    np_model <- lm(Reaction ~ days_deprived + Subject + days_deprived:Subject,
+               data = sleep2)
+  summary(np_model)
    
+
+
+
    
+Call:
+lm(formula = Reaction ~ days_deprived + Subject + days_deprived:Subject, 
+    data = sleep2)
+
+Residuals:
+     Min       1Q   Median       3Q      Max 
+-106.521   -8.541    1.143    8.889  128.545 
+
+Coefficients:
+                         Estimate Std. Error t value Pr(>|t|)    
+(Intercept)              288.2175    16.4772  17.492  < 2e-16 ***
+days_deprived             21.6905     3.9388   5.507 2.49e-07 ***
+Subject309               -87.9262    23.3023  -3.773 0.000264 ***
+Subject310               -62.2856    23.3023  -2.673 0.008685 ** 
+Subject330               -14.9533    23.3023  -0.642 0.522422    
+Subject331                 9.9658    23.3023   0.428 0.669740    
+Subject332                27.8157    23.3023   1.194 0.235215    
+Subject333                -2.7581    23.3023  -0.118 0.906000    
+Subject334               -50.2051    23.3023  -2.155 0.033422 *  
+Subject335               -25.3429    23.3023  -1.088 0.279207    
+Subject337                24.6143    23.3023   1.056 0.293187    
+Subject349               -59.2183    23.3023  -2.541 0.012464 *  
+Subject350               -40.2023    23.3023  -1.725 0.087343 .  
+Subject351               -24.2467    23.3023  -1.041 0.300419    
+Subject352                43.0655    23.3023   1.848 0.067321 .  
+Subject369               -21.5040    23.3023  -0.923 0.358154    
+Subject370               -53.3072    23.3023  -2.288 0.024107 *  
+Subject371               -30.4896    23.3023  -1.308 0.193504    
+Subject372                 2.4772    23.3023   0.106 0.915535    
+days_deprived:Subject309 -17.3334     5.5703  -3.112 0.002380 ** 
+days_deprived:Subject310 -17.7915     5.5703  -3.194 0.001839 ** 
+days_deprived:Subject330 -13.6849     5.5703  -2.457 0.015613 *  
+days_deprived:Subject331 -16.8231     5.5703  -3.020 0.003154 ** 
+days_deprived:Subject332 -19.2947     5.5703  -3.464 0.000765 ***
+days_deprived:Subject333 -10.8151     5.5703  -1.942 0.054796 .  
+days_deprived:Subject334  -3.5745     5.5703  -0.642 0.522423    
+days_deprived:Subject335 -25.8995     5.5703  -4.650 9.47e-06 ***
+days_deprived:Subject337   0.7518     5.5703   0.135 0.892895    
+days_deprived:Subject349  -5.2644     5.5703  -0.945 0.346731    
+days_deprived:Subject350   1.6007     5.5703   0.287 0.774382    
+days_deprived:Subject351 -13.1681     5.5703  -2.364 0.019867 *  
+days_deprived:Subject352 -14.4019     5.5703  -2.585 0.011057 *  
+days_deprived:Subject369  -7.8948     5.5703  -1.417 0.159273    
+days_deprived:Subject370  -1.0495     5.5703  -0.188 0.850912    
+days_deprived:Subject371  -9.3443     5.5703  -1.678 0.096334 .  
+days_deprived:Subject372 -10.6041     5.5703  -1.904 0.059613 .  
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Residual standard error: 25.53 on 108 degrees of freedom
+Multiple R-squared:  0.849, Adjusted R-squared:  0.8001 
+F-statistic: 17.35 on 35 and 108 DF,  p-value: < 2.2e-16
    
+
+
+
    all_intercepts <- c(coef(np_model)["(Intercept)"],
+                    coef(np_model)[3:19] + coef(np_model)["(Intercept)"])
+all_slopes  <- c(coef(np_model)["days_deprived"],
+                 coef(np_model)[20:36] + coef(np_model)["days_deprived"])
+ids <- sleep2 %>% pull(Subject) %>% levels() %>% factor()
+np_coef <- tibble(Subject = ids,
+                  intercept = all_intercepts,
+                  slope = all_slopes)
    
+
+
+
+
+
+
+
    ggplot(sleep2, aes(x = days_deprived, y = Reaction)) +
+  geom_abline(data = np_coef,
+              mapping = aes(intercept = intercept,
+                            slope = slope),
+              color = '#f4cae2', size = 1.5) +
+  geom_point() + theme_bw() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject, nrow=3) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
    
+
+
+
    
+
+
+
+
    Partial Pooling Model
    
+
+
+
+
    pp_mod <- lmer(Reaction ~ days_deprived + (days_deprived | Subject), sleep2)
+summary(pp_mod)
    
+
+
+
    Linear mixed model fit by REML ['lmerMod']
+Formula: Reaction ~ days_deprived + (days_deprived | Subject)
+   Data: sleep2
+
+REML criterion at convergence: 1404.1
+
+Scaled residuals: 
+    Min      1Q  Median      3Q     Max 
+-4.0157 -0.3541  0.0069  0.4681  5.0732 
+
+Random effects:
+ Groups   Name          Variance Std.Dev. Corr
+ Subject  (Intercept)   958.35   30.957       
+          days_deprived  45.78    6.766   0.18
+ Residual               651.60   25.526       
+Number of obs: 144, groups:  Subject, 18
+
+Fixed effects:
+              Estimate Std. Error t value
+(Intercept)    267.967      8.266  32.418
+days_deprived   11.435      1.845   6.197
+
+Correlation of Fixed Effects:
+            (Intr)
+days_deprvd -0.062
    
+
+
+
    newdata <- crossing(
+  Subject = sleep2 %>% pull(Subject) %>% levels() %>% factor(),
+  days_deprived = 0:7)
+newdata2 <- newdata %>%
+  mutate(Reaction = predict(pp_mod, newdata))
    
+
+
+
+
+
+
+
    ggplot(sleep2, aes(x = days_deprived, y = Reaction)) +
+  geom_line(data = newdata2,
+            color = '#f4cae2', size = 1.5) +
+  geom_point() + theme_bw() +
+  scale_x_continuous(breaks = 0:7) +
+  facet_wrap(~Subject, nrow = 3) +
+  labs(y = "Reaction Time", 
+       x = "Days deprived of sleep (0 = baseline)")
    
+
+
+
    
+
+
+
+
    ##Research Question We’re interested in the relationship between
+pitch and politeness (Winter & Grawunder, 2012).
    
+
    Politeness: formal/polite and informal register (categorical factor)
+multiple measures per subject (inter-dependent!)
    
+
+
+
+
    data = read.csv("http://www.bodowinter.com/tutorial/politeness_data.csv")
+head(data)
    
+
+
+
    
+
+
    
+
+
+
+
    Convert attitude, gender, subject into factors.
    
+
+
+
+
    data = data %>% mutate(attitude=as.factor(attitude), gender=as.factor(gender), subject=as.factor(subject))
+
    
+
+
+
+
    Random Intercept Models
    
+
+
+
+
    summary(politeness.model0)
+
    
+
+
+
    Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
+Formula: frequency ~ attitude + (1 | subject) + (1 | scenario)
+   Data: data
+
+REML criterion at convergence: 793.5
+
+Scaled residuals: 
+    Min      1Q  Median      3Q     Max 
+-2.2006 -0.5817 -0.0639  0.5625  3.4385 
+
+Random effects:
+ Groups   Name        Variance Std.Dev.
+ scenario (Intercept)  219     14.80   
+ subject  (Intercept) 4015     63.36   
+ Residual              646     25.42   
+Number of obs: 83, groups:  scenario, 7; subject, 6
+
+Fixed effects:
+            Estimate Std. Error      df t value Pr(>|t|)    
+(Intercept)  202.588     26.754   5.575   7.572 0.000389 ***
+attitudepol  -19.695      5.585  70.022  -3.527 0.000748 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Correlation of Fixed Effects:
+            (Intr)
+attitudepol -0.103
    
+
+
+
+
+
+
+
    politeness.model = lmer(frequency ~ attitude + gender + (1|subject) + (1|scenario), data=data)
+summary(politeness.model)
    
+
+
+
    Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
+Formula: frequency ~ attitude + gender + (1 | subject) + (1 | scenario)
+   Data: data
+
+REML criterion at convergence: 775.5
+
+Scaled residuals: 
+    Min      1Q  Median      3Q     Max 
+-2.2591 -0.6236 -0.0772  0.5388  3.4795 
+
+Random effects:
+ Groups   Name        Variance Std.Dev.
+ scenario (Intercept) 219.5    14.81   
+ subject  (Intercept) 615.6    24.81   
+ Residual             645.9    25.41   
+Number of obs: 83, groups:  scenario, 7; subject, 6
+
+Fixed effects:
+            Estimate Std. Error       df t value Pr(>|t|)    
+(Intercept)  256.846     16.116    5.432  15.938 9.06e-06 ***
+attitudepol  -19.721      5.584   70.054  -3.532 0.000735 ***
+genderM     -108.516     21.013    4.007  -5.164 0.006647 ** 
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Correlation of Fixed Effects:
+            (Intr) atttdp
+attitudepol -0.173       
+genderM     -0.652  0.004
    
+
+
+
+
    Likelihood Ratio Test
    
+
+
+
+
    politeness.null = lmer(frequency ~ gender + (1|subject) + (1|scenario), data=data, REML=FALSE)
+politeness.full = lmer(frequency ~ attitude + gender + (1|subject) + (1|scenario), data=data, REML=FALSE)
+anova(politeness.null, politeness.full)
    
+
+
+
    Data: data
+Models:
+politeness.null: frequency ~ gender + (1 | subject) + (1 | scenario)
+politeness.full: frequency ~ attitude + gender + (1 | subject) + (1 | scenario)
+                npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)    
+politeness.null    5 816.72 828.81 -403.36   806.72                         
+politeness.full    6 807.10 821.61 -397.55   795.10 11.618  1  0.0006532 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
+
+
+
+
    Random Slope Model
    
+
+
+
+
    politeness.model1 = lmer(frequency~attitude + gender + (1+attitude|subject) + (1+attitude|scenario), data = data)
    
+
+
+
    boundary (singular) fit: see help('isSingular')
    
+
+
+
    coef(politeness.model1)
    
+
+
+
    $scenario
+  (Intercept) attitudepol   genderM
+1    244.4740   -19.00296 -111.1058
+2    261.9447   -12.87473 -111.1058
+3    270.9290   -23.46233 -111.1058
+4    277.0651   -15.90595 -111.1058
+5    255.8277   -18.72597 -111.1058
+6    247.0421   -22.37916 -111.1058
+7    249.7042   -25.93003 -111.1058
+
+$subject
+   (Intercept) attitudepol   genderM
+F1    243.2804   -20.49940 -111.1058
+F2    267.1173   -19.30447 -111.1058
+F3    260.2849   -19.64697 -111.1058
+M3    287.1024   -18.30263 -111.1058
+M4    264.6698   -19.42716 -111.1058
+M7    226.3911   -21.34605 -111.1058
+
+attr(,"class")
+[1] "coef.mer"
    
+
+
+
+
+
+
    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From 8e2e723b8d878a9f4811c5cf7ee4fa82287660e4 Mon Sep 17 00:00:00 2001
From: Chetan Kandpal 
Date: Fri, 4 Aug 2023 12:02:13 +0530
Subject: [PATCH 40/55] Shifted Module 1 & 2 in same folder

---
 Module 1/Notebook for chapter 1.Rmd     |   109 +
 Module 1/Notebook for chapter 1.nb.html |   457 +
 Module 1/R notebook tutorial-2.Rmd      |    45 +
 Module 1/R notebook tutorial.Rmd        |   171 +
 Module 1/R notebook tutorial.nb.html    |   423 +
 Module 1/README.md                      |     1 +
 Module 2/README.md                      |    10 +
 Module 2/StatewiseTestingDetails.csv    | 14099 ++++++++++++++++++++++
 Module 2/Tidyverse.Rmd                  |   207 +
 Module 2/Tidyverse.nb.html              |  2309 ++++
 Module 2/project.Rproj                  |    13 +
 Module 2/using ggplot.Rmd               |    59 +
 Module 2/using-ggplot.html              |   311 +
 13 files changed, 18214 insertions(+)
 create mode 100644 Module 1/Notebook for chapter 1.Rmd
 create mode 100644 Module 1/Notebook for chapter 1.nb.html
 create mode 100644 Module 1/R notebook tutorial-2.Rmd
 create mode 100644 Module 1/R notebook tutorial.Rmd
 create mode 100644 Module 1/R notebook tutorial.nb.html
 create mode 100644 Module 1/README.md
 create mode 100644 Module 2/README.md
 create mode 100644 Module 2/StatewiseTestingDetails.csv
 create mode 100644 Module 2/Tidyverse.Rmd
 create mode 100644 Module 2/Tidyverse.nb.html
 create mode 100644 Module 2/project.Rproj
 create mode 100644 Module 2/using ggplot.Rmd
 create mode 100644 Module 2/using-ggplot.html

diff --git a/Module 1/Notebook for chapter 1.Rmd b/Module 1/Notebook for chapter 1.Rmd
new file mode 100644
index 00000000..651600df
--- /dev/null
+++ b/Module 1/Notebook for chapter 1.Rmd	
@@ -0,0 +1,109 @@
+---
+title: "R Notebook"
+output: html_notebook
+---
+
+This is an [R Markdown](http://rmarkdown.rstudio.com) Notebook. When you execute code within the notebook, the results appear beneath the code. 
+
+Try executing this chunk by clicking the *Run* button within the chunk or by placing your cursor inside it and pressing *Ctrl+Shift+Enter*. 
+
+```{r}
+plot(cars)
+```
+Basic commands on R
+Assigning variable 
+```{r}
+x<- 2*8
+x
+```
+Numeric Vector and operation on it
+```{r}
+y <- c(2.3, 1, 5)
+y
+```
+
+```{r}
+length(y)
+mode(y)
+class(y)
+```
+sequence of integers storing in vector
+```{r}
+mynums <- 10:1
+mynums 
+```
+operating on Numeric vector
+```{r}
+sum(mynums)
+min(mynums)
+max(mynums)
+range(mynums)
+```
+standard deviation, mean and median 
+```{r}
+mean(mynums)
+sd(mynums)
+median(mynums)
+```
+Indexing Numeric vector
+```{r}
+mynums[2]
+mynums[1:4]	
+mynums[-4] # retrieve everything except fourth position
+```
+character vectors
+```{r}
+gender <- c('F', 'M', 'M', 'F', 'F')
+gender
+class(gender)
+```
+Finding repitition of a character
+```{r}
+gender[gender == 'F']
+```
+operating on Data frames
+```{r}
+participant <- c('louis', 'paula', 'vincenzo')
+mydf <- data.frame(participant, score = c(67, 85, 32)) 
+mydf
+mydf$score
+mean(mydf$score)
+```
+Indexing on Data frame
+```{r}
+mydf[1,] # first row
+mydf[, 1][2]
+```
+Indexing
+```{r}
+mydf[2, ] # 2nd column
+```
+PLOTING
+```{r}
+mean(mydf$score)
+str(mydf)
+summary(mydf)
+```
+Loading files
+```{r}
+covid_19 <- read.csv('india_covid_19_statewise_status.csv')
+covid_19
+```
+ASSIGNMENT
+Assignment for chapter -1 
+
+1.	Create a Numeric vector with 10 elements ranging between 20 to 30, name it Mynums. Find maximum and minimum element of the vector. Compute Sum operation on the vector put it in a variable z. Find out the 4,5,6 th element of the vector.
+
+2.	Create a 10 element numeric vector and compute mean, median and Standard deviation of the vector. 
+
+3.	Create a dataframe with 5 participants and their math score. Calculate mean of their score.
+
+
+Student name	Math score
+Louis	67
+Paul	86
+Vincenzo	80
+Tim	56
+Dorothy	91
+
+
diff --git a/Module 1/Notebook for chapter 1.nb.html b/Module 1/Notebook for chapter 1.nb.html
new file mode 100644
index 00000000..117d05d6
--- /dev/null
+++ b/Module 1/Notebook for chapter 1.nb.html	
@@ -0,0 +1,457 @@
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+R Notebook
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+
    
+
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+
+
+
+
    This is an R Markdown Notebook. When you execute code within the notebook, the results appear beneath the code.
    
+
    Try executing this chunk by clicking the Run button within the chunk or by placing your cursor inside it and pressing Ctrl+Shift+Enter.
    
+
+
+
+
    plot(cars)
    
+
+
+
+
    Basic commands on R Assigning variable
    
+
+
+
+
    x<- 2*8
+x
    
+
+
+
+
    Numeric Vector and operation on it
    
+
+
+
+
    y <- c(2.3, 1, 5)
+y
    
+
+
+
    [1] 2.3 1.0 5.0
    
+
+
+
+
+
+
+
    length(y)
+mode(y)
+class(y)
    
+
+
+
+
    sequence of integers storing in vector
    
+
+
+
+
    mynums <- 10:1
+mynums 
    
+
+
+
+
    operating on Numeric vector
    
+
+
+
+
    sum(mynums)
    
+
+
+
    [1] 55
    
+
+
+
    min(mynums)
    
+
+
+
    [1] 1
    
+
+
+
    max(mynums)
    
+
+
+
    [1] 10
    
+
+
+
    range(mynums)
    
+
+
+
    [1]  1 10
    
+
+
+
+
    standard deviation, mean and median
    
+
+
+
+
    mean(mynums)
+sd(mynums)
+median(mynums)
    
+
+
+
+
    Indexing Numeric vector
    
+
+
+
+
    mynums[2]
+mynums[1:4] 
+mynums[-4] # retrieve everything except fourth position
    
+
+
+
+
    character vectors
    
+
+
+
+
    gender <- c('F', 'M', 'M', 'F', 'F')
+gender
+class(gender)
    
+
+
+
+
    Finding repetation of a character
    
+
+
+
+
    gender[gender == 'F']
    
+
+
+
+
    operating on Data frames
    
+
+
+
+
    participant <- c('louis', 'paula', 'vincenzo')
+mydf <- data.frame(participant, score = c(67, 85, 32)) 
+mydf
+mydf$score
+mean(mydf$score)
    
+
+
+
+
    Indexing on Data frame
    
+
+
+
+
    mydf[1,] # first row
+mydf[, 1][2]
    
+
+
+
+
    Indexing
    
+
+
+
+
    mydf[2, ] # 2nd column
    
+
+
+
+
    PLOTING
    
+
+
+
+
    mean(mydf$score)
+str(mydf)
+summary(mydf)
    
+
+
+
+
    Loading files
    
+
+
+
+
    covid_19 <- read.csv('india_covid_19_statewise_status.csv')
+covid_19
    
+
+
+
+
    ASSIGNMENT Assignment for chapter -1
    
+
      
+
    1. Create a Numeric vector with 10 elements ranging between 20 to 30, name it Mynums. Find maximum and minimum element of the vector. Compute Sum operation on the vector put it in a variable z. Find out the 4,5,6 th element of the vector.
      2. 
+
    2. Create a 10 element numeric vector and compute mean, median and Standard deviation of the vector.
      3. 
+
    3. Create a dataframe with 5 participants and their math score. Calculate mean of their score.
      4. 
+
    
+
    Student name Math score Louis 67 Paul 86 Vincenzo 80 Tim 56 Dorothy 91
    
+
    Add a new chunk by clicking the Insert Chunk button on the toolbar or by pressing Ctrl+Alt+I.
    
+
    When you save the notebook, an HTML file containing the code and output will be saved alongside it (click the Preview button or press Ctrl+Shift+K to preview the HTML file).
    
+
    The preview shows you a rendered HTML copy of the contents of the editor. Consequently, unlike Knit, Preview does not run any R code chunks. Instead, the output of the chunk when it was last run in the editor is displayed.
    
+
+
+
    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diff --git a/Module 1/R notebook tutorial-2.Rmd b/Module 1/R notebook tutorial-2.Rmd
new file mode 100644
index 00000000..7727cd42
--- /dev/null
+++ b/Module 1/R notebook tutorial-2.Rmd	
@@ -0,0 +1,45 @@
+---
+title: "R Notebook"
+output: html_notebook
+---
+
+---
+title: "Getting used to R notebooks"
+output: html_document
+---
+
+### Hi all, Welcome to statistics with R
+### This file is intended to make you familiar with R notebooks, if you are already an R user - thats good, but still you should have quick view of this tutorial, you may learn something new.
+
+##### This is an R Markdown notebook file, you might have noticed that this file has the format of *.Rmd*
+##### R Markdown or .Rmd is a file format for making dynamic documents with R. An R Markdown document is written in markdown (an easy-to-write plain text format) and contains chunks of embedded R code. Hence, this tutorial file can itself has plain text as well as embeded code. Currently, these instructions you are reading are in the markdown format, and if you wish to insert a chunck of code below it, you can do so by clicking the *Insert Chunk* button on the toolbar or by pressing *Ctrl+Alt+I*. So, why to wait, lets write a code chuck for printing "hello". 
+```{r}
+print('Hello')
+```
+
+##### Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. There can be several different output formats for this markdown file and you can mention it at the start of this file under output:, for this notebook it is mentioned 'html document' which outputs a .html file with the same name as of the .Rmd file. You might be already reading this in a .html output file in your browser, if not then you can always preview that html file in you browser to have a look.
+
+To know more about R Markdown you can visit this [link](http://rmarkdown.rstudio.com). 
+
+[R Markdown interface](https://rmarkdown.rstudio.com/lesson-2.html) notebook. 
+
+Try executing this chunk by clicking the *Run* button within the chunk or by placing your cursor inside it and pressing *Ctrl+Shift+Enter*. 
+
+### Knitting and converting
+When you click the **Knit** button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. You can embed an R code chunk like this:
+
+### Some basic markdown commands to make your text look good
+
+```{r cars}
+summary(cars)
+```
+
+## Including Plots
+
+You can also embed plots, for example:
+
+```{r pressure, echo=FALSE}
+plot(pressure)
+```
+
+Note that the `echo = FALSE` parameter was added to the code chunk to prevent printing of the R code that generated the plot.
diff --git a/Module 1/R notebook tutorial.Rmd b/Module 1/R notebook tutorial.Rmd
new file mode 100644
index 00000000..125a9b99
--- /dev/null
+++ b/Module 1/R notebook tutorial.Rmd	
@@ -0,0 +1,171 @@
+---
+title: "R Notebook"
+output: html_notebook
+---
+
+---
+title: "Introduction to R notebooks"
+output: html_document
+---
+
+### Welcome to statistics with R!
+
+
+This tutorial notebook is intended to introduce and make you familiar with **R notebooks**. Even if you have used R previously, you can take a quick glance at the tutorial and might learn something new.
+
+This is an R Markdown notebook file, you might have noticed that this file has the format of *.Rmd*.
+First things first,
+
+##### What the heck is Markdown?
+Markdown is a [markup language](https://en.wikipedia.org/wiki/Markup_language) for creating formatted text using a plain-text editor. The idea and terminology evolved from the "marking up" of paper manuscripts (i.e., the revision instructions by editors), which is traditionally written with a red or blue pen on authors' manuscripts. Markdown using any language including R is just a *digital* version of such blue and red pen annotations.
+
+##### But what is an R Notebook then?
+R Notebook is simply an R Markdown document (a document written in the *language* R Markdown) with chunks that can be executed independently and interactively, with output visible immediately beneath the input. It is an implementation of [Literate Programming](https://en.wikipedia.org/wiki/Literate_programming) that allows for direct interaction with R while producing a reproducible document with publication-quality output. A notebook can therefore be thought of as a special execution mode for R Markdown documents.
+
+With the .Rmd file format, you can make such dynamic documents with R. In fact, this tutorial file itself is an R Notebook file with both plain text (that you're currently reading) and embeded code (which you'll insert below). 
+
+Before moving to code insertion, so far we've learnt that the R Notebook (R Markdown document)  is written in R markdown (an easy-to-write markup language) and contains chunks of embedded R code. 
+
+Now if you wish to insert a code chunk, you can do so by clicking the *Insert Chunk* button on the toolbar or you can by press *Ctrl+Alt+I*. (*mac: Cmd + option + I*). Let's write our first code chunk for printing "hello". *Try it yourself:* Insert a new code chunk to print "My First Markdown File".
+```{r}
+print('Hello')
+```
+
+Try executing the inserted chunk(s) by clicking the *Run* button (green arrowhead) within the chunk or by placing your cursor inside it and pressing *Ctrl+Shift+Enter*. 
+
+##### Some more information on R Markdown
+Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. There can be several different output formats for this markdown file and you can mention it at the start of this file under output:, for R notebooks, the output type is 'html document' which outputs a .html file with the same name as of the .Rmd file. You might be already reading this in a .html output file in your browser, if not then you can always *Preview* that html file in you browser to have a look.
+
+To know more about R Markdown you can visit this [link](http://rmarkdown.rstudio.com). 
+
+[R Markdown interface](https://rmarkdown.rstudio.com/lesson-2.html) notebook. 
+
+**_Suggested reading:_** Section 3.2 (R Notebook) and Sections 2.5, 2.6 (Markdown syntax) from the book - [R Markdown: The definitive guide](https://bookdown.org/yihui/rmarkdown/)
+
+#### R Markdown syntax
+The text in an R Markdown document is written with the Markdown syntax. More precisely, it is [Pandoc’s Markdown](https://pandoc.org/MANUAL.html). The following information has been adapted form the above mentioned book.
+
+_Note_ : It is suggested to view the file as HTML document for this section in order to see the effect of formatting. You can do so by clicking the Preview button in the toolbar.
+
+##### Inline formatting
+
+Inline text will be _italic_ if surrounded by underscores or asterisks, e.g., `_text_` or `*text*`. **Bold** text is produced using a pair of double asterisks (`**text**`). A pair of tildes (~) turn text to a subscript (e.g., `H~3~PO~4~` renders H~3~PO~4). A pair of carets (^) produce a superscript (e.g., `Cu^2+^` renders Cu^2+^). 
+
+To mark text as inline code, use a pair of backticks, e.g., `` `code` ``.
+
+Hyperlinks are created using the syntax `[text](link)`, e.g., `[RStudio](https://www.rstudio.com)` will output [RStudio](https://www.rstudio.com). The syntax for images is similar: just add an exclamation mark, e.g., `![alt text or image title](path/to/image)`. Footnotes are put inside the square brackets after a caret `^[]`, e.g., `^[This is a footnote.]`
+
+##### Block level elements
+Section headers can be written after a number of hashtags, e.g.,
+
+`# First-level header`
+
+# First-level header
+
+`## Second-level header`
+
+## Second-level header
+
+`### Third-level header`
+
+### Third-level header
+
+If you do not want a certain heading to be numbered, you can add {-} or {.unnumbered} after the heading, e.g.,
+
+`# Preface {-}`
+
+Unordered list items start with *, -, or +, and you can nest one list within another list by indenting the sub-list, e.g.,
+
+```
+- one item
+- one item
+- one item
+    - one more item
+    - one more item
+    - one more item
+```
+
+The output is:
+
+- one item
+
+- one item
+
+- one item
+
+  - one more item
+  
+  - one more item
+  
+  - one more item
+
+Ordered list items start with numbers (you can also nest lists within lists), e.g.,
+
+```
+1. the first item
+2. the second item
+3. the third item
+    - one unordered item
+    - one unordered item
+```
+
+The output does not look too much different with the Markdown source:
+
+1. the first item
+
+2. the second item
+
+3. the third item
+
+    - one unordered item
+    
+    - one unordered item
+    
+Plain code blocks can be written after three or more backticks.
+````
+```
+Just like this
+```
+````
+
+#### All about Code chunks
+You inserted a code chunk in the beginning and therefore you now know that code chunks can be inserted by either using the RStudio toolbar (the `Insert` button) or the keyboard shortcut `Ctrl + Alt + I` (`Cmd + Option + I` on macOS). There are a lot of things you can do in a code chunk: you can produce text output, tables, or graphics. You have fine control over all these output via chunk options, which can be provided inside the curly braces (between ```` ```{r and }````). For example, you can choose hide text output via the chunk option `results = 'hide'`, or set the figure height to 4 inches via `fig.height = 4`. Chunk options are separated by commas, e.g.,
+
+**```{r, chunk-label, results='hide', fig.height=4}**
+
+A few of the options are:
+
+`echo = FALSE` Whether to echo the source code in the output document (someone may not prefer reading your smart source code but only results)
+
+`include = FALSE` prevents code and results from appearing in the finished file. R Markdown still runs the code in the chunk, and the results can be used by other chunks.
+
+`message = FALSE` prevents messages that are generated by code from appearing in the finished file.
+
+`warning = FALSE` prevents warnings that are generated by code from appearing in the finished.
+
+`fig.cap = "..."` adds a caption to graphical results.
+
+##### Plotting figures:
+By default, figures produced by R code will be placed immediately after the code chunk they were generated from. For example:
+
+```{r pressure, echo=FALSE}
+plot(pressure)
+```
+Notice the `echo = 'FALSE'` option added in the code in the source file. That is why the code for the above plot is not visible in the HTML output file.
+
+##### Creating tables
+
+```{r tables-mtcars}
+knitr::kable(iris[1:5, ], caption = 'A caption')
+```
+
+You might have noticed `knitr` in the above code chunk. So let's briefly know about Knitr here. 
+
+##### Knitr - 
+It is a package in the programming language R that enables integration of R code into LaTeX, LyX, HTML, Markdown, AsciiDoc, and reStructuredText documents. _Note_ - Packages will be introduced in a separate tutorial.
+
+In the R Studio toolbar, where you can see the `Preview` button, if you click on the drop down arrow next to it, you will find the options `Knit to HTML`, `Knit to PDF`, `Knit to Word` etc. When you click the **Knit** button the specific output document will be generated that includes both content as well as the result of any embedded R code chunks within the notebook.
+
+Before, we end this introductory tutorial, here's a [cheatsheet](https://rmarkdown.rstudio.com/lesson-15.html) for working with R Markdown and Notebooks for easy and quick reference.
+
+Now that you've learnt about R Notebooks, you are ready to create your own and get started : Go to `File -> New File -> R Notebook`, or if you have opened an `R markdown` file, you can specify the output type as `html_notebook` in the document’s YAML metadata. Have fun working with R Notebooks!
diff --git a/Module 1/R notebook tutorial.nb.html b/Module 1/R notebook tutorial.nb.html
new file mode 100644
index 00000000..2cb5cd0a
--- /dev/null
+++ b/Module 1/R notebook tutorial.nb.html	
@@ -0,0 +1,423 @@
+
+
+
+
+
+
+
+
+
+
+
+
+
+Introduction to R notebooks
+
+
+
+
+
+
+
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+
+
+
+
+
+
+
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+
+
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+
    
+
    Welcome to statistics with R!
    
+
    This tutorial notebook is intended to introduce and make you familiar with R notebooks. Even if you have used R previously, you can take a quick glance at the tutorial and might learn something new.
    
+
    This is an R Markdown notebook file, you might have noticed that this file has the format of .Rmd. First things first,
    
+
    
+
    What the heck is Markdown?
    
+
    Markdown is a markup language for creating formatted text using a plain-text editor. The idea and terminology evolved from the “marking up” of paper manuscripts (i.e., the revision instructions by editors), which is traditionally written with a red or blue pen on authors’ manuscripts. Markdown using any language including R is just a digital version of such blue and red pen annotations.
    
+
    
+
    
+
    But what is an R Notebook then?
    
+
    R Notebook is simply an R Markdown document (a document written in the language R Markdown) with chunks that can be executed independently and interactively, with output visible immediately beneath the input. It is an implementation of Literate Programming that allows for direct interaction with R while producing a reproducible document with publication-quality output. A notebook can therefore be thought of as a special execution mode for R Markdown documents.
    
+
    With the .Rmd file format, you can make such dynamic documents with R. In fact, this tutorial file itself is an R Notebook file with both plain text (that you’re currently reading) and embeded code (which you’ll insert below).
    
+
    Before moving to code insertion, so far we’ve learnt that the R Notebook (R Markdown document) is written in R markdown (an easy-to-write markup language) and contains chunks of embedded R code.
    
+
    Now if you wish to insert a code chunk, you can do so by clicking the Insert Chunk button on the toolbar or you can by press Ctrl+Alt+I. (mac: Cmd + option + I). Let’s write our first code chunk for printing “hello”. Try it yourself: Insert a new code chunk to print “My First Markdown File”.
    
+
+
+
+
    print('Hello')
    
+
+
+
+
    Try executing the inserted chunk(s) by clicking the Run button (green arrowhead) within the chunk or by placing your cursor inside it and pressing Ctrl+Shift+Enter.
    
+
    
+
    
+
    Some more information on R Markdown
    
+
    Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. There can be several different output formats for this markdown file and you can mention it at the start of this file under output:, for R notebooks, the output type is ‘html document’ which outputs a .html file with the same name as of the .Rmd file. You might be already reading this in a .html output file in your browser, if not then you can always Preview that html file in you browser to have a look.
    
+
    To know more about R Markdown you can visit this link.
    
+
    R Markdown interface notebook.
    
+
    Suggested reading: Section 3.2 (R Notebook) and Sections 2.5, 2.6 (Markdown syntax) from the book - R Markdown: The definitive guide
    
+
    
+
    
+
    R Markdown syntax
    
+
    The text in an R Markdown document is written with the Markdown syntax. More precisely, it is Pandoc’s Markdown. The following information has been adapted form the above mentioned book.
    
+
    Note : It is suggested to view the file as HTML document for this section in order to see the effect of formatting. You can do so by clicking the Preview button in the toolbar.
    
+
    
+
    Inline formatting
    
+
    Inline text will be italic if surrounded by underscores or asterisks, e.g., _text_ or *text*. Bold text is produced using a pair of double asterisks (**text**). A pair of tildes (~) turn text to a subscript (e.g., H~3~PO~4~ renders H3PO~4). A pair of carets (^) produce a superscript (e.g., Cu^2+^ renders Cu2+).
    
+
    To mark text as inline code, use a pair of backticks, e.g., `code`.
    
+
    Hyperlinks are created using the syntax [text](link), e.g., [RStudio](https://www.rstudio.com) will output RStudio. The syntax for images is similar: just add an exclamation mark, e.g., ![alt text or image title](path/to/image). Footnotes are put inside the square brackets after a caret ^[], e.g., ^[This is a footnote.]
    
+
    
+
    
+
    Block level elements
    
+
    Section headers can be written after a number of hashtags, e.g.,
    
+
    # First-level header
    
+
    
+
    
+
    
+
    
+
    First-level header
    
+
    ## Second-level header
    
+
    
+
    Second-level header
    
+
    ### Third-level header
    
+
    
+
    Third-level header
    
+
    If you do not want a certain heading to be numbered, you can add {-} or {.unnumbered} after the heading, e.g.,
    
+
    # Preface {-}
    
+
    Unordered list items start with *, -, or +, and you can nest one list within another list by indenting the sub-list, e.g.,
    
+
    - one item
+- one item
+- one item
+    - one more item
+    - one more item
+    - one more item
    
+
    The output is:
    
+
      
+
    • one item
      • 
+
    • one item
      • 
+
    • one item
      
+
        
+
      • one more item
        • 
+
      • one more item
        • 
+
      • one more item
        • 
+
      • 
+
    
+
    Ordered list items start with numbers (you can also nest lists within lists), e.g.,
    
+
    1. the first item
+2. the second item
+3. the third item
+    - one unordered item
+    - one unordered item
    
+
    The output does not look too much different with the Markdown source:
    
+
      
+
    1. the first item
      2. 
+
    2. the second item
      3. 
+
    3. the third item
      
+
        
+
      • one unordered item
        • 
+
      • one unordered item
        • 
+
      4. 
+
    
+
    Plain code blocks can be written after three or more backticks.
    
+
    ```
+Just like this
+```
    
+
    
+
    All about Code chunks
    
+
    You inserted a code chunk in the beginning and therefore you now know that code chunks can be inserted by either using the RStudio toolbar (the Insert button) or the keyboard shortcut Ctrl + Alt + I (Cmd + Option + I on macOS). There are a lot of things you can do in a code chunk: you can produce text output, tables, or graphics. You have fine control over all these output via chunk options, which can be provided inside the curly braces (between ```{r and }). For example, you can choose hide text output via the chunk option results = 'hide', or set the figure height to 4 inches via fig.height = 4. Chunk options are separated by commas, e.g.,
    
+
    ```{r, chunk-label, results=‘hide’, fig.height=4}
    
+
    A few of the options are:
    
+
    echo = FALSE Whether to echo the source code in the output document (someone may not prefer reading your smart source code but only results)
    
+
    include = FALSE prevents code and results from appearing in the finished file. R Markdown still runs the code in the chunk, and the results can be used by other chunks.
    
+
    message = FALSE prevents messages that are generated by code from appearing in the finished file.
    
+
    warning = FALSE prevents warnings that are generated by code from appearing in the finished.
    
+
    fig.cap = "..." adds a caption to graphical results.
    
+
    
+
    Plotting figures:
    
+
    By default, figures produced by R code will be placed immediately after the code chunk they were generated from. For example:
    
+
+
+
+
    
+
+
+
+
    Notice the echo = 'FALSE' option added in the code in the source file. That is why the code for the above plot is not visible in the HTML output file.
    
+
    
+
    
+
    Creating tables
    
+
+
+
+
    knitr::kable(iris[1:5, ], caption = 'A caption')
    
+
+
+
+
    You might have noticed knitr in the above code chunk. So let’s briefly know about Knitr here.
    
+
    
+
    
+
    Knitr -
    
+
    It is a package in the programming language R that enables integration of R code into LaTeX, LyX, HTML, Markdown, AsciiDoc, and reStructuredText documents. Note - Packages will be introduced in a separate tutorial.
    
+
    In the R Studio toolbar, where you can see the Preview button, if you click on the drop down arrow next to it, you will find the options Knit to HTML, Knit to PDF, Knit to Word etc. When you click the Knit button the specific output document will be generated that includes both content as well as the result of any embedded R code chunks within the notebook.
    
+
    Before, we end this introductory tutorial, here’s a cheatsheet for working with R Markdown and Notebooks for easy and quick reference.
    
+
    Now that you’ve learnt about R Notebooks, you are ready to create your own and get started : Go to File -> New File -> R Notebook, or if you have opened an R markdown file, you can specify the output type as html_notebook in the document’s YAML metadata. Have fun working with R Notebooks!
    
+
+
    
+
    
+
    
+
    
+
    
+
+
    ---
title: "R Notebook"
output: html_notebook
---

---
title: "Introduction to R notebooks"
output: html_document
---

### Welcome to statistics with R!


This tutorial notebook is intended to introduce and make you familiar with **R notebooks**. Even if you have used R previously, you can take a quick glance at the tutorial and might learn something new.

This is an R Markdown notebook file, you might have noticed that this file has the format of *.Rmd*.
First things first,

##### What the heck is Markdown?
Markdown is a [markup language](https://en.wikipedia.org/wiki/Markup_language) for creating formatted text using a plain-text editor. The idea and terminology evolved from the "marking up" of paper manuscripts (i.e., the revision instructions by editors), which is traditionally written with a red or blue pen on authors' manuscripts. Markdown using any language including R is just a *digital* version of such blue and red pen annotations.

##### But what is an R Notebook then?
R Notebook is simply an R Markdown document (a document written in the *language* R Markdown) with chunks that can be executed independently and interactively, with output visible immediately beneath the input. It is an implementation of [Literate Programming](https://en.wikipedia.org/wiki/Literate_programming) that allows for direct interaction with R while producing a reproducible document with publication-quality output. A notebook can therefore be thought of as a special execution mode for R Markdown documents.

With the .Rmd file format, you can make such dynamic documents with R. In fact, this tutorial file itself is an R Notebook file with both plain text (that you're currently reading) and embeded code (which you'll insert below). 

Before moving to code insertion, so far we've learnt that the R Notebook (R Markdown document)  is written in R markdown (an easy-to-write markup language) and contains chunks of embedded R code. 

Now if you wish to insert a code chunk, you can do so by clicking the *Insert Chunk* button on the toolbar or you can by press *Ctrl+Alt+I*. (*mac: Cmd + option + I*). Let's write our first code chunk for printing "hello". *Try it yourself:* Insert a new code chunk to print "My First Markdown File".
```{r}
print('Hello')
```

Try executing the inserted chunk(s) by clicking the *Run* button (green arrowhead) within the chunk or by placing your cursor inside it and pressing *Ctrl+Shift+Enter*. 

##### Some more information on R Markdown
Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. There can be several different output formats for this markdown file and you can mention it at the start of this file under output:, for R notebooks, the output type is 'html document' which outputs a .html file with the same name as of the .Rmd file. You might be already reading this in a .html output file in your browser, if not then you can always *Preview* that html file in you browser to have a look.

To know more about R Markdown you can visit this [link](http://rmarkdown.rstudio.com). 

[R Markdown interface](https://rmarkdown.rstudio.com/lesson-2.html) notebook. 

**_Suggested reading:_** Section 3.2 (R Notebook) and Sections 2.5, 2.6 (Markdown syntax) from the book - [R Markdown: The definitive guide](https://bookdown.org/yihui/rmarkdown/)

#### R Markdown syntax
The text in an R Markdown document is written with the Markdown syntax. More precisely, it is [Pandoc’s Markdown](https://pandoc.org/MANUAL.html). The following information has been adapted form the above mentioned book.

_Note_ : It is suggested to view the file as HTML document for this section in order to see the effect of formatting. You can do so by clicking the Preview button in the toolbar.

##### Inline formatting

Inline text will be _italic_ if surrounded by underscores or asterisks, e.g., `_text_` or `*text*`. **Bold** text is produced using a pair of double asterisks (`**text**`). A pair of tildes (~) turn text to a subscript (e.g., `H~3~PO~4~` renders H~3~PO~4). A pair of carets (^) produce a superscript (e.g., `Cu^2+^` renders Cu^2+^). 

To mark text as inline code, use a pair of backticks, e.g., `` `code` ``.

Hyperlinks are created using the syntax `[text](link)`, e.g., `[RStudio](https://www.rstudio.com)` will output [RStudio](https://www.rstudio.com). The syntax for images is similar: just add an exclamation mark, e.g., `![alt text or image title](path/to/image)`. Footnotes are put inside the square brackets after a caret `^[]`, e.g., `^[This is a footnote.]`

##### Block level elements
Section headers can be written after a number of hashtags, e.g.,

`# First-level header`

# First-level header

`## Second-level header`

## Second-level header

`### Third-level header`

### Third-level header

If you do not want a certain heading to be numbered, you can add {-} or {.unnumbered} after the heading, e.g.,

`# Preface {-}`

Unordered list items start with *, -, or +, and you can nest one list within another list by indenting the sub-list, e.g.,

```
- one item
- one item
- one item
    - one more item
    - one more item
    - one more item
```

The output is:

- one item

- one item

- one item

  - one more item
  
  - one more item
  
  - one more item

Ordered list items start with numbers (you can also nest lists within lists), e.g.,

```
1. the first item
2. the second item
3. the third item
    - one unordered item
    - one unordered item
```

The output does not look too much different with the Markdown source:

1. the first item

2. the second item

3. the third item

    - one unordered item
    
    - one unordered item
    
Plain code blocks can be written after three or more backticks.
````
```
Just like this
```
````

#### All about Code chunks
You inserted a code chunk in the beginning and therefore you now know that code chunks can be inserted by either using the RStudio toolbar (the `Insert` button) or the keyboard shortcut `Ctrl + Alt + I` (`Cmd + Option + I` on macOS). There are a lot of things you can do in a code chunk: you can produce text output, tables, or graphics. You have fine control over all these output via chunk options, which can be provided inside the curly braces (between ```` ```{r and }````). For example, you can choose hide text output via the chunk option `results = 'hide'`, or set the figure height to 4 inches via `fig.height = 4`. Chunk options are separated by commas, e.g.,

**```{r, chunk-label, results='hide', fig.height=4}**

A few of the options are:

`echo = FALSE` Whether to echo the source code in the output document (someone may not prefer reading your smart source code but only results)

`include = FALSE` prevents code and results from appearing in the finished file. R Markdown still runs the code in the chunk, and the results can be used by other chunks.

`message = FALSE` prevents messages that are generated by code from appearing in the finished file.

`warning = FALSE` prevents warnings that are generated by code from appearing in the finished.

`fig.cap = "..."` adds a caption to graphical results.

##### Plotting figures:
By default, figures produced by R code will be placed immediately after the code chunk they were generated from. For example:

```{r pressure, echo=FALSE}
plot(pressure)
```
Notice the `echo = 'FALSE'` option added in the code in the source file. That is why the code for the above plot is not visible in the HTML output file.

##### Creating tables

```{r tables-mtcars}
knitr::kable(iris[1:5, ], caption = 'A caption')
```

You might have noticed `knitr` in the above code chunk. So let's briefly know about Knitr here. 

##### Knitr - 
It is a package in the programming language R that enables integration of R code into LaTeX, LyX, HTML, Markdown, AsciiDoc, and reStructuredText documents. _Note_ - Packages will be introduced in a separate tutorial.

In the R Studio toolbar, where you can see the `Preview` button, if you click on the drop down arrow next to it, you will find the options `Knit to HTML`, `Knit to PDF`, `Knit to Word` etc. When you click the **Knit** button the specific output document will be generated that includes both content as well as the result of any embedded R code chunks within the notebook.

Before, we end this introductory tutorial, here's a [cheatsheet](https://rmarkdown.rstudio.com/lesson-15.html) for working with R Markdown and Notebooks for easy and quick reference.

Now that you've learnt about R Notebooks, you are ready to create your own and get started : Go to `File -> New File -> R Notebook`, or if you have opened an `R markdown` file, you can specify the output type as `html_notebook` in the document’s YAML metadata. Have fun working with R Notebooks!

    
+
+
+
+
    
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/Module 1/README.md b/Module 1/README.md
new file mode 100644
index 00000000..ccb19cd1
--- /dev/null
+++ b/Module 1/README.md	
@@ -0,0 +1 @@
+# BSE658_chapter1
\ No newline at end of file
diff --git a/Module 2/README.md b/Module 2/README.md
new file mode 100644
index 00000000..90701a98
--- /dev/null
+++ b/Module 2/README.md	
@@ -0,0 +1,10 @@
+## BSE658: Chapter 2
+
+### This repository covers two important things:
+
+1.  Tidyverse
+2.  ggplot2
+
+`Tidyverse.rmd` file explains a few of the `tidyverse` packages in order to handle data frames. `ggplot.rmd` explains how to use `ggplot2` package to create beautiful plots. 
+
+The html files can be downloaded and opened to view in your browser.
diff --git a/Module 2/StatewiseTestingDetails.csv b/Module 2/StatewiseTestingDetails.csv
new file mode 100644
index 00000000..079c7f75
--- /dev/null
+++ b/Module 2/StatewiseTestingDetails.csv	
@@ -0,0 +1,14099 @@
+Date,State,TotalSamples,Negative,Positive
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+2020-11-09,Andaman and Nicobar Islands,100025.0,,4463.0
+2020-11-10,Andaman and Nicobar Islands,101226.0,,4477.0
+2020-11-11,Andaman and Nicobar Islands,102433.0,,4494.0
+2020-11-12,Andaman and Nicobar Islands,103599.0,,4507.0
+2020-11-13,Andaman and Nicobar Islands,104785.0,,4518.0
+2020-11-14,Andaman and Nicobar Islands,105699.0,,4527.0
+2020-11-15,Andaman and Nicobar Islands,106734.0,,4536.0
+2020-11-16,Andaman and Nicobar Islands,108080.0,,4557.0
+2020-11-17,Andaman and Nicobar Islands,109646.0,,4574.0
+2020-11-18,Andaman and Nicobar Islands,111166.0,,4593.0
+2020-11-19,Andaman and Nicobar Islands,112792.0,,4604.0
+2020-11-20,Andaman and Nicobar Islands,114392.0,,4617.0
+2020-11-21,Andaman and Nicobar Islands,115968.0,,4631.0
+2020-11-22,Andaman and Nicobar Islands,117093.0,,4641.0
+2020-11-23,Andaman and Nicobar Islands,118654.0,,4656.0
+2020-11-24,Andaman and Nicobar Islands,120324.0,,4667.0
+2020-11-25,Andaman and Nicobar Islands,121931.0,,4677.0
+2020-11-26,Andaman and Nicobar Islands,123165.0,,4682.0
+2020-11-27,Andaman and Nicobar Islands,124832.0,,4689.0
+2020-11-28,Andaman and Nicobar Islands,126331.0,,4698.0
+2020-11-29,Andaman and Nicobar Islands,127912.0,,4704.0
+2020-11-30,Andaman and Nicobar Islands,129389.0,,4710.0
+2020-12-01,Andaman and Nicobar Islands,131080.0,,4718.0
+2020-12-02,Andaman and Nicobar Islands,132784.0,,4723.0
+2020-12-03,Andaman and Nicobar Islands,134355.0,,4729.0
+2020-12-04,Andaman and Nicobar Islands,135952.0,,4742.0
+2020-12-05,Andaman and Nicobar Islands,137648.0,,4747.0
+2020-12-06,Andaman and Nicobar Islands,139326.0,,4758.0
+2020-12-07,Andaman and Nicobar Islands,140961.0,,4773.0
+2020-12-08,Andaman and Nicobar Islands,142667.0,,4778.0
+2020-12-09,Andaman and Nicobar Islands,144401.0,,4783.0
+2020-12-10,Andaman and Nicobar Islands,146143.0,,4791.0
+2020-12-11,Andaman and Nicobar Islands,147806.0,,4798.0
+2020-12-12,Andaman and Nicobar Islands,149926.0,,4805.0
+2020-12-13,Andaman and Nicobar Islands,152004.0,,4818.0
+2020-12-14,Andaman and Nicobar Islands,153685.0,,4834.0
+2020-12-15,Andaman and Nicobar Islands,155454.0,,4842.0
+2020-12-16,Andaman and Nicobar Islands,157242.0,,4850.0
+2020-12-17,Andaman and Nicobar Islands,159027.0,,4860.0
+2020-12-18,Andaman and Nicobar Islands,160856.0,,4869.0
+2020-12-19,Andaman and Nicobar Islands,162821.0,,4875.0
+2020-12-20,Andaman and Nicobar Islands,165024.0,,4881.0
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+2020-12-24,Andaman and Nicobar Islands,172756.0,,4901.0
+2020-12-25,Andaman and Nicobar Islands,174482.0,,4912.0
+2020-12-26,Andaman and Nicobar Islands,175578.0,,4912.0
+2020-12-27,Andaman and Nicobar Islands,176898.0,,4924.0
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+2020-12-29,Andaman and Nicobar Islands,179444.0,,4936.0
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+2021-01-03,Andaman and Nicobar Islands,184955.0,,4948.0
+2021-01-04,Andaman and Nicobar Islands,186140.0,,4948.0
+2021-01-05,Andaman and Nicobar Islands,187115.0,,4949.0
+2021-01-06,Andaman and Nicobar Islands,188351.0,,4949.0
+2021-01-07,Andaman and Nicobar Islands,190031.0,,4958.0
+2021-01-08,Andaman and Nicobar Islands,191206.0,,4959.0
+2021-01-09,Andaman and Nicobar Islands,192560.0,,4959.0
+2021-01-10,Andaman and Nicobar Islands,193719.0,,4960.0
+2021-01-11,Andaman and Nicobar Islands,195351.0,,4963.0
+2021-01-12,Andaman and Nicobar Islands,196859.0,,4966.0
+2021-01-13,Andaman and Nicobar Islands,198478.0,,4969.0
+2021-01-14,Andaman and Nicobar Islands,199805.0,,4976.0
+2021-01-15,Andaman and Nicobar Islands,200953.0,,4979.0
+2021-01-16,Andaman and Nicobar Islands,202521.0,,4982.0
+2021-01-17,Andaman and Nicobar Islands,203741.0,,4983.0
+2021-01-18,Andaman and Nicobar Islands,205411.0,,4988.0
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+2021-01-20,Andaman and Nicobar Islands,208179.0,,4991.0
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+2021-01-24,Andaman and Nicobar Islands,213467.0,,4993.0
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+2021-01-28,Andaman and Nicobar Islands,218629.0,,4994.0
+2021-01-29,Andaman and Nicobar Islands,220484.0,,4994.0
+2021-01-30,Andaman and Nicobar Islands,221876.0,,4994.0
+2021-01-31,Andaman and Nicobar Islands,223137.0,,4994.0
+2021-02-01,Andaman and Nicobar Islands,224852.0,,4994.0
+2021-02-02,Andaman and Nicobar Islands,226584.0,,4994.0
+2021-02-03,Andaman and Nicobar Islands,228290.0,,4994.0
+2021-02-04,Andaman and Nicobar Islands,230133.0,,4994.0
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+2021-02-12,Andaman and Nicobar Islands,243631.0,,5007.0
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+2021-03-02,Andaman and Nicobar Islands,271193.0,,5020.0
+2021-03-03,Andaman and Nicobar Islands,273010.0,,5022.0
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+2021-03-05,Andaman and Nicobar Islands,276471.0,,5024.0
+2021-03-06,Andaman and Nicobar Islands,278437.0,,5024.0
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+2021-03-09,Andaman and Nicobar Islands,283492.0,,5028.0
+2021-03-10,Andaman and Nicobar Islands,285087.0,,5028.0
+2021-03-11,Andaman and Nicobar Islands,286505.0,,5028.0
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+2021-03-13,Andaman and Nicobar Islands,290341.0,,5030.0
+2021-03-14,Andaman and Nicobar Islands,291842.0,,5031.0
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+2021-03-30,Andaman and Nicobar Islands,318382.0,,5081.0
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+2021-04-01,Andaman and Nicobar Islands,321456.0,,5084.0
+2021-04-02,Andaman and Nicobar Islands,323001.0,,5084.0
+2021-04-03,Andaman and Nicobar Islands,324207.0,,5098.0
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+2021-04-05,Andaman and Nicobar Islands,327270.0,,5116.0
+2021-04-06,Andaman and Nicobar Islands,328737.0,,5123.0
+2021-04-07,Andaman and Nicobar Islands,330481.0,,5131.0
+2021-04-08,Andaman and Nicobar Islands,332119.0,,5149.0
+2021-04-09,Andaman and Nicobar Islands,333671.0,,5161.0
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+2021-05-02,Andaman and Nicobar Islands,373007.0,,6084.0
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+2021-05-15,Andaman and Nicobar Islands,379096.0,,6568.0
+2021-05-16,Andaman and Nicobar Islands,379674.0,,6603.0
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+2021-06-03,Andaman and Nicobar Islands,388486.0,,7070.0
+2021-06-04,Andaman and Nicobar Islands,389152.0,,7088.0
+2021-06-05,Andaman and Nicobar Islands,389785.0,,7105.0
+2021-06-06,Andaman and Nicobar Islands,390574.0,,7119.0
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+2020-04-09,Dadra and Nagar Haveli and Daman and Diu,80.0,80,0.0
+2020-04-10,Dadra and Nagar Haveli and Daman and Diu,130.0,130,0.0
+2020-04-11,Dadra and Nagar Haveli and Daman and Diu,211.0,211,0.0
+2020-04-15,Dadra and Nagar Haveli and Daman and Diu,356.0,356,0.0
+2020-04-16,Dadra and Nagar Haveli and Daman and Diu,382.0,382,0.0
+2020-04-21,Dadra and Nagar Haveli and Daman and Diu,474.0,474,0.0
+2020-04-22,Dadra and Nagar Haveli and Daman and Diu,674.0,674,0.0
+2020-04-23,Dadra and Nagar Haveli and Daman and Diu,913.0,913,0.0
+2020-04-25,Dadra and Nagar Haveli and Daman and Diu,1386.0,1386,0.0
+2020-04-26,Dadra and Nagar Haveli and Daman and Diu,1901.0,1901,0.0
+2020-04-27,Dadra and Nagar Haveli and Daman and Diu,2379.0,2379,0.0
+2020-04-29,Dadra and Nagar Haveli and Daman and Diu,2921.0,2921,0.0
+2020-04-30,Dadra and Nagar Haveli and Daman and Diu,3464.0,3464,0.0
+2020-05-01,Dadra and Nagar Haveli and Daman and Diu,4003.0,4003,0.0
+2020-05-03,Dadra and Nagar Haveli and Daman and Diu,4328.0,4328,0.0
+2020-05-04,Dadra and Nagar Haveli and Daman and Diu,4434.0,4434,0.0
+2020-05-05,Dadra and Nagar Haveli and Daman and Diu,4781.0,4780,1.0
+2020-05-06,Dadra and Nagar Haveli and Daman and Diu,4967.0,4966,1.0
+2020-05-07,Dadra and Nagar Haveli and Daman and Diu,5272.0,5271,1.0
+2020-05-08,Dadra and Nagar Haveli and Daman and Diu,5430.0,5429,1.0
+2020-05-10,Dadra and Nagar Haveli and Daman and Diu,5581.0,5580,1.0
+2020-05-11,Dadra and Nagar Haveli and Daman and Diu,5763.0,5762,1.0
+2020-05-12,Dadra and Nagar Haveli and Daman and Diu,5977.0,5976,1.0
+2020-05-13,Dadra and Nagar Haveli and Daman and Diu,6294.0,6293,1.0
+2020-05-14,Dadra and Nagar Haveli and Daman and Diu,6554.0,6553,1.0
+2020-05-15,Dadra and Nagar Haveli and Daman and Diu,6916.0,6915,1.0
+2020-05-17,Dadra and Nagar Haveli and Daman and Diu,7157.0,7156,1.0
+2020-05-18,Dadra and Nagar Haveli and Daman and Diu,7571.0,7570,1.0
+2020-05-19,Dadra and Nagar Haveli and Daman and Diu,7829.0,7828,1.0
+2020-05-20,Dadra and Nagar Haveli and Daman and Diu,8137.0,8136,1.0
+2020-05-21,Dadra and Nagar Haveli and Daman and Diu,8456.0,8455,1.0
+2020-05-22,Dadra and Nagar Haveli and Daman and Diu,8979.0,8978,1.0
+2020-05-23,Dadra and Nagar Haveli and Daman and Diu,9298.0,9296,2.0
+2020-05-24,Dadra and Nagar Haveli and Daman and Diu,9678.0,9676,2.0
+2020-05-25,Dadra and Nagar Haveli and Daman and Diu,10085.0,10083,2.0
+2020-05-26,Dadra and Nagar Haveli and Daman and Diu,10388.0,10386,2.0
+2020-05-27,Dadra and Nagar Haveli and Daman and Diu,10872.0,10870,2.0
+2020-05-28,Dadra and Nagar Haveli and Daman and Diu,11177.0,11175,2.0
+2020-05-29,Dadra and Nagar Haveli and Daman and Diu,11477.0,11475,2.0
+2020-06-01,Dadra and Nagar Haveli and Daman and Diu,11693.0,11690,3.0
+2020-06-03,Dadra and Nagar Haveli and Daman and Diu,12064.0,12052,12.0
+2020-06-04,Dadra and Nagar Haveli and Daman and Diu,12375.0,12361,14.0
+2020-06-05,Dadra and Nagar Haveli and Daman and Diu,13667.0,13653,14.0
+2020-06-06,Dadra and Nagar Haveli and Daman and Diu,14780.0,14761,19.0
+2020-06-07,Dadra and Nagar Haveli and Daman and Diu,15546.0,14926,20.0
+2020-06-08,Dadra and Nagar Haveli and Daman and Diu,16277.0,15655,22.0
+2020-06-10,Dadra and Nagar Haveli and Daman and Diu,17179.0,16552,27.0
+2020-06-13,Dadra and Nagar Haveli and Daman and Diu,17649.0,17021,28.0
+2020-06-14,Dadra and Nagar Haveli and Daman and Diu,18445.0,17813,31.0
+2020-06-15,Dadra and Nagar Haveli and Daman and Diu,23835.0,23072,38.0
+2020-06-16,Dadra and Nagar Haveli and Daman and Diu,24492.0,23695,51.0
+2020-06-17,Dadra and Nagar Haveli and Daman and Diu,25575.0,24650,58.0
+2020-06-18,Dadra and Nagar Haveli and Daman and Diu,26752.0,25461,69.0
+2020-06-20,Dadra and Nagar Haveli and Daman and Diu,27617.0,26901,93.0
+2020-06-21,Dadra and Nagar Haveli and Daman and Diu,27839.0,27241,97.0
+2020-06-22,Dadra and Nagar Haveli and Daman and Diu,28364.0,27572,112.0
+2020-06-23,Dadra and Nagar Haveli and Daman and Diu,29072.0,28096,121.0
+2020-06-24,Dadra and Nagar Haveli and Daman and Diu,29403.0,28697,134.0
+2020-06-25,Dadra and Nagar Haveli and Daman and Diu,30188.0,29015,154.0
+2020-06-26,Dadra and Nagar Haveli and Daman and Diu,30602.0,29864,169.0
+2020-06-27,Dadra and Nagar Haveli and Daman and Diu,31015.0,30316,184.0
+2020-06-28,Dadra and Nagar Haveli and Daman and Diu,31321.0,30781,188.0
+2020-06-29,Dadra and Nagar Haveli and Daman and Diu,31602.0,30977,203.0
+2020-06-30,Dadra and Nagar Haveli and Daman and Diu,32031.0,31396,213.0
+2020-07-01,Dadra and Nagar Haveli and Daman and Diu,32435.0,31783,229.0
+2020-07-02,Dadra and Nagar Haveli and Daman and Diu,32834.0,32157,254.0
+2020-07-03,Dadra and Nagar Haveli and Daman and Diu,33618.0,32899,304.0
+2020-07-04,Dadra and Nagar Haveli and Daman and Diu,33995.0,33156,333.0
+2020-07-05,Dadra and Nagar Haveli and Daman and Diu,34179.0,33341,371.0
+2020-07-06,Dadra and Nagar Haveli and Daman and Diu,34444.0,33441,398.0
+2020-07-07,Dadra and Nagar Haveli and Daman and Diu,34761.0,33926,423.0
+2020-07-08,Dadra and Nagar Haveli and Daman and Diu,35234.0,34176,440.0
+2020-07-09,Dadra and Nagar Haveli and Daman and Diu,35391.0,34585,454.0
+2020-07-10,Dadra and Nagar Haveli and Daman and Diu,35829.0,34827,468.0
+2020-07-11,Dadra and Nagar Haveli and Daman and Diu,36166.0,35279,480.0
+2020-07-12,Dadra and Nagar Haveli and Daman and Diu,36310.0,35611,495.0
+2020-07-13,Dadra and Nagar Haveli and Daman and Diu,36621.0,35696,520.0
+2020-07-14,Dadra and Nagar Haveli and Daman and Diu,36905.0,35963,536.0
+2020-07-15,Dadra and Nagar Haveli and Daman and Diu,37392.0,36195,552.0
+2020-07-16,Dadra and Nagar Haveli and Daman and Diu,37880.0,36548,570.0
+2020-07-17,Dadra and Nagar Haveli and Daman and Diu,38176.0,36898,601.0
+2020-07-18,Dadra and Nagar Haveli and Daman and Diu,38313.0,37291,646.0
+2020-07-19,Dadra and Nagar Haveli and Daman and Diu,38640.0,37520,684.0
+2020-07-20,Dadra and Nagar Haveli and Daman and Diu,38880.0,37789,705.0
+2020-07-21,Dadra and Nagar Haveli and Daman and Diu,39199.0,38022,733.0
+2020-07-24,Dadra and Nagar Haveli and Daman and Diu,40425.0,39019,854.0
+2020-07-25,Dadra and Nagar Haveli and Daman and Diu,40881.0,39431,890.0
+2020-07-26,Dadra and Nagar Haveli and Daman and Diu,41143.0,39815,940.0
+2020-07-28,Dadra and Nagar Haveli and Daman and Diu,41588.0,40258,1020.0
+2020-07-29,Dadra and Nagar Haveli and Daman and Diu,41895.0,40411,1058.0
+2020-07-30,Dadra and Nagar Haveli and Daman and Diu,42148.0,40564,1093.0
+2020-07-31,Dadra and Nagar Haveli and Daman and Diu,42403.0,40775,1136.0
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+2020-08-02,Dadra and Nagar Haveli and Daman and Diu,42718.0,41132,1232.0
+2020-08-03,Dadra and Nagar Haveli and Daman and Diu,42926.0,41402,1265.0
+2020-08-04,Dadra and Nagar Haveli and Daman and Diu,43191.0,41514,1304.0
+2020-08-05,Dadra and Nagar Haveli and Daman and Diu,43478.0,41781,1368.0
+2020-08-06,Dadra and Nagar Haveli and Daman and Diu,43899.0,42059,1412.0
+2020-08-07,Dadra and Nagar Haveli and Daman and Diu,44456.0,42608,1456.0
+2020-08-08,Dadra and Nagar Haveli and Daman and Diu,44777.0,42805,1505.0
+2020-08-09,Dadra and Nagar Haveli and Daman and Diu,45253.0,43165,
+2020-08-10,Dadra and Nagar Haveli and Daman and Diu,45724.0,43579,
+2020-08-11,Dadra and Nagar Haveli and Daman and Diu,46149.0,43919,
+2020-08-12,Dadra and Nagar Haveli and Daman and Diu,46516.0,44328,
+2020-08-13,Dadra and Nagar Haveli and Daman and Diu,46937.0,44712,
+2020-08-14,Dadra and Nagar Haveli and Daman and Diu,47272.0,45021,1797.0
+2020-08-15,Dadra and Nagar Haveli and Daman and Diu,47643.0,45335,1843.0
+2020-08-16,Dadra and Nagar Haveli and Daman and Diu,48212.0,45676,1878.0
+2020-08-17,Dadra and Nagar Haveli and Daman and Diu,48710.0,46224,1908.0
+2020-08-18,Dadra and Nagar Haveli and Daman and Diu,49290.0,46665,1951.0
+2020-08-19,Dadra and Nagar Haveli and Daman and Diu,49879.0,47184,1995.0
+2020-08-20,Dadra and Nagar Haveli and Daman and Diu,50293.0,47703,2030.0
+2020-08-21,Dadra and Nagar Haveli and Daman and Diu,50923.0,48105,2082.0
+2020-08-22,Dadra and Nagar Haveli and Daman and Diu,51285.0,48524,2120.0
+2020-08-23,Dadra and Nagar Haveli and Daman and Diu,51580.0,48995,2161.0
+2020-08-24,Dadra and Nagar Haveli and Daman and Diu,51843.0,49322,2181.0
+2020-08-25,Dadra and Nagar Haveli and Daman and Diu,52160.0,49538,2203.0
+2020-08-26,Dadra and Nagar Haveli and Daman and Diu,52524.0,49806,2228.0
+2020-08-27,Dadra and Nagar Haveli and Daman and Diu,52859.0,50236,2266.0
+2020-08-28,Dadra and Nagar Haveli and Daman and Diu,53227.0,50539,2290.0
+2020-08-29,Dadra and Nagar Haveli and Daman and Diu,53527.0,50809,2306.0
+2020-08-30,Dadra and Nagar Haveli and Daman and Diu,53769.0,51141,2340.0
+2020-08-31,Dadra and Nagar Haveli and Daman and Diu,54194.0,48426,
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+2020-09-15,Dadra and Nagar Haveli and Daman and Diu,59530.0,56305,2783.0
+2020-09-16,Dadra and Nagar Haveli and Daman and Diu,59771.0,56646,2810.0
+2020-09-17,Dadra and Nagar Haveli and Daman and Diu,60237.0,56905,2831.0
+2020-09-18,Dadra and Nagar Haveli and Daman and Diu,60738.0,57338,2859.0
+2020-09-19,Dadra and Nagar Haveli and Daman and Diu,61270.0,57759,2887.0
+2020-09-20,Dadra and Nagar Haveli and Daman and Diu,61482.0,58254,2924.0
+2020-09-21,Dadra and Nagar Haveli and Daman and Diu,62055.0,58459,2933.0
+2020-09-23,Dadra and Nagar Haveli and Daman and Diu,63047.0,59438,2962.0
+2020-09-24,Dadra and Nagar Haveli and Daman and Diu,63512.0,59995,2977.0
+2020-09-26,Dadra and Nagar Haveli and Daman and Diu,64336.0,60848,2996.0
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+2020-09-28,Dadra and Nagar Haveli and Daman and Diu,65134.0,61618,3012.0
+2020-09-29,Dadra and Nagar Haveli and Daman and Diu,65539.0,62045,3032.0
+2020-09-30,Dadra and Nagar Haveli and Daman and Diu,65953.0,62439,3040.0
+2020-10-01,Dadra and Nagar Haveli and Daman and Diu,66337.0,62840,3054.0
+2020-10-02,Dadra and Nagar Haveli and Daman and Diu,66566.0,63228,3063.0
+2020-10-03,Dadra and Nagar Haveli and Daman and Diu,67029.0,63509,3070.0
+2020-10-04,Dadra and Nagar Haveli and Daman and Diu,67272.0,63873,3087.0
+2020-10-05,Dadra and Nagar Haveli and Daman and Diu,67666.0,64129,3092.0
+2020-10-06,Dadra and Nagar Haveli and Daman and Diu,68056.0,64498,3103.0
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+2020-10-09,Dadra and Nagar Haveli and Daman and Diu,69192.0,65638,3140.0
+2020-10-10,Dadra and Nagar Haveli and Daman and Diu,69529.0,65980,3152.0
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+2020-10-14,Dadra and Nagar Haveli and Daman and Diu,70815.0,67297,3171.0
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+2020-10-16,Dadra and Nagar Haveli and Daman and Diu,71386.0,67909,3177.0
+2020-10-17,Dadra and Nagar Haveli and Daman and Diu,71664.0,68164,3180.0
+2020-10-18,Dadra and Nagar Haveli and Daman and Diu,71807.0,68439,3183.0
+2020-10-19,Dadra and Nagar Haveli and Daman and Diu,72136.0,68578,3185.0
+2020-10-20,Dadra and Nagar Haveli and Daman and Diu,72410.0,68894,3194.0
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+2021-04-22,Himachal Pradesh,1419314.0,1334242,
+2021-04-23,Himachal Pradesh,1429699.0,1340858,
+2021-04-24,Himachal Pradesh,1440233.0,1349906,
+2021-04-25,Himachal Pradesh,1447397.0,1354023,
+2021-04-26,Himachal Pradesh,1456795.0,1361455,
+2021-04-27,Himachal Pradesh,1467876.0,1369301,
+2021-04-28,Himachal Pradesh,1482357.0,1380796,
+2021-04-29,Himachal Pradesh,1497617.0,1392000,
+2021-04-30,Himachal Pradesh,1509568.0,1398490,
+2021-05-01,Himachal Pradesh,1521350.0,1408695,
+2021-05-02,Himachal Pradesh,1531034.0,1416323,
+2021-05-03,Himachal Pradesh,1539545.0,1424035,
+2021-05-04,Himachal Pradesh,1555499.0,1436089,
+2021-05-05,Himachal Pradesh,1571240.0,1446837,
+2021-05-06,Himachal Pradesh,1590372.0,1459013,
+2021-05-07,Himachal Pradesh,1605831.0,1470211,
+2021-05-08,Himachal Pradesh,1624556.0,1484123,
+2021-05-09,Himachal Pradesh,1633923.0,1491904,
+2021-05-10,Himachal Pradesh,1645146.0,1502326,
+2021-05-11,Himachal Pradesh,1662432.0,1514018,
+2021-05-12,Himachal Pradesh,1679291.0,1527372,
+2021-05-13,Himachal Pradesh,1696695.0,1540650,
+2021-05-14,Himachal Pradesh,1708083.0,1549618,
+2021-05-15,Himachal Pradesh,1723544.0,1562292,
+2021-05-16,Himachal Pradesh,1733730.0,1569635,
+2021-05-17,Himachal Pradesh,1747264.0,1581759,
+2021-05-18,Himachal Pradesh,1763099.0,1592882,
+2021-05-19,Himachal Pradesh,1777957.0,1605855,
+2021-05-20,Himachal Pradesh,1792460.0,1616797,
+2021-05-21,Himachal Pradesh,1806900.0,1629443,
+2021-05-22,Himachal Pradesh,1822120.0,1642876,
+2021-05-23,Himachal Pradesh,1829865.0,1648850,
+2021-05-24,Himachal Pradesh,1841031.0,1659536,
+2021-05-25,Himachal Pradesh,1856038.0,1670759,
+2021-05-26,Himachal Pradesh,1864817.0,1678501,
+2021-05-27,Himachal Pradesh,1875554.0,1688775,
+2021-05-28,Himachal Pradesh,1888929.0,1699998,
+2021-05-29,Himachal Pradesh,1902322.0,1711309,
+2021-05-30,Himachal Pradesh,1910855.0,1720705,
+2021-05-31,Himachal Pradesh,1922211.0,1731490,
+2021-06-01,Himachal Pradesh,1935704.0,1743195,
+2021-06-02,Himachal Pradesh,1950718.0,1756703,
+2021-06-03,Himachal Pradesh,1966968.0,1772896,
+2021-06-04,Himachal Pradesh,1986041.0,1791377,
+2021-06-05,Himachal Pradesh,2005214.0,1809866,
+2021-06-06,Himachal Pradesh,2017058.0,1820983,
+2021-06-07,Himachal Pradesh,2035502.0,1839411,
+2020-04-05,Jammu and Kashmir,1551.0,1429,106.0
+2020-04-10,Jammu and Kashmir,2961.0,2754,207.0
+2020-04-11,Jammu and Kashmir,3206.0,2982,224.0
+2020-04-12,Jammu and Kashmir,3600.0,3355,245.0
+2020-04-13,Jammu and Kashmir,4065.0,3795,270.0
+2020-04-14,Jammu and Kashmir,4619.0,4341,278.0
+2020-04-15,Jammu and Kashmir,5171.0,4871,300.0
+2020-04-16,Jammu and Kashmir,5680.0,5366,314.0
+2020-04-17,Jammu and Kashmir,6438.0,6110,328.0
+2020-04-18,Jammu and Kashmir,6937.0,6596,341.0
+2020-04-19,Jammu and Kashmir,7895.0,7545,350.0
+2020-04-20,Jammu and Kashmir,8612.0,8244,368.0
+2020-04-21,Jammu and Kashmir,9220.0,8840,380.0
+2020-04-22,Jammu and Kashmir,10039.0,9632,407.0
+2020-04-23,Jammu and Kashmir,10977.0,10550,427.0
+2020-04-24,Jammu and Kashmir,11764.0,11310,454.0
+2020-04-25,Jammu and Kashmir,12835.0,12341,494.0
+2020-04-26,Jammu and Kashmir,13959.0,13436,523.0
+2020-04-27,Jammu and Kashmir,14988.0,14442,546.0
+2020-04-28,Jammu and Kashmir,16619.0,16054,565.0
+2020-04-29,Jammu and Kashmir,18450.0,17869,581.0
+2020-04-30,Jammu and Kashmir,19746.0,19132,614.0
+2020-05-01,Jammu and Kashmir,21695.0,21056,639.0
+2020-05-02,Jammu and Kashmir,23406.0,22794,666.0
+2020-05-03,Jammu and Kashmir,26038.0,25337,701.0
+2020-05-04,Jammu and Kashmir,28199.0,27473,726.0
+2020-05-05,Jammu and Kashmir,31312.0,30571,741.0
+2020-05-06,Jammu and Kashmir,34277.0,33502,775.0
+2020-05-07,Jammu and Kashmir,37706.0,36913,793.0
+2020-05-08,Jammu and Kashmir,42427.0,41604,823.0
+2020-05-09,Jammu and Kashmir,44753.0,43917,836.0
+2020-05-10,Jammu and Kashmir,47080.0,46291,861.0
+2020-05-11,Jammu and Kashmir,51334.0,50455,879.0
+2020-05-12,Jammu and Kashmir,53726.0,52792,934.0
+2020-05-13,Jammu and Kashmir,57111.0,56140,971.0
+2020-05-14,Jammu and Kashmir,63515.0,62532,983.0
+2020-05-15,Jammu and Kashmir,70306.0,69293,1013.0
+2020-05-16,Jammu and Kashmir,76191.0,75070,1121.0
+2020-05-17,Jammu and Kashmir,80934.0,79751,1183.0
+2020-05-18,Jammu and Kashmir,88601.0,87312,1289.0
+2020-05-19,Jammu and Kashmir,96826.0,95509,1317.0
+2020-05-20,Jammu and Kashmir,101950.0,100560,1390.0
+2020-05-21,Jammu and Kashmir,107108.0,105659,1449.0
+2020-05-22,Jammu and Kashmir,114859.0,113370,1489.0
+2020-05-23,Jammu and Kashmir,124074.0,122505,1569.0
+2020-05-24,Jammu and Kashmir,130433.0,128812,1621.0
+2020-05-25,Jammu and Kashmir,134188.0,132520,1668.0
+2020-05-26,Jammu and Kashmir,140962.0,139203,1759.0
+2020-05-27,Jammu and Kashmir,145162.0,143241,1921.0
+2020-05-28,Jammu and Kashmir,153522.0,151486,2036.0
+2020-05-29,Jammu and Kashmir,158729.0,156565,2164.0
+2020-05-30,Jammu and Kashmir,164581.0,162240,2341.0
+2020-05-31,Jammu and Kashmir,171045.0,168599,2446.0
+2020-06-01,Jammu and Kashmir,176309.0,173708,2601.0
+2020-06-02,Jammu and Kashmir,183067.0,180349,2718.0
+2020-06-03,Jammu and Kashmir,189364.0,186507,2857.0
+2020-06-04,Jammu and Kashmir,195677.0,192535,3142.0
+2020-06-05,Jammu and Kashmir,202257.0,198933,3324.0
+2020-06-06,Jammu and Kashmir,211880.0,208413,3467.0
+2020-06-07,Jammu and Kashmir,218481.0,214394,4087.0
+2020-06-08,Jammu and Kashmir,227906.0,223621,4285.0
+2020-06-09,Jammu and Kashmir,235816.0,231470,4346.0
+2020-06-10,Jammu and Kashmir,241891.0,237384,4507.0
+2020-06-11,Jammu and Kashmir,247267.0,242693,4574.0
+2020-06-12,Jammu and Kashmir,254059.0,249329,4730.0
+2020-06-13,Jammu and Kashmir,260098.0,255220,4878.0
+2020-06-14,Jammu and Kashmir,266163.0,261122,5041.0
+2020-06-15,Jammu and Kashmir,271416.0,266196,5220.0
+2020-06-16,Jammu and Kashmir,276174.0,270876,5298.0
+2020-06-17,Jammu and Kashmir,282268.0,276862,5406.0
+2020-06-18,Jammu and Kashmir,289027.0,283472,5555.0
+2020-06-19,Jammu and Kashmir,295202.0,289522,5680.0
+2020-06-20,Jammu and Kashmir,301209.0,295375,5834.0
+2020-06-21,Jammu and Kashmir,307638.0,301682,5956.0
+2020-06-22,Jammu and Kashmir,313687.0,307599,6088.0
+2020-06-23,Jammu and Kashmir,319664.0,313428,6236.0
+2020-06-24,Jammu and Kashmir,326430.0,320008,6422.0
+2020-06-25,Jammu and Kashmir,332445.0,325896,6549.0
+2020-06-26,Jammu and Kashmir,338903.0,332141,6762.0
+2020-06-27,Jammu and Kashmir,345426.0,338460,6966.0
+2020-06-28,Jammu and Kashmir,351865.0,344772,7093.0
+2020-06-29,Jammu and Kashmir,358530.0,351293,7237.0
+2020-06-30,Jammu and Kashmir,365058.0,357561,7497.0
+2020-07-01,Jammu and Kashmir,371486.0,363791,7695.0
+2020-07-02,Jammu and Kashmir,377961.0,370112,7849.0
+2020-07-03,Jammu and Kashmir,385501.0,377482,8019.0
+2020-07-04,Jammu and Kashmir,392919.0,384673,8246.0
+2020-07-05,Jammu and Kashmir,399385.0,390956,8429.0
+2020-07-06,Jammu and Kashmir,406212.0,397537,8675.0
+2020-07-07,Jammu and Kashmir,413358.0,404427,8931.0
+2020-07-08,Jammu and Kashmir,421571.0,412310,9261.0
+2020-07-09,Jammu and Kashmir,429787.0,420286,9501.0
+2020-07-10,Jammu and Kashmir,437928.0,428040,9888.0
+2020-07-11,Jammu and Kashmir,445169.0,435013,10156.0
+2020-07-12,Jammu and Kashmir,452455.0,441942,10513.0
+2020-07-13,Jammu and Kashmir,459703.0,448876,10827.0
+2020-07-14,Jammu and Kashmir,466333.0,455160,11173.0
+2020-07-15,Jammu and Kashmir,474149.0,462483,11666.0
+2020-07-16,Jammu and Kashmir,481452.0,469296,12156.0
+2020-07-17,Jammu and Kashmir,489382.0,476625,12757.0
+2020-07-18,Jammu and Kashmir,498007.0,484809,13198.0
+2020-07-19,Jammu and Kashmir,507678.0,493779,13899.0
+2020-07-20,Jammu and Kashmir,518029.0,503379,14650.0
+2020-07-21,Jammu and Kashmir,528946.0,513688,15258.0
+2020-07-22,Jammu and Kashmir,537891.0,522180,15711.0
+2020-07-23,Jammu and Kashmir,548877.0,532448,16429.0
+2020-07-24,Jammu and Kashmir,559399.0,542617,16782.0
+2020-07-25,Jammu and Kashmir,570508.0,553203,17305.0
+2020-07-26,Jammu and Kashmir,581707.0,563787,17920.0
+2020-07-27,Jammu and Kashmir,592482.0,574092,18390.0
+2020-07-28,Jammu and Kashmir,603728.0,584849,18879.0
+2020-07-29,Jammu and Kashmir,615380.0,595961,19419.0
+2020-07-30,Jammu and Kashmir,627387.0,607518,19869.0
+2020-07-31,Jammu and Kashmir,637515.0,617156,20359.0
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+2020-08-02,Jammu and Kashmir,654117.0,632701,21416.0
+2020-08-03,Jammu and Kashmir,662941.0,640935,22006.0
+2020-08-04,Jammu and Kashmir,671413.0,649017,22396.0
+2020-08-05,Jammu and Kashmir,679415.0,656460,22955.0
+2020-08-06,Jammu and Kashmir,686808.0,663354,23454.0
+2020-08-07,Jammu and Kashmir,695620.0,671693,23927.0
+2020-08-08,Jammu and Kashmir,706780.0,682390,24390.0
+2020-08-09,Jammu and Kashmir,717110.0,692213,24897.0
+2020-08-10,Jammu and Kashmir,725542.0,700175,25367.0
+2020-08-11,Jammu and Kashmir,738203.0,712272,
+2020-08-12,Jammu and Kashmir,750847.0,724434,26413.0
+2020-08-13,Jammu and Kashmir,763211.0,736262,26949.0
+2020-08-14,Jammu and Kashmir,775333.0,747844,27489.0
+2020-08-15,Jammu and Kashmir,785475.0,757454,28021.0
+2020-08-16,Jammu and Kashmir,793537.0,765067,
+2020-08-17,Jammu and Kashmir,801329.0,772437,
+2020-08-18,Jammu and Kashmir,811167.0,781841,
+2020-08-19,Jammu and Kashmir,822574.0,792540,
+2020-08-20,Jammu and Kashmir,833403.0,802686,
+2020-08-21,Jammu and Kashmir,844641.0,813270,31371.0
+2020-08-22,Jammu and Kashmir,855991.0,824010,
+2020-08-23,Jammu and Kashmir,868594.0,835947,
+2020-08-24,Jammu and Kashmir,877836.0,844761,
+2020-08-25,Jammu and Kashmir,888127.0,854351,
+2020-08-26,Jammu and Kashmir,902677.0,868197,
+2020-08-27,Jammu and Kashmir,915226.0,880091,
+2020-08-28,Jammu and Kashmir,929733.0,893902,
+2020-08-29,Jammu and Kashmir,943981.0,907604,
+2020-08-30,Jammu and Kashmir,956733.0,919570,
+2020-08-31,Jammu and Kashmir,966412.0,928714,
+2020-09-01,Jammu and Kashmir,978882.0,940659,
+2020-09-02,Jammu and Kashmir,996481.0,957617,
+2020-09-03,Jammu and Kashmir,1012892.0,972949,
+2020-09-04,Jammu and Kashmir,1031316.0,990326,
+2020-09-05,Jammu and Kashmir,1051826.0,1009585,
+2020-09-06,Jammu and Kashmir,1074998.0,1031441,
+2020-09-07,Jammu and Kashmir,1088652.0,1044082,
+2020-09-08,Jammu and Kashmir,1107406.0,1061481,
+2020-09-09,Jammu and Kashmir,1131076.0,1083534,
+2020-09-10,Jammu and Kashmir,1152563.0,1103429,
+2020-09-11,Jammu and Kashmir,1177773.0,1127061,
+2020-09-12,Jammu and Kashmir,1203799.0,1151389,
+2020-09-13,Jammu and Kashmir,1231698.0,1177602,
+2020-09-14,Jammu and Kashmir,1248495.0,1193170,
+2020-09-15,Jammu and Kashmir,1270310.0,1213656,
+2020-09-16,Jammu and Kashmir,1297329.0,1239085,
+2020-09-17,Jammu and Kashmir,1320240.0,1260529,
+2020-09-18,Jammu and Kashmir,1345750.0,1284709,
+2020-09-19,Jammu and Kashmir,1369702.0,1307169,
+2020-09-20,Jammu and Kashmir,1396729.0,1332739,
+2020-09-21,Jammu and Kashmir,1415364.0,1350338,
+2020-09-22,Jammu and Kashmir,1436409.0,1370148,
+2020-09-23,Jammu and Kashmir,1464565.0,1397055,
+2020-09-24,Jammu and Kashmir,1487962.0,1419348,
+2020-09-25,Jammu and Kashmir,1513224.0,1443392,
+2020-09-26,Jammu and Kashmir,1539284.0,1468235,
+2020-09-27,Jammu and Kashmir,1563309.0,1491119,
+2020-09-28,Jammu and Kashmir,1581606.0,1508592,
+2020-09-29,Jammu and Kashmir,1600606.0,1526511,
+2020-09-30,Jammu and Kashmir,1622775.0,1547705,
+2020-10-01,Jammu and Kashmir,1643722.0,1567559,
+2020-10-02,Jammu and Kashmir,1667587.0,1590334,
+2020-10-03,Jammu and Kashmir,1685809.0,1607581,
+2020-10-04,Jammu and Kashmir,1707503.0,1628397,
+2020-10-05,Jammu and Kashmir,1723337.0,1643599,
+2020-10-06,Jammu and Kashmir,1741300.0,1660824,
+2020-10-07,Jammu and Kashmir,1761792.0,1680695,
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+2020-10-10,Jammu and Kashmir,1818753.0,1735689,
+2020-10-11,Jammu and Kashmir,1839765.0,1756132,
+2020-10-12,Jammu and Kashmir,1854486.0,1770455,
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+2020-10-14,Jammu and Kashmir,1902884.0,1817475,
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+2020-10-16,Jammu and Kashmir,1952850.0,1866096,
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+2020-10-30,Jammu and Kashmir,2284588.0,2190258,
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+2020-11-02,Jammu and Kashmir,2343543.0,2247833,
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+2020-11-04,Jammu and Kashmir,2391065.0,2294365,
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+2020-11-10,Jammu and Kashmir,2529786.0,2429942,
+2020-11-11,Jammu and Kashmir,2551946.0,2451595,
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+2020-11-27,Jammu and Kashmir,2940302.0,2831431,
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+2020-12-08,Jammu and Kashmir,3218875.0,3104837,
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+2020-12-25,Jammu and Kashmir,3667726.0,3548098,
+2020-12-26,Jammu and Kashmir,3693803.0,3573926,
+2020-12-27,Jammu and Kashmir,3716877.0,3596740,
+2020-12-28,Jammu and Kashmir,3739623.0,3619330,
+2020-12-29,Jammu and Kashmir,3768849.0,3648322,
+2020-12-30,Jammu and Kashmir,3792251.0,3671507,
+2020-12-31,Jammu and Kashmir,3822674.0,3701703,
+2021-01-01,Jammu and Kashmir,3851827.0,3730600,
+2021-01-02,Jammu and Kashmir,3885403.0,3763932,
+2021-01-03,Jammu and Kashmir,3916848.0,3795195,
+2021-01-04,Jammu and Kashmir,3943822.0,3822036,
+2021-01-05,Jammu and Kashmir,3970242.0,3848319,
+2021-01-06,Jammu and Kashmir,3989145.0,3867096,
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+2020-04-01,Kerala,7965.0,7256,265.0
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+2020-04-03,Kerala,9139.0,8126,295.0
+2020-04-04,Kerala,9744.0,8586,306.0
+2020-04-05,Kerala,10221.0,9300,314.0
+2020-04-06,Kerala,10716.0,9607,327.0
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diff --git a/Module 2/Tidyverse.Rmd b/Module 2/Tidyverse.Rmd
new file mode 100644
index 00000000..31a2c756
--- /dev/null
+++ b/Module 2/Tidyverse.Rmd	
@@ -0,0 +1,207 @@
+---
+title: "Working with Packages -- Tidyverse"
+output: html_notebook
+---
+
+This notebook aims to explain installing and working with tidyverse packages with some example functions implemented on data frames. Have fun!
+
+### What's a package in R?
+
+> In R, the fundamental unit of shareable code is the package. A package bundles together code, data, documentation, and tests, and is easy to share with others. - [Chapter 1: Introduction, R packages (2nd edition)](https://r-pkgs.org/Introduction.html)
+
+If you need to use a package, you need to first download it using `install.packages('package_name')` and then load the package using `library(package_name)`. This makes sure the installed package is fetched from **C**omprehensive **R** **A**rchive **N**etwork, or [**CRAN**](https://cran.r-project.org).
+
+*Q1. How would you install a package from a GitHub repository?*
+
+### Handling data using base R functions
+
+```{r}
+# Importing covid testing data 
+covid_testdata <- read.csv('StatewiseTestingDetails.csv')
+```
+
+```{r}
+# Using head function to view first few rows of the data
+head(covid_testdata)
+```
+
+```{r}
+# Using nrow()  and ncol() base R functions to view the number of rows and columns for the data
+nrow(covid_testdata)
+ncol(covid_testdata)
+```
+
+*Q2. How would you check the number of `NA` rows in the `Negative` column?*
+
+```{r Q2}
+sum(is.na(covid_testdata$Negative))
+```
+
+### [Tidyverse package](https://www.tidyverse.org)
+
+Tidyverse package contains a list of packages useful for working with data.
+
+```{r}
+# install.packages("tidyverse")
+library(tidyverse)
+```
+
+We are going to use `tibble` package from `tidyverse`: \`tibble is a modern package to work with data frames in a better way than base R functions.
+
+```{r}
+# converting the data to tibble
+covid_data <- as_tibble(covid_testdata)
+covid_data
+```
+
+We saw here that using *tibble* package made things easier: we had to separately use base R functions such as *head*, *nrow* and *ncol* functions to get the necessary information from our data frame, but these features are by default shown by *as_tibble*.
+
+So, you might ask how could I directly read a file as a tibble? Easy: all you have to do is use read_csv (instead of read.csv)
+
+```{r}
+covid_data <- read_csv('StatewiseTestingDetails.csv')
+covid_data
+```
+
+Now we will look at another tidyverse package, [**dplyr**](https://dplyr.tidyverse.org): it is used for editing tibbles.
+
+```{r}
+# library(dplyr)
+```
+
+We will look at `filter` and `select` functions from the dplyr package:
+
+> `filter` function works on rows e.g., if we want to reduce our data only to the days when there were more than 5000 covid positive cases, we can do that using:
+
+```{r}
+filter(covid_data, Positive > 5000)
+```
+
+Let's try `filter` to view data only for the state of Uttarakhand:
+
+```{r}
+filter(covid_data, State == 'Uttarakhand')
+```
+
+```{r}
+filter(covid_data, State == 'Uttarakhand' & Positive > 5000)
+```
+
+> `select` function is used to select columns e.g., if we just want to view the states and the total Samples collected
+
+```{r}
+select(covid_data, State, TotalSamples)
+select(covid_data, c(State, TotalSamples))
+```
+
+If we want to view all columns other than the column `Negative`, we can use '-' minus to eliminate that column:
+
+```{r}
+select(covid_data, -Negative)
+```
+
+Or you can select a few contiguous coloumns using `:` operator:
+
+```{r}
+select(covid_data, State:Positive)
+```
+
+> `rename` can be used to easily rename columns with an argument suggesting that *new_column_name* = *old_column_name*.
+
+```{r}
+rename(covid_data, Neg = Negative)
+covid_data
+```
+
+*Q3. Oops, what did go wrong here? Why can't we see the column name changed?*
+
+> `mutate` can be used to edit/manipulate the content of the tibble.
+
+Let's say if we want the proportion of positive cases, we can use `mutate` as shown below:
+
+```{r}
+newdata <- mutate(covid_data, prop = Positive/TotalSamples)
+newdata
+```
+
+*Q4. Can you add a new column in the tibble without using `mutate`? If you can, why do you think we'll still need `mutate` for editing tibble?*
+
+> `arrange` function from dplyr package is used to arrange tibble data in an order.
+
+Let's arrange rows in the ascending order of `Date`.
+
+```{r}
+arrange(covid_data, Date)
+```
+
+By looking at the output of the above code, it seems like `arrange` function sorts the column in an ascending order. *Q5. How will you confirm that the default sorting order of `arrange` function?*
+
+Let's arrange the rows in the descending order of number of positive cases:
+
+```{r}
+arrange(covid_data, desc(Positive))
+```
+
+### Writing and reading files
+
+Now you may want to save this modified file for future use. Make sure you have saved the changes made in the file by using the assignment operator.
+
+To save this file, you can simply use the write command:
+
+```{r}
+write.csv(covid_data, file= "covid_data_copy.csv")
+```
+
+You may then later want to read the file into your workspace. For this you can use the read command.
+
+```{r}
+covid_data <- read.csv(file= "covid_data_copy.csv")
+```
+
+To find your current working directory and see the files in the directory:
+
+```{r}
+getwd() #current directory
+list.files() # list all files in the directory
+```
+
+### What's in a package for you?
+
+> But packages are useful even if you never share your code. As Hilary Parker says in her [introduction to packages](https://hilaryparker.com/2014/04/29/writing-an-r-package-from-scratch/): "Seriously, it doesn't have to be about sharing your code (although that is an added benefit!). It is about saving yourself time." Organising code in a package makes your life easier because packages come with conventions. For example, you put R code in `R/`, you put tests in `tests/` and you put data in `data/`... - [Chapter 1: Introduction, R packages (2nd edition)](https://r-pkgs.org/Introduction.html)
+
+### The pipe operator, `%>%`:
+
+Imagine an ordered set of manipulations you would want to do on your data frame stored in the current work space as a tibble. Would you really like to save the output of each one of those manipulations as a new (or maybe the same) variable? May be not. The pipe operator `%>%` helps you to do such a one-shot ordered manipulation.
+
+Let's directly take a look at an example:
+
+```{r}
+covid_data %>%
+  filter(State == 'Mizoram')
+```
+
+Whoa! It seems like `%>%` (referred as pipe operator) connects or pours the tibble to the function as an input.
+
+```{r}
+covid_data %>%
+  filter(State == "Mizoram") %>%
+  select(Date, Positive)
+```
+
+You can use the assignment operator to store the *product* of these manipulations:
+
+```{r}
+covid_data %>%
+  filter(State == "Mizoram") %>%
+  select(Date, Positive) ->
+  new_df
+```
+
+*Q6. Sometimes, it's better to use the variable name of the product in the first line of code chunk in which `%>%` is used multiple times. How would you do that?*
+
+### References:
+
+-   [What are the differences between "=" and "\<-" assignment operators?](https://stackoverflow.com/questions/1741820/what-are-the-differences-between-and-assignment-operators)
+-   [Understanding basic data types in R](https://resbaz.github.io/2014-r-materials/lessons/01-intro_r/data-structures.html)
+-   [Tidyverse](https://www.tidyverse.org)
+-   [Statistics from Linguists: An Introduction Using R](https://appliedstatisticsforlinguists.org/bwinter_stats_proofs.pdf)
diff --git a/Module 2/Tidyverse.nb.html b/Module 2/Tidyverse.nb.html
new file mode 100644
index 00000000..df1af17e
--- /dev/null
+++ b/Module 2/Tidyverse.nb.html	
@@ -0,0 +1,2309 @@
+
+
+
+
+
+
+
+
+
+
+
+
+
+Working with Packages – Tidyverse
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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+
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+
+
+
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+
+
+
+
+
+
+
+
+
+
+
    
+
+
+
+
+
+
+
+
+
    This notebook aims to explain installing and working with tidyverse
+packages with some example functions implemented on data frames. Have
+fun!
    
+
    
+
    What’s a package in R?
    
+
    
+
    In R, the fundamental unit of shareable code is the package. A
+package bundles together code, data, documentation, and tests, and is
+easy to share with others. - Chapter 1: Introduction, R
+packages (2nd edition)
    
+
    
+
    If you need to use a package, you need to first download it using
+install.packages('package_name') and then load the package
+using library(package_name). This makes sure the installed
+package is fetched from Comprehensive
+R Archive Network, or
+CRAN.
    
+
    Q1. How would you install a package from a GitHub
+repository?
    
+
    
+
    
+
    Handling data using base R functions
    
+
+
+
+
    # Importing covid testing data 
+covid_testdata <- read.csv('StatewiseTestingDetails.csv')
    
+
+
+
+
+
+
+
    # Using head function to view first few rows of the data
+head(covid_testdata)
    
+
+
+
    
+
+
    
+
+
+
+
+
+
+
    # Using nrow()  and ncol() base R functions to view the number of rows and columns for the data
+nrow(covid_testdata)
    
+
+
+
    [1] 14098
    
+
+
+
    ncol(covid_testdata)
    
+
+
+
    [1] 5
    
+
+
+
+
    Q2. How would you check the number of NA rows in the
+Negative column?
    
+
+
+
+
    sum(is.na(covid_testdata$Negative))
    
+
+
+
    [1] 7751
    
+
+
+
+
    
+
    
+
    Tidyverse package
    
+
    Tidyverse package contains a list of packages useful for working with
+data.
    
+
+
+
+
    # install.packages("tidyverse")
+library(tidyverse)
    
+
+
+
+
    We are going to use tibble package from
+tidyverse: `tibble is a modern package to work with data
+frames in a better way than base R functions.
    
+
+
+
+
    # converting the data to tibble
+covid_data <- as_tibble(covid_testdata)
+covid_data
    
+
+
+
    
+
+
    
+
+
+
+
    We saw here that using tibble package made things easier: we
+had to separately use base R functions such as head,
+nrow and ncol functions to get the necessary
+information from our data frame, but these features are by default shown
+by as_tibble.
    
+
    So, you might ask how could I directly read a file as a tibble? Easy:
+all you have to do is use read_csv (instead of read.csv)
    
+
+
+
+
    covid_data <- read_csv('StatewiseTestingDetails.csv')
    
+
+
+
    Rows: 14098 Columns: 5── Column specification ──────────────────────────────────────────────────────────────────────────────────────────
+Delimiter: ","
+chr  (1): State
+dbl  (3): TotalSamples, Negative, Positive
+date (1): Date
+ℹ Use `spec()` to retrieve the full column specification for this data.
+ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
    
+
+
+
    covid_data
    
+
+
+
    
+
+
    
+
+
+
+
    Now we will look at another tidyverse package, dplyr: it is
+used for editing tibbles.
    
+
+
+
+
    # library(dplyr)
    
+
+
+
+
    We will look at filter and select functions
+from the dplyr package:
    
+
    
+
    filter function works on rows e.g., if we want to reduce
+our data only to the days when there were more than 5000 covid positive
+cases, we can do that using:
    
+
    
+
+
+
+
    filter(covid_data, Positive > 5000)
    
+
+
+
    
+
+
    
+
+
+
+
    Let’s try filter to view data only for the state of
+Uttarakhand:
    
+
+
+
+
    filter(covid_data, State == 'Uttarakhand')
    
+
+
+
    
+
+
    
+
+
+
+
+
+
+
    filter(covid_data, State == 'Uttarakhand' & Positive > 5000)
    
+
+
+
    
+
+
    
+
+
+
+
    
+
    select function is used to select columns e.g., if we
+just want to view the states and the total Samples collected
    
+
    
+
+
+
+
    select(covid_data, State, TotalSamples)
    
+
+
+
    
+
+
    
+
+
+
    select(covid_data, c(State, TotalSamples))
    
+
+
+
    
+
+
    
+
+
+
+
    If we want to view all columns other than the column
+Negative, we can use ‘-’ minus to eliminate that
+column:
    
+
+
+
+
    select(covid_data, -Negative)
    
+
+
+
    
+
+
    
+
+
+
+
    Or you can select a few contiguous coloumns using :
+operator:
    
+
+
+
+
    select(covid_data, State:Positive)
    
+
+
+
    
+
+
    
+
+
+
+
    
+
    rename can be used to easily rename columns with an
+argument suggesting that new_column_name =
+old_column_name.
    
+
    
+
+
+
+
    rename(covid_data, Neg = Negative)
    
+
+
+
    
+
+
    
+
+
+
    covid_data
    
+
+
+
    
+
+
    
+
+
+
+
    Q3. Oops, what did go wrong here? Why can’t we see the column
+name changed?
    
+
    
+
    mutate can be used to edit/manipulate the content of the
+tibble.
    
+
    
+
    Let’s say if we want the proportion of positive cases, we can use
+mutate as shown below:
    
+
+
+
+
    newdata <- mutate(covid_data, prop = Positive/TotalSamples)
+newdata
    
+
+
+
    
+
+
    
+
+
+
+
    Q4. Can you add a new column in the tibble without using
+mutate? If you can, why do you think we’ll still need
+mutate for editing tibble?
    
+
    
+
    arrange function from dplyr package is used to arrange
+tibble data in an order.
    
+
    
+
    Let’s arrange rows in the ascending order of Date.
    
+
+
+
+
    arrange(covid_data, Date)
    
+
+
+
    
+
+
    
+
+
+
+
    By looking at the output of the above code, it seems like
+arrange function sorts the column in an ascending order.
+Q5. How will you confirm that the default sorting order of
+arrange function?
    
+
    Let’s arrange the rows in the descending order of number of positive
+cases:
    
+
+
+
+
    arrange(covid_data, desc(Positive))
    
+
+
+
    
+
+
    
+
+
+
+
    
+
    
+
    Writing and reading files
    
+
    Now you may want to save this modified file for future use. Make sure
+you have saved the changes made in the file by using the assignment
+operator.
    
+
    To save this file, you can simply use the write command:
    
+
+
+
+
    write.csv(covid_data, file= "covid_data_copy.csv")
    
+
+
+
+
    You may then later want to read the file into your workspace. For
+this you can use the read command.
    
+
+
+
+
    covid_data <- read.csv(file= "covid_data_copy.csv")
    
+
+
+
+
    To find your current working directory and see the files in the
+directory:
    
+
+
+
+
    getwd() #current directory
    
+
+
+
    [1] "/Users/takhil/Lab-related/BSE658_chapter2"
    
+
+
+
    list.files() # list all files in the directory
    
+
+
+
    [1] "BSE658_Chapter1.Rmd"         "covid_data_copy.csv"         "project.Rproj"              
+[4] "README.md"                   "StatewiseTestingDetails.csv" "Tidyverse.nb.html"          
+[7] "Tidyverse.Rmd"               "using ggplot.Rmd"            "using-ggplot.html"          
    
+
+
+
+
    
+
    
+
    What’s in a package for you?
    
+
    
+
    But packages are useful even if you never share your code. As Hilary
+Parker says in her introduction
+to packages: “Seriously, it doesn’t have to be about sharing your
+code (although that is an added benefit!). It is about saving yourself
+time.” Organising code in a package makes your life easier because
+packages come with conventions. For example, you put R code in
+R/, you put tests in tests/ and you put data
+in data/… - Chapter 1: Introduction, R
+packages (2nd edition)
    
+
    
+
    
+
    
+
    The pipe operator, %>%:
    
+
    Imagine an ordered set of manipulations you would want to do on your
+data frame stored in the current work space as a tibble. Would you
+really like to save the output of each one of those manipulations as a
+new (or maybe the same) variable? May be not. The pipe operator
+%>% helps you to do such a one-shot ordered
+manipulation.
    
+
    Let’s directly take a look at an example:
    
+
+
+
+
    
+
+
    
+
+
+
+
    Whoa! It seems like %>% (referred as pipe operator)
+connects or pours the tibble to the function as an input.
    
+
+
+
+
    covid_data %>%
+  filter(State == "Mizoram") %>%
+  select(Date, Positive)
    
+
+
+
+
    You can use the assignment operator to store the product of
+these manipulations:
    
+
+
+
+
    covid_data %>%
+  filter(State == "Mizoram") %>%
+  select(Date, Positive) ->
+  new_df
    
+
+
+
+
    Q6. Sometimes, it’s better to use the variable name of the
+product in the first line of code chunk in which %>% is
+used multiple times. How would you do that?
    
+
    
+
+
+
    ---
title: "Working with Packages -- Tidyverse"
output: html_notebook
---

This notebook aims to explain installing and working with tidyverse packages with some example functions implemented on data frames. Have fun!

### What's a package in R?

> In R, the fundamental unit of shareable code is the package. A package bundles together code, data, documentation, and tests, and is easy to share with others. - [Chapter 1: Introduction, R packages (2nd edition)](https://r-pkgs.org/Introduction.html)

If you need to use a package, you need to first download it using `install.packages('package_name')` and then load the package using `library(package_name)`. This makes sure the installed package is fetched from **C**omprehensive **R** **A**rchive **N**etwork, or [**CRAN**](https://cran.r-project.org).

*Q1. How would you install a package from a GitHub repository?*

### Handling data using base R functions

```{r}
# Importing covid testing data 
covid_testdata <- read.csv('StatewiseTestingDetails.csv')
```

```{r}
# Using head function to view first few rows of the data
head(covid_testdata)
```

```{r}
# Using nrow()  and ncol() base R functions to view the number of rows and columns for the data
nrow(covid_testdata)
ncol(covid_testdata)
```

*Q2. How would you check the number of `NA` rows in the `Negative` column?*

```{r Q2}
sum(is.na(covid_testdata$Negative))
```

### [Tidyverse package](https://www.tidyverse.org)

Tidyverse package contains a list of packages useful for working with data.

```{r}
# install.packages("tidyverse")
library(tidyverse)
```

We are going to use `tibble` package from `tidyverse`: \`tibble is a modern package to work with data frames in a better way than base R functions.

```{r}
# converting the data to tibble
covid_data <- as_tibble(covid_testdata)
covid_data
```

We saw here that using *tibble* package made things easier: we had to separately use base R functions such as *head*, *nrow* and *ncol* functions to get the necessary information from our data frame, but these features are by default shown by *as_tibble*.

So, you might ask how could I directly read a file as a tibble? Easy: all you have to do is use read_csv (instead of read.csv)

```{r}
covid_data <- read_csv('StatewiseTestingDetails.csv')
covid_data
```

Now we will look at another tidyverse package, [**dplyr**](https://dplyr.tidyverse.org): it is used for editing tibbles.

```{r}
# library(dplyr)
```

We will look at `filter` and `select` functions from the dplyr package:

> `filter` function works on rows e.g., if we want to reduce our data only to the days when there were more than 5000 covid positive cases, we can do that using:

```{r}
filter(covid_data, Positive > 5000)
```

Let's try `filter` to view data only for the state of Uttarakhand:

```{r}
filter(covid_data, State == 'Uttarakhand')
```

```{r}
filter(covid_data, State == 'Uttarakhand' & Positive > 5000)
```

> `select` function is used to select columns e.g., if we just want to view the states and the total Samples collected

```{r}
select(covid_data, State, TotalSamples)
select(covid_data, c(State, TotalSamples))
```

If we want to view all columns other than the column `Negative`, we can use '-' minus to eliminate that column:

```{r}
select(covid_data, -Negative)
```

Or you can select a few contiguous coloumns using `:` operator:

```{r}
select(covid_data, State:Positive)
```

> `rename` can be used to easily rename columns with an argument suggesting that *new_column_name* = *old_column_name*.

```{r}
rename(covid_data, Neg = Negative)
covid_data
```

*Q3. Oops, what did go wrong here? Why can't we see the column name changed?*

> `mutate` can be used to edit/manipulate the content of the tibble.

Let's say if we want the proportion of positive cases, we can use `mutate` as shown below:

```{r}
newdata <- mutate(covid_data, prop = Positive/TotalSamples)
newdata
```

*Q4. Can you add a new column in the tibble without using `mutate`? If you can, why do you think we'll still need `mutate` for editing tibble?*

> `arrange` function from dplyr package is used to arrange tibble data in an order.

Let's arrange rows in the ascending order of `Date`.

```{r}
arrange(covid_data, Date)
```

By looking at the output of the above code, it seems like `arrange` function sorts the column in an ascending order. *Q5. How will you confirm that the default sorting order of `arrange` function?*

Let's arrange the rows in the descending order of number of positive cases:

```{r}
arrange(covid_data, desc(Positive))
```

### Writing and reading files

Now you may want to save this modified file for future use. Make sure you have saved the changes made in the file by using the assignment operator.

To save this file, you can simply use the write command:

```{r}
write.csv(covid_data, file= "covid_data_copy.csv")
```

You may then later want to read the file into your workspace. For this you can use the read command.

```{r}
covid_data <- read.csv(file= "covid_data_copy.csv")
```

To find your current working directory and see the files in the directory:

```{r}
getwd() #current directory
list.files() # list all files in the directory
```

### What's in a package for you?

> But packages are useful even if you never share your code. As Hilary Parker says in her [introduction to packages](https://hilaryparker.com/2014/04/29/writing-an-r-package-from-scratch/): "Seriously, it doesn't have to be about sharing your code (although that is an added benefit!). It is about saving yourself time." Organising code in a package makes your life easier because packages come with conventions. For example, you put R code in `R/`, you put tests in `tests/` and you put data in `data/`... - [Chapter 1: Introduction, R packages (2nd edition)](https://r-pkgs.org/Introduction.html)

### The pipe operator, `%>%`:

Imagine an ordered set of manipulations you would want to do on your data frame stored in the current work space as a tibble. Would you really like to save the output of each one of those manipulations as a new (or maybe the same) variable? May be not. The pipe operator `%>%` helps you to do such a one-shot ordered manipulation.

Let's directly take a look at an example:

```{r}
covid_data %>%
  filter(State == 'Mizoram')
```

Whoa! It seems like `%>%` (referred as pipe operator) connects or pours the tibble to the function as an input.

```{r}
covid_data %>%
  filter(State == "Mizoram") %>%
  select(Date, Positive)
```

You can use the assignment operator to store the *product* of these manipulations:

```{r}
covid_data %>%
  filter(State == "Mizoram") %>%
  select(Date, Positive) ->
  new_df
```

*Q6. Sometimes, it's better to use the variable name of the product in the first line of code chunk in which `%>%` is used multiple times. How would you do that?*

### References:

-   [What are the differences between "=" and "\<-" assignment operators?](https://stackoverflow.com/questions/1741820/what-are-the-differences-between-and-assignment-operators)
-   [Understanding basic data types in R](https://resbaz.github.io/2014-r-materials/lessons/01-intro_r/data-structures.html)
-   [Tidyverse](https://www.tidyverse.org)
-   [Statistics from Linguists: An Introduction Using R](https://appliedstatisticsforlinguists.org/bwinter_stats_proofs.pdf)

    
+
+
+
+
    
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/Module 2/project.Rproj b/Module 2/project.Rproj
new file mode 100644
index 00000000..8e3c2ebc
--- /dev/null
+++ b/Module 2/project.Rproj	
@@ -0,0 +1,13 @@
+Version: 1.0
+
+RestoreWorkspace: Default
+SaveWorkspace: Default
+AlwaysSaveHistory: Default
+
+EnableCodeIndexing: Yes
+UseSpacesForTab: Yes
+NumSpacesForTab: 2
+Encoding: UTF-8
+
+RnwWeave: Sweave
+LaTeX: pdfLaTeX
diff --git a/Module 2/using ggplot.Rmd b/Module 2/using ggplot.Rmd
new file mode 100644
index 00000000..3676edc4
--- /dev/null
+++ b/Module 2/using ggplot.Rmd	
@@ -0,0 +1,59 @@
+---
+title: "ggplot"
+output: html_document
+---
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+```
+### This notebook aims to explore working with ggplot2 package, ggplot enables fast and efficient plotting. 
+
+```{r}
+# Loading the packages
+library(tidyverse)
+library(dplyr)
+library(tibble)
+library(ggplot2)
+```
+
+
+```{r}
+# Importing covid testing data using tibble read_csv function
+covid_data <- read_csv('StatewiseTestingDetails.csv')
+```
+```{r}
+covid_data
+```
+### We will use ggplot2 to view the trend in covid positive cases in delhi state
+```{r}
+state_data = filter(covid_data, State=='Uttarakhand')
+state_data = arrange(state_data,Date)
+state_data
+```
+### Now that we have the data for Delhi arranged in ascending order of date, we will plot the number of cases 
+```{r}
+ggplot(state_data) + geom_point(mapping = aes(x=Date, y=Positive))
+```
+
+
+```{r}
+ggplot(state_data) + geom_text(mapping = aes(x=Date, y=Positive, label = Positive))
+```
+### Saving a plot
+```{r}
+ggsave('UK.png', width = 8, height = 6)
+```
+
+### Creating double plots
+#### For creating double plot we need to load an additional library _gridExtra_
+#### Let's create two plots for total samples and total positive cases for a particular state
+
+```{r}
+plot1 <- ggplot(state_data) + geom_point(mapping = aes(x=Date, y=TotalSamples))
+plot2 <- ggplot(state_data) + geom_point(mapping = aes(x=Date, y=Positive))
+library(gridExtra)
+grid.arrange(plot1, plot2, ncol = 2)
+```
+
+
+
diff --git a/Module 2/using-ggplot.html b/Module 2/using-ggplot.html
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+
    This notebook aims to explore working with ggplot2 package, ggplot enables fast and efficient plotting.
    
+
    # Loading the packages
+library(tidyverse)
    
+
    ## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──
    
+
    ## ✓ ggplot2 3.3.4     ✓ purrr   0.3.4
+## ✓ tibble  3.1.2     ✓ dplyr   1.0.6
+## ✓ tidyr   1.1.3     ✓ stringr 1.4.0
+## ✓ readr   1.4.0     ✓ forcats 0.5.1
    
+
    ## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
+## x dplyr::filter() masks stats::filter()
+## x dplyr::lag()    masks stats::lag()
    
+
    library(dplyr)
+library(tibble)
+library(ggplot2)
    
+
    # Importing covid testing data using tibble read_csv function
+covid_data <- read_csv('../StatewiseTestingDetails.csv')
    
+
    ## 
+## ── Column specification ────────────────────────────────────────────────────────
+## cols(
+##   Date = col_date(format = ""),
+##   State = col_character(),
+##   TotalSamples = col_double(),
+##   Negative = col_double(),
+##   Positive = col_double()
+## )
    
+
    covid_data
    
+
    ## # A tibble: 14,098 x 5
+##    Date       State                       TotalSamples Negative Positive
+##    <date>     <chr>                              <dbl>    <dbl>    <dbl>
+##  1 2020-04-17 Andaman and Nicobar Islands         1403     1210       12
+##  2 2020-04-24 Andaman and Nicobar Islands         2679       NA       27
+##  3 2020-04-27 Andaman and Nicobar Islands         2848       NA       33
+##  4 2020-05-01 Andaman and Nicobar Islands         3754       NA       33
+##  5 2020-05-16 Andaman and Nicobar Islands         6677       NA       33
+##  6 2020-05-19 Andaman and Nicobar Islands         6965       NA       33
+##  7 2020-05-20 Andaman and Nicobar Islands         7082       NA       33
+##  8 2020-05-21 Andaman and Nicobar Islands         7167       NA       33
+##  9 2020-05-22 Andaman and Nicobar Islands         7263       NA       33
+## 10 2020-05-23 Andaman and Nicobar Islands         7327       NA       33
+## # … with 14,088 more rows
    
+
    
+
    
+
    We will use ggplot2 to view the trend in covid positive cases in delhi state
    
+
    state_data = filter(covid_data, State=='Uttarakhand')
+state_data = arrange(state_data,Date)
+state_data
    
+
    ## # A tibble: 427 x 5
+##    Date       State       TotalSamples Negative Positive
+##    <date>     <chr>              <dbl>    <dbl>    <dbl>
+##  1 2020-04-02 Uttarakhand          678      554        7
+##  2 2020-04-07 Uttarakhand         1289     1092       32
+##  3 2020-04-09 Uttarakhand         1531     1235       35
+##  4 2020-04-10 Uttarakhand         1688     1320       35
+##  5 2020-04-11 Uttarakhand         1705     1340       35
+##  6 2020-04-12 Uttarakhand         1820     1452       35
+##  7 2020-04-13 Uttarakhand         1998     1665       35
+##  8 2020-04-14 Uttarakhand         2174     1838       35
+##  9 2020-04-15 Uttarakhand         2413     2022       37
+## 10 2020-04-16 Uttarakhand         2593     2210       37
+## # … with 417 more rows
    
+
    
+
    
+
    Now that we have the data for Delhi arranged in ascending order of date, we will plot the number of cases
    
+
    ggplot(state_data) + geom_point(mapping = aes(x=Date, y=Positive))
    
+
    ## Warning: Removed 295 rows containing missing values (geom_point).
    
+
    
+
    ggplot(state_data) + geom_text(mapping = aes(x=Date, y=Positive, label = Positive))
    
+
    ## Warning: Removed 295 rows containing missing values (geom_text).
    
+
     ### Saving a plot
    
+
    ggsave('UK.png', width = 8, height = 6)
    
+
    ## Warning: Removed 295 rows containing missing values (geom_text).
    
+
    
+
    
+
    Creating double plots
    
+
    
+
    For creating double plot we need to load an additional library gridExtra
    
+
    
+
    
+
    Let’s create two plots for total samples and total positive cases for a particular state
    
+
    plot1 <- ggplot(state_data) + geom_point(mapping = aes(x=Date, y=TotalSamples))
+plot2 <- ggplot(state_data) + geom_point(mapping = aes(x=Date, y=Positive))
+library(gridExtra)
    
+
    ## 
+## Attaching package: 'gridExtra'
    
+
    ## The following object is masked from 'package:dplyr':
+## 
+##     combine
    
+
    grid.arrange(plot1, plot2, ncol = 2)
    
+
    ## Warning: Removed 295 rows containing missing values (geom_point).
    
+
    
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From a1d15e622713674ee8477c26857feef9e3ac9a19 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Fri, 18 Aug 2023 09:03:29 +0530
Subject: [PATCH 41/55] Module 6 last set of changes

Done last year
---
 Module 6/ANOVA.Rmd                 |   32 +
 Module 6/Regression.Rmd            |    2 +
 Module 6/linearMixedModels.Rmd     |    1 -
 Module 6/linearMixedModels.nb.html |  118 +-
 Module 6/testcode.Rmd              |  268 +++
 Module 6/testcode.nb.html          | 2429 ++++++++++++++++++++++++++++
 6 files changed, 2740 insertions(+), 110 deletions(-)
 create mode 100644 Module 6/testcode.Rmd
 create mode 100644 Module 6/testcode.nb.html

diff --git a/Module 6/ANOVA.Rmd b/Module 6/ANOVA.Rmd
index 0c1b8478..d9cc960c 100644
--- a/Module 6/ANOVA.Rmd	
+++ b/Module 6/ANOVA.Rmd	
@@ -329,4 +329,36 @@ summary(model.3)
 etaSquared( model.3 )
 ```
 
+```{r}
+data(ToothGrowth)
+ToothGrowth$dose <- as.factor(ToothGrowth$dose)
+head(ToothGrowth)
+```
+
+
+
+```{r}
+model.3a <- aov( len ~ supp + dose + supp:dose, ToothGrowth )  
+summary(model.3a)
+```
+
+```{r}
+TukeyHSD( model.3a )
+```
+
+```{r}
+etaSquared( model.3a )
+```
+
+```{r}
+my.anova.residuals <- residuals( object = model.3a )   # extract the residuals
+hist( x = my.anova.residuals )           # plot a histogram (similar to Figure @ref{fig:normalityanova}a)
+qqnorm( y = my.anova.residuals )         # draw a QQ plot (similar to Figure @ref{fig:normalityanova}b)
+shapiro.test( x = my.anova.residuals )   # run Shapiro-Wilk test
+```
+```{r}
+library(car)
+leveneTest(y = len ~ dose, data = ToothGrowth)   # y is a formula in this case
+```
+
 
diff --git a/Module 6/Regression.Rmd b/Module 6/Regression.Rmd
index 0a9c0d3d..a55fa92b 100644
--- a/Module 6/Regression.Rmd	
+++ b/Module 6/Regression.Rmd	
@@ -201,3 +201,5 @@ AIC( M0, M1 )
 anova( M0, M1 )
 ```
 
+
+
diff --git a/Module 6/linearMixedModels.Rmd b/Module 6/linearMixedModels.Rmd
index e401d899..8909613f 100644
--- a/Module 6/linearMixedModels.Rmd	
+++ b/Module 6/linearMixedModels.Rmd	
@@ -121,7 +121,6 @@ head(data)
 Convert attitude, gender, subject into factors.
 ```{r}
 data = data %>% mutate(attitude=as.factor(attitude), gender=as.factor(gender), subject=as.factor(subject))
-
 ```
 
 
diff --git a/Module 6/linearMixedModels.nb.html b/Module 6/linearMixedModels.nb.html
index dc575050..20345add 100644
--- a/Module 6/linearMixedModels.nb.html	
+++ b/Module 6/linearMixedModels.nb.html	
@@ -2046,48 +2046,18 @@ 
    R Notebook
    
 
    Convert attitude, gender, subject into factors.
    
 
 
-
-
    data = data %>% mutate(attitude=as.factor(attitude), gender=as.factor(gender), subject=as.factor(subject))
-
    
+
+
    data = data %>% mutate(attitude=as.factor(attitude), gender=as.factor(gender), subject=as.factor(subject))
    
 
 
 
 
    Random Intercept Models
    
 
 
-
-
    summary(politeness.model0)
-
    
+
+
    politeness.model0 = lmer(frequency ~ attitude + (1|subject) + (1|scenario), data=data)
+summary(politeness.model0)
    
 
-
-
    Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
-Formula: frequency ~ attitude + (1 | subject) + (1 | scenario)
-   Data: data
-
-REML criterion at convergence: 793.5
-
-Scaled residuals: 
-    Min      1Q  Median      3Q     Max 
--2.2006 -0.5817 -0.0639  0.5625  3.4385 
-
-Random effects:
- Groups   Name        Variance Std.Dev.
- scenario (Intercept)  219     14.80   
- subject  (Intercept) 4015     63.36   
- Residual              646     25.42   
-Number of obs: 83, groups:  scenario, 7; subject, 6
-
-Fixed effects:
-            Estimate Std. Error      df t value Pr(>|t|)    
-(Intercept)  202.588     26.754   5.575   7.572 0.000389 ***
-attitudepol  -19.695      5.585  70.022  -3.527 0.000748 ***
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-
-Correlation of Fixed Effects:
-            (Intr)
-attitudepol -0.103
    
-
 
 
 
@@ -2096,37 +2066,6 @@ 
    R Notebook
    
 
    politeness.model = lmer(frequency ~ attitude + gender + (1|subject) + (1|scenario), data=data)
 summary(politeness.model)
    
 
-
-
    Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
-Formula: frequency ~ attitude + gender + (1 | subject) + (1 | scenario)
-   Data: data
-
-REML criterion at convergence: 775.5
-
-Scaled residuals: 
-    Min      1Q  Median      3Q     Max 
--2.2591 -0.6236 -0.0772  0.5388  3.4795 
-
-Random effects:
- Groups   Name        Variance Std.Dev.
- scenario (Intercept) 219.5    14.81   
- subject  (Intercept) 615.6    24.81   
- Residual             645.9    25.41   
-Number of obs: 83, groups:  scenario, 7; subject, 6
-
-Fixed effects:
-            Estimate Std. Error       df t value Pr(>|t|)    
-(Intercept)  256.846     16.116    5.432  15.938 9.06e-06 ***
-attitudepol  -19.721      5.584   70.054  -3.532 0.000735 ***
-genderM     -108.516     21.013    4.007  -5.164 0.006647 ** 
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-
-Correlation of Fixed Effects:
-            (Intr) atttdp
-attitudepol -0.173       
-genderM     -0.652  0.004
    
-
 
 
 
    Likelihood Ratio Test
    
@@ -2137,59 +2076,20 @@ 
    R Notebook
    
 politeness.full = lmer(frequency ~ attitude + gender + (1|subject) + (1|scenario), data=data, REML=FALSE)
 anova(politeness.null, politeness.full)
    
 
-
-
    Data: data
-Models:
-politeness.null: frequency ~ gender + (1 | subject) + (1 | scenario)
-politeness.full: frequency ~ attitude + gender + (1 | subject) + (1 | scenario)
-                npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)    
-politeness.null    5 816.72 828.81 -403.36   806.72                         
-politeness.full    6 807.10 821.61 -397.55   795.10 11.618  1  0.0006532 ***
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
-
 
 
 
    Random Slope Model
    
 
 
-
-
    politeness.model1 = lmer(frequency~attitude + gender + (1+attitude|subject) + (1+attitude|scenario), data = data)
    
-
-
-
    boundary (singular) fit: see help('isSingular')
    
-
-
-
    coef(politeness.model1)
    
+
+
    politeness.model1 = lmer(frequency~attitude + gender + (1+attitude|subject) + (1+attitude|scenario), data = data)
+coef(politeness.model1)
    
 
-
-
    $scenario
-  (Intercept) attitudepol   genderM
-1    244.4740   -19.00296 -111.1058
-2    261.9447   -12.87473 -111.1058
-3    270.9290   -23.46233 -111.1058
-4    277.0651   -15.90595 -111.1058
-5    255.8277   -18.72597 -111.1058
-6    247.0421   -22.37916 -111.1058
-7    249.7042   -25.93003 -111.1058
-
-$subject
-   (Intercept) attitudepol   genderM
-F1    243.2804   -20.49940 -111.1058
-F2    267.1173   -19.30447 -111.1058
-F3    260.2849   -19.64697 -111.1058
-M3    287.1024   -18.30263 -111.1058
-M4    264.6698   -19.42716 -111.1058
-M7    226.3911   -21.34605 -111.1058
-
-attr(,"class")
-[1] "coef.mer"
    
-
 
 
 
 
-
    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+
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   
 
 
 
diff --git a/Module 6/testcode.Rmd b/Module 6/testcode.Rmd
new file mode 100644
index 00000000..828769c9
--- /dev/null
+++ b/Module 6/testcode.Rmd	
@@ -0,0 +1,268 @@
+---
+title: "R Notebook"
+output: html_notebook
+---
+
+Testing
+
+```{r}
+data()
+```
+
+
+```{r}
+data(package = .packages(all.available = TRUE))
+```
+
+```{r}
+library(lme4)
+data("InstEval")
+head(InstEval)
+
+```
+```{r}
+library(lmerTest)
+```
+```{r}
+InstEval$studage <- as.numeric(InstEval$studage)
+InstEval$lectage <- as.numeric(InstEval$lectage)
+InstEval$service <- as.numeric(InstEval$service) 
+InstEval$dept <- as.numeric(InstEval$dept)
+```
+
+
+
+
+
+```{r}
+m1 <- lmer( formula = y ~ studage + lectage + service + dept + (1|s) + (1|d),  
+                     data = InstEval )
+summary( m1 )
+```
+
+```{r}
+library(lme4)
+data("sleepstudy")
+head(sleepstudy)
+
+```
+
+```{r}
+library(tidyverse)
+library(magrittr)
+```
+
+
+```{r}
+dep_1 <- sleepstudy %>% filter(Days == 1) %>% select(Reaction)
+dep_2 <- sleepstudy %>% filter(Days == 2) %>% select(Reaction)
+dep_3 <- sleepstudy %>% filter(Days == 3) %>% select(Reaction)
+dep_4 <- sleepstudy %>% filter(Days == 4) %>% select(Reaction)
+dep_5 <- sleepstudy %>% filter(Days == 5) %>% select(Reaction)
+dep_6 <- sleepstudy %>% filter(Days == 6) %>% select(Reaction)
+dep_7 <- sleepstudy %>% filter(Days == 7) %>% select(Reaction)
+dep_8 <- sleepstudy %>% filter(Days == 8) %>% select(Reaction)
+dep_9 <- sleepstudy %>% filter(Days == 9) %>% select(Reaction)
+df <- data.frame(dep_1,dep_2, dep_3, dep_4, dep_5, dep_6, dep_7, dep_8, dep_9)
+```
+
+```{r}
+df2 <- sleepstudy %>% filter(Days %in% c(1, 5))
+```
+
+
+```{r}
+diff <- df$Reaction.5 - df$Reaction
+shapiro.test(diff)
+```
+```{r}
+df2$Days <- as.factor(df2$Days)
+leveneTest(Reaction ~ Days, df2)
+```
+
+
+```{r}
+t.test(df$Reaction, df$Reaction.5, paired = TRUE, alternative = "two.sided", var.equal = TRUE)
+```
+
+```{r}
+m2 <- lmer( formula = Reaction ~ Days + (1|Subject),  
+                     data = sleepstudy )
+summary( m2 )
+```
+
+```{r}
+m3 <- lmer( formula = Reaction ~ Days + (1 + Days|Subject),  
+                     data = sleepstudy )
+summary( m3 )
+```
+```{r}
+AIC(m2, m3)
+```
+
+```{r}
+anova(m2,m3)
+```
+
+
+
+
+```{r}
+library(easystats)
+
+```
+
+
+```{r}
+check_model(m2)
+```
+
+```{r}
+check_heteroscedasticity(m3)
+```
+
+```{r}
+library(MASS)
+data("HairEyeColor")
+head(HairEyeColor)
+```
+```{r}
+tab_n <- c(124,151,105,149)
+```
+
+```{r}
+library(lsr)
+cramersV( tab_n )
+```
+
+
+```{r}
+chisq.test( tab_n  )
+
+```
+```{r}
+library(pwr)
+pwr.chisq.test(w = 0.08, df = 3, sig.level = 0.05, power = 0.8 )
+```
+
+
+
+
+
+
+
+
+
+
+```{r}
+data("mtcars")
+head(mtcars)
+```
+
+```{r}
+m2 <- lm( formula = mpg ~ hp + wt,  
+                     data = mtcars )
+summary( m2 )
+```
+```{r}
+library(car)
+```
+```{r}
+plot(x = m1, which = 3)
+
+```
+
+```{r}
+ncvTest( m1 )
+```
+
+
+```{r}
+vif( mod = m1 )
+```
+```{r}
+residualPlots( model = m1 ) 
+```
+```{r}
+plot( x = m2, which = 2) 
+
+```
+```{r}
+shapiro.test(residuals( m1 ))
+```
+
+
+
+
+
+```{r}
+
+m3 <- lm( formula = log(mpg) ~ log(hp) + log(wt),  
+                     data = mtcars )
+summary( m3 )
+```
+```{r}
+residualPlots( model = m3 ) 
+```
+```{r}
+ncvTest( m3 )
+```
+
+
+```{r}
+vif( mod = m3 )
+```
+
+```{r}
+library(MASS)
+```
+
+
+```{r}
+#bc <- boxcox(m3)
+bc.car <- powerTransform(m2)
+lambda <- bc.car$lambda
+
+#lambda <- bc$x[which.max(bc$y)]
+#lambda
+```
+
+fit new linear regression model using the Box-Cox transformation
+
+```{r}
+m4 <- lm(((mpg^lambda-1)/lambda) ~ hp + wt,data = mtcars)
+summary(m4)
+```
+
+```{r}
+residualPlots( model = m4 ) 
+```
+
+```{r}
+ncvTest( m4 )
+```
+
+
+```{r}
+vif( mod = m4 )
+```
+
+```{r}
+plot( x = m3, which = 2) 
+
+```
+```{r}
+shapiro.test(residuals( m4 ))
+```
+
+
+```{r}
+data()
+```
+
+```{r}
+library(lme4)
+data("sleepstudy")
+head(sleepstudy)
+```
+
diff --git a/Module 6/testcode.nb.html b/Module 6/testcode.nb.html
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+
+
+
+
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+
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+
+
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+
+
+
+R Notebook
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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+
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+
    
+
+
+
+
+
+
+
+
+
    Testing
    
+
+
+
+
    data()
    
+
+
+
+
+
+
+
    data(package = .packages(all.available = TRUE))
    
+
+
+
+
+
+
+
    
+
+
    
+
+
+
    head(InstEval)
+
    
+
+
+
+
+
    library(lmerTest)
    
+
+
+
    Warning: package ‘lmerTest’ was built under R version 4.2.2
+Attaching package: ‘lmerTest’
+
+The following object is masked from ‘package:lme4’:
+
+    lmer
+
+The following object is masked from ‘package:stats’:
+
+    step
    
+
+
+
+
+
    InstEval$studage <- as.numeric(InstEval$studage)
+InstEval$lectage <- as.numeric(InstEval$lectage)
+InstEval$service <- as.numeric(InstEval$service) 
+InstEval$dept <- as.numeric(InstEval$dept)
    
+
+
+
+
+
+
+
    m1 <- lmer( formula = y ~ studage + lectage + service + dept + (1|s) + (1|d),  
+                     data = InstEval )
+summary( m1 )
    
+
+
+
    Linear mixed model fit by REML. t-tests use Satterthwaite's method [lmerModLmerTest
+]
+Formula: y ~ studage + lectage + service + dept + (1 | s) + (1 | d)
+   Data: InstEval
+
+REML criterion at convergence: 237616.9
+
+Scaled residuals: 
+     Min       1Q   Median       3Q      Max 
+-3.08727 -0.74646  0.04076  0.77073  3.15558 
+
+Random effects:
+ Groups   Name        Variance Std.Dev.
+ s        (Intercept) 0.1065   0.3264  
+ d        (Intercept) 0.2673   0.5170  
+ Residual             1.3837   1.1763  
+Number of obs: 73421, groups:  s, 2972; d, 1128
+
+Fixed effects:
+              Estimate Std. Error         df t value Pr(>|t|)    
+(Intercept)  3.342e+00  4.427e-02  2.661e+03  75.492  < 2e-16 ***
+studage      4.383e-02  8.393e-03  5.304e+03   5.222 1.84e-07 ***
+lectage     -4.683e-02  3.787e-03  5.433e+04 -12.367  < 2e-16 ***
+service     -7.086e-02  1.344e-02  5.901e+04  -5.273 1.35e-07 ***
+dept         8.121e-04  4.655e-03  1.255e+03   0.174    0.862    
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Correlation of Fixed Effects:
+        (Intr) studag lectag servic
+studage -0.442                     
+lectage  0.085 -0.448              
+service -0.330  0.066 -0.120       
+dept    -0.690  0.028 -0.032 -0.109
    
+
+
+
+
+
+
+
    library(lme4)
+data("sleepstudy")
+head(sleepstudy)
    
+
+
+
    
+
+
    
+
+
+
    NA
    
+
+
+
+
+
+
+
    library(magrittr)
+
    
+
+
+
    
+Attaching package: ‘magrittr’
+
+The following object is masked from ‘package:purrr’:
+
+    set_names
+
+The following object is masked from ‘package:tidyr’:
+
+    extract
    
+
+
+
+
+
+
+
    df <- dataframe(dep_1,dep_2, dep_3)
+
    
+
+
+
    Error in dataframe(dep_1, dep_2, dep_3) : 
+  could not find function "dataframe"
    
+
+
+
+
+
+
+
    df2 <- sleepstudy %>% filter(Days %in% c(1, 5))
    
+
+
+
+
+
+
+
    diff <- df$Reaction.5 - df$Reaction
+shapiro.test(diff)
    
+
+
+
    
+    Shapiro-Wilk normality test
+
+data:  diff
+W = 0.9041, p-value = 0.06775
    
+
+
+
+
+
    leveneTest(Reaction ~ Days, df2)
+
    
+
+
+
    Levene's Test for Homogeneity of Variance (center = median)
+      Df F value Pr(>F)
+group  1  2.0914 0.1573
+      34               
    
+
+
+
+
+
+
+
    t.test(df$Reaction, df$Reaction.5, paired = TRUE, alternative = "two.sided", var.equal = TRUE)
    
+
+
+
    
+    Paired t-test
+
+data:  df$Reaction and df$Reaction.5
+t = -3.2912, df = 17, p-value = 0.004311
+alternative hypothesis: true mean difference is not equal to 0
+95 percent confidence interval:
+ -78.24917 -17.11583
+sample estimates:
+mean difference 
+       -47.6825 
    
+
+
+
+
+
+
+
    m2 <- lmer( formula = Reaction ~ Days + (1|Subject),  
+                     data = sleepstudy )
+summary( m2 )
    
+
+
+
    Linear mixed model fit by REML. t-tests use Satterthwaite's method [
+lmerModLmerTest]
+Formula: Reaction ~ Days + (1 | Subject)
+   Data: sleepstudy
+
+REML criterion at convergence: 1786.5
+
+Scaled residuals: 
+    Min      1Q  Median      3Q     Max 
+-3.2257 -0.5529  0.0109  0.5188  4.2506 
+
+Random effects:
+ Groups   Name        Variance Std.Dev.
+ Subject  (Intercept) 1378.2   37.12   
+ Residual              960.5   30.99   
+Number of obs: 180, groups:  Subject, 18
+
+Fixed effects:
+            Estimate Std. Error       df t value Pr(>|t|)    
+(Intercept) 251.4051     9.7467  22.8102   25.79   <2e-16 ***
+Days         10.4673     0.8042 161.0000   13.02   <2e-16 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Correlation of Fixed Effects:
+     (Intr)
+Days -0.371
    
+
+
+
+
+
+
+
    m3 <- lmer( formula = Reaction ~ Days + (1 + Days|Subject),  
+                     data = sleepstudy )
+summary( m3 )
    
+
+
+
    Linear mixed model fit by REML. t-tests use Satterthwaite's method [
+lmerModLmerTest]
+Formula: Reaction ~ Days + (1 + Days | Subject)
+   Data: sleepstudy
+
+REML criterion at convergence: 1743.6
+
+Scaled residuals: 
+    Min      1Q  Median      3Q     Max 
+-3.9536 -0.4634  0.0231  0.4634  5.1793 
+
+Random effects:
+ Groups   Name        Variance Std.Dev. Corr
+ Subject  (Intercept) 612.10   24.741       
+          Days         35.07    5.922   0.07
+ Residual             654.94   25.592       
+Number of obs: 180, groups:  Subject, 18
+
+Fixed effects:
+            Estimate Std. Error      df t value Pr(>|t|)    
+(Intercept)  251.405      6.825  17.000  36.838  < 2e-16 ***
+Days          10.467      1.546  17.000   6.771 3.26e-06 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Correlation of Fixed Effects:
+     (Intr)
+Days -0.138
    
+
+
+
+
+
    AIC(m2, m3)
    
+
+
+
    
+
+
    
+
+
+
+
+
+
+
    anova(m2,m3)
    
+
+
+
    refitting model(s) with ML (instead of REML)
    
+
+
+
    Data: sleepstudy
+Models:
+m2: Reaction ~ Days + (1 | Subject)
+m3: Reaction ~ Days + (1 + Days | Subject)
+   npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)    
+m2    4 1802.1 1814.8 -897.04   1794.1                         
+m3    6 1763.9 1783.1 -875.97   1751.9 42.139  2  7.072e-10 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
+
+
+
+
+
+
+
    library(easystats)
    
+
+
+
    Warning: package ‘easystats’ was built under R version 4.2.2# Attaching packages: easystats 0.5.2
+✔ insight     0.18.7   ✔ datawizard  0.6.4 
+✔ bayestestR  0.13.0   ✔ parameters  0.19.0
+✔ effectsize  0.8.2    ✔ modelbased  0.8.5 
+✔ correlation 0.8.3    ✔ see         0.7.3 
+✔ report      0.5.5    
    
+
+
+
+
+
+
+
    check_model(m2)
    
+
+
+
    
+
+
+
    
+
+
+
+
+
+
+
    check_heteroscedasticity(m3)
    
+
+
+
    Warning: Heteroscedasticity (non-constant error variance) detected (p < .001).
    
+
+
+
+
+
+
+
    data("HairEyeColor")
+head(HairEyeColor)
+
    
+
+
+
    , , Sex = Male
+
+       Eye
+Hair    Brown Blue Hazel Green
+  Black    32   11    10     3
+  Brown    53   50    25    15
+  Red      10   10     7     7
+  Blond     3   30     5     8
+
+, , Sex = Female
+
+       Eye
+Hair    Brown Blue Hazel Green
+  Black    36    9     5     2
+  Brown    66   34    29    14
+  Red      16    7     7     7
+  Blond     4   64     5     8
    
+
+
+
+
+
    tab_n <- c(124,151,105,149)
    
+
+
+
+
+
+
+
    library(lsr)
+cramersV( tab_n )
    
+
+
+
    [1] 0.08291044
    
+
+
+
+
+
+
+
    chisq.test( tab_n  )
    
+
+
+
    
+    Chi-squared test for given probabilities
+
+data:  tab_n
+X-squared = 10.909, df = 3, p-value = 0.01223
    
+
+
+
+
+
    library(pwr)
+pwr.chisq.test(w = 0.08, df = 3, sig.level = 0.05, power = 0.8 )
    
+
+
+
    
+     Chi squared power calculation 
+
+              w = 0.08
+              N = 1703.526
+             df = 3
+      sig.level = 0.05
+          power = 0.8
+
+NOTE: N is the number of observations
    
+
+
+
+
+
+
+
    data("mtcars")
+head(mtcars)
    
+
+
+
+
+
+
+
    m2 <- lm( formula = mpg ~ hp + wt,  
+                     data = mtcars )
+summary( m2 )
    
+
+
+
+
+
    library(car)
    
+
+
+
+
+
    plot(x = m1, which = 3)
+
    
+
+
+
+
+
+
+
    ncvTest( m1 )
    
+
+
+
+
+
+
+
    vif( mod = m1 )
    
+
+
+
+
+
    residualPlots( model = m1 ) 
    
+
+
+
+
+
    plot( x = m2, which = 2) 
+
    
+
+
+
+
+
    shapiro.test(residuals( m1 ))
    
+
+
+
+
+
+
+
    
+m3 <- lm( formula = log(mpg) ~ log(hp) + log(wt),  
+                     data = mtcars )
+summary( m3 )
    
+
+
+
+
+
    residualPlots( model = m3 ) 
    
+
+
+
+
+
    ncvTest( m3 )
    
+
+
+
+
+
+
+
    vif( mod = m3 )
    
+
+
+
+
+
+
+
    library(MASS)
    
+
+
+
+
+
+
+
    #bc <- boxcox(m3)
+bc.car <- powerTransform(m2)
+lambda <- bc.car$lambda
+
+#lambda <- bc$x[which.max(bc$y)]
+#lambda
    
+
+
+
+
    fit new linear regression model using the Box-Cox transformation
    
+
+
+
+
    m4 <- lm(((mpg^lambda-1)/lambda) ~ hp + wt,data = mtcars)
+summary(m4)
    
+
+
+
+
+
+
+
    residualPlots( model = m4 ) 
    
+
+
+
+
+
+
+
    ncvTest( m4 )
    
+
+
+
+
+
+
+
    vif( mod = m4 )
    
+
+
+
+
+
+
+
    plot( x = m3, which = 2) 
+
    
+
+
+
+
+
    shapiro.test(residuals( m4 ))
    
+
+
+
+
+
+
+
    data()
    
+
+
+
+
+
+
+
    library(lme4)
+data("sleepstudy")
+head(sleepstudy)
    
+
+
+
+
+
+
    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From 461b084cd8ee10f29a08e9e371aae880f41c5e49 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Fri, 1 Sep 2023 13:33:15 +0530
Subject: [PATCH 42/55] Changes to distribution parameters

---
 Module 3/Notebooks/Distributions.Rmd     |   37 +-
 Module 3/Notebooks/Distributions.nb.html | 2126 ++++++++++++++++--
 Module 3/Notebooks/Module3_Nb2.Rmd       |   20 +-
 Module 3/Notebooks/Module3_Nb2.nb.html   | 2539 +++++++++++++++++++---
 4 files changed, 4232 insertions(+), 490 deletions(-)

diff --git a/Module 3/Notebooks/Distributions.Rmd b/Module 3/Notebooks/Distributions.Rmd
index f38d5a7e..0225228d 100644
--- a/Module 3/Notebooks/Distributions.Rmd	
+++ b/Module 3/Notebooks/Distributions.Rmd	
@@ -107,8 +107,10 @@ Some basic terminology - We’ll let `N` denote the number of dice rolls in our
 Let's generate a binomial distribution in R:
 
 ```{r}
-dbinom( x = 1, size = 20, prob = 1/6)
+dbinom( x = 4, size = 10, prob = 1/2)
 ```
+
+
 The above command calculates the probability of getting x = 4 skulls, from an experiment of size = 20 trials, in which the probability of getting a skull on any one trial is prob = 1/6.
 
 What if the dice is replaced by a coin in the above example? How will the probability change? 
@@ -122,7 +124,7 @@ If we want to find the probability of obtaining an outcome smaller than or equal
 
 ```{r}
 #Find the probability of rolling 0 skulls or 1 skull or 2 skulls or 3 skulls or 4 skulls
-pbinom( q= 3, size = 20, prob = 1/6)
+pbinom( q= 4, size = 10, prob = 1/2)
 
 #Practice - Find probability of getting 0-5 heads in 50 trials of coin flip
 ```
@@ -131,7 +133,7 @@ In other words, value of 4 is actually the 76.9th percentile of this binomial di
 Now let’s say we want to calculate the 75th percentile of the binomial distribution.
 
 ```{r}
-qbinom( p = 0.566, size = 20, prob = 1/6 )
+qbinom( p = 0.376, size = 10, prob = 1/2 )
 
 #Practice - Find the 40th percentile
 ```
@@ -139,9 +141,9 @@ qbinom( p = 0.566, size = 20, prob = 1/6 )
 We've found different quantities. What if we want to simulate the above experiments. We specify how many times R should “simulate” the experiment using the n argument, and it will generate random outcomes from the binomial distribution using the `rbinom` function.
 
 ```{r}
-z <- rbinom( n = 1000000, size = 100, prob = 1/2 )
+z <- rbinom( n = 10000, size = 10, prob = 1/2 )
 #Let's also plot this and see how it looks
-hist(z, breaks=15, col = 'steelblue')
+hist(z, breaks=25, col = 'steelblue')
 ```
 #Try plotting the distributions in above examples and vary the size, trial number and probability to generate different plots.
 ```{r}
@@ -165,7 +167,7 @@ qbinom(p = 0.025, size = 10, prob = 1/2)
 ```
 
 ```{r}
-binom.test( x=62, n=100, p=.5 )
+binom.test( x=6, n=10, p=1/2 )
 ```
 
 
@@ -232,7 +234,7 @@ Basically, whenever you have accumulation of data at the center, fewer extreme v
 
 mean = 0; sd = 1 -> standard normal distribution
 ```{r}
-normal_distribution <- rnorm(10000, mean = 10, sd = 5) 
+normal_distribution <- rnorm(10000, mean = 165, sd = 20) 
 histogram_normal_distribution <- hist(normal_distribution)
 plot(histogram_normal_distribution$mids,histogram_normal_distribution$density, type="l", col="blue", lwd=1)
 
@@ -243,32 +245,41 @@ Note: Normal distribution is sometimes referred to as the bell curve or Gaussian
 
 The notation for a normal distribution is: X ∼ Normal(μ,σ) 
 
+```{r}
+no_test <- rnorm(10000, mean = 165, sd = 20)
+
+```
+
+
 
 dnorm tells you the probability of getting a particular outcome
 ```{r}
-dnorm(x=10, mean=10, sd=5)
+dnorm(x=165, mean=165, sd=20)
 
 ```
 Cumulative normal distribution
 ```{r}
-pnorm(q = 19.8, mean = 10, sd = 5)
+pnorm(q = 175, mean = 165, sd = 20)
 ```
 
 ```{r}
-qnorm(0.25 ,mean = 0 , sd = 1)
+qnorm(0.75 ,mean = 165 , sd = 20)
 ```
 
 *Checking for normality using the Shapiro-Wilk Test*
 ```{r}
-norm <- rnorm(50, mean = 0, sd = 1) 
+norm <- rnorm(1000, mean = 0, sd = 1) 
 shapiro.test(norm)
 
-binom <- rbinom(100, 20, 1/6)
-shapiro.test(binom)
+
 
 ```
 
 
+```{r}
+binom <- rbinom(100, 10, 1/2)
+shapiro.test(binom)
+```
 
 
 
diff --git a/Module 3/Notebooks/Distributions.nb.html b/Module 3/Notebooks/Distributions.nb.html
index 97a872b7..d40c7fa8 100644
--- a/Module 3/Notebooks/Distributions.nb.html	
+++ b/Module 3/Notebooks/Distributions.nb.html	
@@ -13,13 +13,49 @@
 
 Inferential Statistics: Probability & Distributions - 1
 
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+
 
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+
 
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+
 
 
 
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+
 
 
 
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+
 
 
 
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+
 
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+
 
 
 
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+
    This is an R Markdown
+Notebook. When you execute code within the notebook, the results appear
+beneath the code.
    
+
    Try executing this chunk by clicking the Run button within
+the chunk or by placing your cursor inside it and pressing
+Ctrl+Shift+Enter.
    
+
    Regression with categorical predictors: Basically, using regression
+for a t-test!
    
+
    Reference: Bodo Winter Chapter 7
    
+
+
+
+
    library(tidyverse)
+library(broom)
    
+
+
+
+
+
+
+
    senses <- read_csv('winter_2016_senses_valence.csv')
    
+
+
+
    Rows: 405 Columns: 6── Column specification ──────────────────────────────────────────────────────────────────────────────────────────
+Delimiter: ","
+chr (2): Word, DominantModality
+dbl (4): Val, AbsVal, Sent, AbsSent
+ℹ Use `spec()` to retrieve the full column specification for this data.
+ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
    
+
+
+
    senses
    
+
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+
    
+
+
    
+
+
+
+
    Preprocessing and visualization to get only the Taste and Smell
+data
    
+
+
+
+
    chem <- filter(senses, DominantModality %in% c('Taste', 'Smell'))
+table(chem$DominantModality)
    
+
+
+
    
+Smell Taste 
+   25    47 
    
+
+
+
+
+
+
+
    chem %>% group_by(DominantModality) %>%
+summarize(M = mean(Val), SD = sd(Val))
    
+
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+
    
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+
    
+
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+
    chem %>% ggplot(aes(x = DominantModality, y = Val, fill = DominantModality)) +
+geom_boxplot() + theme_minimal() +
+scale_fill_brewer(palette = 'Accent')
    
+
+
+
    
+
+
+
+
    If you are interested in choosing the right colors for your plots,
+then here’s a good resource https://ggplot2-book.org/scales-colour
    
+
+
+
+
    chem_mdl <- lm(Val ~ DominantModality, data = chem)
+summary(chem_mdl)
    
+
+
+
    
+Call:
+lm(formula = Val ~ DominantModality, data = chem)
+
+Residuals:
+     Min       1Q   Median       3Q      Max 
+-0.99315 -0.20870  0.04343  0.19115  0.62788 
+
+Coefficients:
+                      Estimate Std. Error t value Pr(>|t|)    
+(Intercept)            5.47101    0.06297  86.889  < 2e-16 ***
+DominantModalityTaste  0.33711    0.07793   4.326 4.95e-05 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Residual standard error: 0.3148 on 70 degrees of freedom
+Multiple R-squared:  0.2109,    Adjusted R-squared:  0.1997 
+F-statistic: 18.71 on 1 and 70 DF,  p-value: 4.951e-05
    
+
+
+
+
    Using the predict() to determine the values associated with each
+predictor
    
+
+
+
+
    chem_preds <- tibble(DominantModality = unique(chem$DominantModality))
+chem_preds$fit <- predict(chem_mdl, chem_preds)
+chem_preds
    
+
+
+
    
+
+
    
+
+
+
+
    Change the reference - from Smell to Taste.
    
+
+
+
+
    chem <- mutate(chem,
+DominantModality = factor(DominantModality),
+ModRe = relevel(DominantModality, ref = 'Taste'))
    
+
+
+
+
    Verify reference levels
    
+
+
+
+
    levels(chem$DominantModality)
+levels(chem$ModRe)
    
+
+
+
+
    Rerun the regression
    
+
+
+
+
    lm(Val ~ ModRe, data = chem)
    
+
+
+
    
+Call:
+lm(formula = Val ~ ModRe, data = chem)
+
+Coefficients:
+(Intercept)   ModReSmell  
+     5.8081      -0.3371  
    
+
+
+
+
    Treatment Coding vs Sum Coding:
    
+
    Treatment Coding: Taste = 0; Smell = 1
    
+
    Sum Coding: Taste = -1; Smell = 1
    
+
    Sum Coding is useful esepcially for interpreting interactions.
    
+
    
+
    Fitting categorical data with more than two levels
    
+
+
+
+
    unique(senses$DominantModality)
    
+
+
+
    [1] "Touch" "Sight" "Taste" "Smell" "Sound"
    
+
+
+
+
+
+
+
    sense_all <- lm(Val ~ DominantModality, data = senses)
+summary(sense_all)
    
+
+
+
    
+Call:
+lm(formula = Val ~ DominantModality, data = senses)
+
+Residuals:
+     Min       1Q   Median       3Q      Max 
+-0.99315 -0.16482 -0.02158  0.15920  1.15734 
+
+Coefficients:
+                      Estimate Std. Error t value Pr(>|t|)    
+(Intercept)            5.57966    0.01889 295.308  < 2e-16 ***
+DominantModalitySmell -0.10865    0.05643  -1.925   0.0549 .  
+DominantModalitySound -0.17447    0.03758  -4.643 4.66e-06 ***
+DominantModalityTaste  0.22846    0.04314   5.296 1.96e-07 ***
+DominantModalityTouch -0.04523    0.03737  -1.210   0.2269    
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+
+Residual standard error: 0.2659 on 400 degrees of freedom
+Multiple R-squared:  0.1455,    Adjusted R-squared:  0.137 
+F-statistic: 17.03 on 4 and 400 DF,  p-value: 6.616e-13
    
+
+
+
+
    Using predict to help with interpretation
    
+
+
+
+
    sense_preds <- tibble(DominantModality = sort(unique(senses$DominantModality))) 
+sense_preds$fit <- predict(sense_all, sense_preds)
+sense_preds
    
+
+
+
    
+
+
    
+
+
+
+
    Lastly, check assumptions:
    
+
    Normality test:
    
+
+
+
+
    library(car)
    
+
+
+
    Loading required package: carData
+
+Attaching package: ‘car’
+
+The following object is masked from ‘package:dplyr’:
+
+    recode
+
+The following object is masked from ‘package:purrr’:
+
+    some
    
+
+
+
+
+
+
+
    shapiro.test(residuals(sense_all))
    
+
+
+
    
+    Shapiro-Wilk normality test
+
+data:  residuals(sense_all)
+W = 0.98434, p-value = 0.0002282
    
+
+
+
+
+
     hist( x = residuals( sense_all ))
+
    
+
+
+
    
+
+
+
+
+
    plot( x = sense_all, which = 2 )
    
+
+
+
    
+
+
+
+
+
+
+
    ncvTest( sense_all )
    
+
+
+
    Non-constant Variance Score Test 
+Variance formula: ~ fitted.values 
+Chisquare = 0.07688406, Df = 1, p = 0.78157
    
+
+
+
+
    Linearity test
    
+
+
+
+
    library(carData)
    
+
+
+
+
+
+
+
    residualPlots( model = sense_all ) 
    
+
+
+
    
+
+
+
+
    Lastly, we assess the variance inflation factor – to diagnose for
+collinearity
    
+
+
+
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    car::vif( mod = sense_all )
    
+
+
+
    Error in vif.default(mod = sense_all) : model contains fewer than 2 terms
    
+
+
+
+
    Add a new chunk by clicking the Insert Chunk button on the
+toolbar or by pressing Ctrl+Alt+I.
    
+
    When you save the notebook, an HTML file containing the code and
+output will be saved alongside it (click the Preview button or
+press Ctrl+Shift+K to preview the HTML file).
    
+
    The preview shows you a rendered HTML copy of the contents of the
+editor. Consequently, unlike Knit, Preview does not
+run any R code chunks. Instead, the output of the chunk when it was last
+run in the editor is displayed.
    
+
+
+
    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diff --git a/Module 6/winter_2016_senses_valence.csv b/Module 6/winter_2016_senses_valence.csv
new file mode 100644
index 00000000..171c3852
--- /dev/null
+++ b/Module 6/winter_2016_senses_valence.csv	
@@ -0,0 +1,406 @@
+Word,Val,AbsVal,Sent,AbsSent,DominantModality
+abrasive,5.398113208,0.843700622,0.194851852,0.486741904,Touch
+absorbent,5.876666667,0.886794227,0.147041667,0.517712259,Sight
+aching,5.233369565,1.187834297,0.054022989,0.581149967,Touch
+acidic,5.539591837,1.094426701,0.140979167,0.550524759,Taste
+acrid,5.173947368,1.198603117,0.27927027,0.752406679,Smell
+adhesive,5.24,0.704821556,0.191047619,0.427764896,Touch
+alcoholic,5.557227723,1.297204591,0.319635417,0.589180028,Taste
+alive,6.040952381,1.15744753,-0.081378788,0.436801603,Sight
+amber,5.720549451,0.98872315,0.509806818,0.679473047,Sight
+angular,5.477615894,0.771201857,0.082993056,0.50379992,Sight
+antiseptic,5.511794872,1.034930669,0.08225641,0.634669083,Smell
+aromatic,5.952941176,1.153099047,0.528716418,0.638933108,Smell
+astringent,5.965,1.184922458,0.409052632,0.842423532,Taste
+audible,5.351283784,1.052645342,-0.020659574,0.558118006,Sound
+banging,5.706666667,1.172051024,0.247222222,0.318783012,Sound
+barbecued,6.052777778,1.240213159,0.645368421,0.917268943,Taste
+barking,5.696315789,1.211740606,0.240058824,0.618296889,Sound
+beautiful,5.819057493,1.094074032,0.30241619,0.609696419,Sight
+beeping,4.898,0.681538771,-0.0416,0.407190578,Sound
+beery,6.072142857,1.711373184,0.300266667,0.859998953,Taste
+beige,5.585794393,0.775966532,0.250476636,0.501792238,Sight
+bent,5.382307692,0.644934697,0.148606299,0.410828379,Sight
+big,5.279962335,1.010275059,0.073129602,0.630690464,Sight
+bitter,5.124433249,1.295432917,-0.088536082,0.712386112,Taste
+black,5.314124016,0.946227064,0.082560516,0.654928223,Sight
+bland,5.752285714,1.030387858,0.230095238,0.720230815,Taste
+blaring,5.275555556,1.121794786,-0.285444444,0.926338568,Sound
+bleeping,5.12,0.838717691,-0.075,0.200338568,Sound
+blonde,5.523850932,1.080950505,0.196423313,0.541344811,Sight
+bloody,5.062922201,1.14708936,0.067457364,0.61476454,Sight
+blotchy,5.692,0.805691843,0.0684,0.163596859,Sight
+blue,5.57398622,0.870116971,0.30159042,0.591372228,Sight
+blunt,5.225740741,0.940279923,-0.14087013,0.56955579,Touch
+boiling,5.467142857,0.968486549,0.443638298,0.710686865,Sight
+booming,5.528,0.94826207,0.13847561,0.635381496,Sound
+bouncy,6.130857143,1.174746656,0.225705882,0.432004619,Touch
+brackish,5.846363636,1.278461046,0.41875,0.891917975,Touch
+branching,5.616551724,0.733182624,0.083285714,0.501002243,Sight
+breakable,5.778333333,0.715768715,0.3748,0.424009422,Sight
+breezy,5.993287671,1.080252315,0.137514706,0.554991298,Sight
+bright,5.744751381,0.975360081,0.292201818,0.58953661,Sight
+brilliant,5.725644531,1.040307563,0.153002012,0.575518398,Sight
+briny,5.854210526,1.410930726,0.601631579,0.860166986,Taste
+bristly,5.487931034,0.710079175,0.21075,0.394683058,Touch
+brittle,5.687473684,0.989100864,0.129706522,0.53992562,Touch
+broad,5.507315951,0.902034811,0.073175079,0.604878549,Sight
+broken,5.542655678,0.83361053,0.210868472,0.539069318,Sight
+brown,5.470278638,0.905534344,0.2127366,0.565539475,Sight
+bulky,5.424375,0.804525698,-0.013833333,0.509406535,Sight
+bumpy,5.566727273,0.779691843,0.411571429,0.459580963,Touch
+burning,5.284299287,1.023708932,0.191967419,0.55767471,Smell
+burnt,5.688061224,1.067432949,0.395278351,0.671138715,Smell
+bursting,5.030666667,0.697897281,0.158214286,0.3746451,Sight
+buttery,5.981587302,1.137374319,0.710060606,0.777586702,Taste
+buzzing,5.304186047,1.024132181,0.1234,0.529800349,Sound
+charred,5.445233645,0.932745857,0.07051,0.61749157,Sight
+cheesy,5.712931034,1.050198459,-0.023553571,0.776446989,Taste
+chewy,6.031666667,1.19535851,0.844375,0.862762759,Taste
+chilly,5.646640625,1.112145094,0.354584746,0.638912635,Touch
+chubby,5.797,1.128784089,0.113319149,0.484297538,Sight
+circular,5.500909091,0.806298707,0.236855124,0.573049099,Sight
+citrusy,6.394166667,1.404294227,0.767166667,1.068507852,Taste
+clammy,5.379714286,0.88707668,0.249882353,0.549476131,Touch
+clamorous,5.485882353,1.151357147,0.316588235,0.53894949,Touch
+clean,5.519521989,0.928112601,0.123771203,0.500923913,Sight
+clear,5.441101591,0.985005812,-0.007715006,0.613408866,Sight
+cloudy,5.802820513,1.018914711,0.382493333,0.547138568,Sight
+cloying,5.882222222,1.263218829,0.277192308,0.798193516,Taste
+cold,5.441072797,1.06014217,0.289351105,0.65307916,Touch
+colorful,5.733112245,0.899834659,0.17256266,0.54775181,Sight
+colossal,4.809586207,1.172870021,-0.147886525,0.695813041,Sight
+compact,5.610840336,0.739411342,0.187937778,0.458038836,Sight
+conical,5.315897436,0.743648618,0.114947368,0.420424359,Sight
+contoured,5.273809524,0.563589412,-0.028590909,0.38749429,Sight
+cooing,5.44625,1.227788268,0.239125,0.815128926,Sound
+cool,5.733410138,1.028034568,0.252304075,0.576385448,Touch
+crackling,5.406304348,0.927474714,0.442822222,0.649450377,Sound
+craggy,5.613863636,0.836940172,0.405756098,0.510248882,Sight
+crashing,5.431944444,1.142115084,0.257647059,0.536945795,Sight
+creaking,5.258,0.758856855,0.057939394,0.510667143,Sound
+creamy,5.915211268,0.969966952,0.596403974,0.759087894,Taste
+creased,5.638333333,0.775512477,0.4309,0.564307852,Sight
+crimson,5.701768707,0.99066938,0.442277778,0.649005453,Sight
+crinkled,5.62125,0.68634567,0.051533333,0.471932286,Sight
+crisp,5.792485207,0.992061413,0.416251429,0.70875958,Touch
+crooked,5.554246575,0.864583454,0.119426573,0.540757331,Sight
+crowded,5.491034483,0.859884711,0.260286822,0.506067006,Sight
+crunching,5.56875,1.014326536,-0.498888889,0.9944427,Sound
+curly,5.797142857,0.908094759,0.377977778,0.595180918,Sight
+curved,5.403806818,0.681708608,0.302923077,0.494359397,Sight
+cute,5.948918919,1.087993251,0.069875,0.440716279,Sight
+damp,5.563317073,0.863320414,0.28277561,0.550488366,Touch
+dank,5.300655738,1.085611411,0.294472727,0.683277581,Sight
+dappled,6.320952381,1.464138494,0.466090909,0.657007138,Sight
+dark,5.51117395,0.993654002,0.261747615,0.585279711,Sight
+dazzling,6.0021875,1.109678899,0.373535135,0.626189471,Sight
+dead,5.540287443,1.069409696,0.131922039,0.556985792,Sight
+deafening,5.128783784,0.946372037,0.186716418,0.600361256,Sound
+deep,5.276958525,1.152975692,0.037859085,0.65502942,Sight
+delicious,5.820344828,1.248790013,0.393941176,0.700223481,Taste
+dim,5.670717949,0.911754952,0.308666667,0.577072356,Sight
+dirty,5.396744186,1.041647502,0.095174935,0.51469587,Sight
+downy,5.875882353,1.152216694,0.4662,0.537807328,Sight
+drab,5.702307692,0.942307176,0.171114286,0.498316529,Sight
+dry,5.469625668,0.913745739,0.243444056,0.625744773,Touch
+dull,5.377754386,1.130005194,0.092355072,0.558965361,Sight
+dusty,5.498056426,0.848374421,0.216871287,0.54573326,Sight
+earthy,5.9436,1.270230212,0.379395833,0.765985383,Sight
+echoing,5.227454545,1.043776069,0.108236364,0.465400857,Sound
+eggy,6.436,1.777691843,1.07675,1.327757852,Taste
+elastic,5.487065217,0.834874229,-0.028510638,0.557829787,Touch
+elegant,5.783939394,0.971478009,0.285729437,0.623574023,Sight
+empty,5.501381669,0.871316108,0.242671491,0.592348224,Sight
+enormous,5.308307692,1.10687257,0.012227683,0.646293771,Sight
+faint,5.395808581,1.107087866,0.193757475,0.644357725,Sight
+falling,5.42992,0.970258208,0.172983471,0.542554108,Sight
+fat,5.380684932,1.036814433,0.045331429,0.629380683,Sight
+fatty,5.365283019,0.945268198,0.203791667,0.693666667,Taste
+fetid,5.210606061,1.234871678,-0.060294118,0.840940253,Smell
+feverish,5.6364,1.145830413,0.180381443,0.502096183,Touch
+filthy,5.32619883,1.025456323,0.107319277,0.555008294,Sight
+fishy,5.7956,1.158830212,0.143425926,0.649982063,Smell
+flaky,5.906,1.082109337,0.118486486,0.632595868,Touch
+flat,5.48552809,0.841512204,0.138405286,0.591208709,Sight
+fleshy,5.603472222,0.875181129,0.1761,0.464273223,Sight
+flickering,5.691851852,0.949068915,0.296746835,0.659257339,Sight
+floppy,5.60125,0.742339292,0.211272727,0.42054831,Sight
+floral,5.783373494,1.030435094,0.394339506,0.751000031,Sight
+flowery,5.849807692,1.109452127,0.27892,0.625002513,Sight
+fluffy,5.941969697,0.943600812,0.410217391,0.567992107,Touch
+foamy,5.396153846,1.188323134,0.335842105,0.770318269,Sight
+foggy,5.673283582,0.909850296,0.425439394,0.629839995,Sight
+forked,5.738125,0.857211103,0.3503,0.544639615,Sight
+fragrant,5.860545455,1.172167354,0.637763636,0.796209993,Smell
+freezing,5.465164835,0.920878789,0.107438202,0.527103417,Touch
+fresh,5.453644159,1.058009421,0.247532468,0.677849549,Smell
+frosty,5.83578125,1.050444204,0.302692308,0.581189689,Sight
+fruity,5.958382353,1.080191784,0.570310811,0.75977791,Taste
+fuzzy,5.703173432,0.951345024,0.167674157,0.522680196,Sight
+garlicky,5.971944444,1.211602048,0.610111111,0.797176263,Taste
+gigantic,5.379981618,1.009014184,0.14768785,0.628786025,Sight
+giggling,5.805238095,1.316776004,0.0212,0.497596859,Sound
+glamorous,5.867992126,1.031383504,0.239468,0.495542513,Sight
+gleaming,5.567815385,0.861000852,0.29075625,0.560880202,Sight
+glistening,5.636195122,0.913354261,0.28328934,0.601567198,Sight
+glittery,5.607974684,0.94395286,0.192556962,0.53871065,Sight
+globular,5.597142857,0.697252194,0.0773,0.501095289,Sight
+glossy,5.69556213,0.870905272,0.267028902,0.582871863,Sight
+glowing,5.707633588,0.926597737,0.262279373,0.569102784,Sight
+gold,5.510550459,0.832349659,0.212587156,0.508773668,Sight
+gooey,5.731956522,1.142809011,0.239909091,0.670343407,Touch
+gorgeous,5.921105882,1.104730692,0.426390995,0.643675677,Sight
+grainy,5.741573034,0.986248459,0.0854,0.592579174,Touch
+grassy,5.638606061,0.81696927,0.219238994,0.474903618,Sight
+gray,5.377326944,0.879013424,0.269956943,0.584393368,Sight
+greasy,5.462868852,0.986809305,0.209275424,0.567668315,Touch
+green,5.522520516,0.918839048,0.229263521,0.637912524,Sight
+gritty,5.712488889,1.063397874,0.246355856,0.537953987,Touch
+groaning,5.3925,1.069697526,0.184928571,0.385146222,Sound
+grotesque,5.153974359,1.224296357,-0.214844749,0.756568267,Sight
+growling,5.283823529,0.979162668,0.049911765,0.529089159,Sound
+grubby,5.602173913,0.973087701,0.169742424,0.426079089,Sight
+gurgling,5.121621622,1.49070684,-0.169567568,0.781133013,Sound
+hairy,5.195744681,0.996947462,-0.020716578,0.509164852,Sight
+handsome,5.71040153,0.990935016,0.145710983,0.537131908,Sight
+happy,5.628561644,1.141903416,0.112581655,0.580655176,Sight
+hard,5.441934884,0.979582548,0.110767956,0.558275558,Touch
+harsh,5.249680171,1.02297013,-0.032264317,0.623418018,Sound
+hazy,5.745,1.064040557,0.384396104,0.66664714,Sight
+heavy,5.205227119,0.939753686,0.099854583,0.633693484,Touch
+high,5.436636829,0.993667823,0.050994819,0.678831267,Sight
+hissing,5.35704918,0.999381903,0.253857143,0.684685282,Sound
+hoarse,5.390217391,1.22841127,-0.056588235,0.696941484,Sound
+hollow,5.444704433,0.933684833,0.134235897,0.517545684,Sight
+honeyed,5.940416667,1.101602048,0.292130435,0.922658319,Taste
+hot,5.528398719,0.999291429,0.264069477,0.631770058,Touch
+howling,4.933522727,1.280970315,0.104747126,0.695069507,Sound
+huge,5.262619254,1.000398121,0.065123412,0.678684885,Sight
+humid,5.586162791,1.110831519,0.338202247,0.562118182,Touch
+hushed,5.672018349,1.247903675,0.122443396,0.691862046,Sound
+husky,5.609876543,1.120711897,0.137644737,0.453882819,Sound
+icy,5.434382022,1.015641898,0.339471698,0.716037537,Sight
+immense,5.405320122,1.139663207,0.044130159,0.664061265,Sight
+insipid,6.155227273,1.431870137,-0.1328,0.669553113,Sight
+itchy,5.224848485,1.02463853,0.142447761,0.460451277,Touch
+jagged,5.306728972,0.843802882,0.246238806,0.538531348,Touch
+jammy,5.517142857,1.41087901,0.192,0.636890634,Taste
+jingling,5.798,1.248461229,0.6238,0.758206281,Sound
+juicy,5.674646465,1.117233463,0.243393204,0.600307655,Taste
+khaki,5.744918033,0.831512721,0.25284127,0.480498793,Sight
+large,5.371699427,0.883640211,0.188187581,0.649644371,Sight
+laughing,5.542653061,1.155933782,-0.030636364,0.51146195,Sound
+leathery,5.649411765,0.746696286,0.337411765,0.524886972,Touch
+lemony,6.201777778,1.355127672,0.780625,0.91405279,Taste
+light,5.461285831,0.901889698,0.230668023,0.61965099,Sight
+lilting,6.17125,1.297114804,0.5034,0.637672948,Sight
+lithe,5.680909091,0.924559004,0.252862745,0.526514423,Sight
+little,5.248460458,1.006403959,0.087386576,0.715857707,Sight
+long,5.283406473,0.923859872,0.134224957,0.63314965,Sight
+loose,5.419716088,0.774770396,0.065110749,0.579434708,Sight
+loud,5.208567639,1.017706465,-0.009465608,0.623614089,Sound
+low,5.38074108,0.946057895,0.014958716,0.651570243,Sight
+lukewarm,6.003924051,1.174507757,0.3914125,0.741891818,Touch
+lumpy,5.75376,0.932042538,0.184432,0.494883392,Touch
+lush,5.970545455,1.049936465,0.354795349,0.669462314,Sight
+malty,5.87,0.967252194,0.71625,1.344011777,Taste
+meaty,5.502743363,0.929706976,0.306575221,0.684809062,Taste
+medicinal,5.528715596,0.978404705,0.125419048,0.558867115,Sight
+mellow,5.997295082,1.073284385,0.412645669,0.78016255,Sight
+melodious,5.941627907,1.300428816,0.319854167,0.648107438,Sound
+melted,5.526966292,0.96280865,0.368268817,0.742424927,Sight
+meowing,5.53,0.466153072,-0.609,0.672984297,Sound
+mild,4.81497553,1.299486783,-0.01188551,0.76000724,Taste
+miniature,5.61950783,0.974394723,0.375247059,0.668382272,Sight
+minty,6.074827586,1.194800501,0.5178,0.750003141,Taste
+misty,5.768053691,1.100541644,0.386066667,0.623565025,Sight
+moaning,5.2115,0.889345922,-0.02215,0.499245289,Sound
+moist,5.59518018,1.02707171,0.324640553,0.625281636,Touch
+motionless,5.533119266,0.9117992,0.102981982,0.408316589,Sight
+mottled,5.611685393,0.938971,0.17777381,0.498183058,Sight
+muddy,5.317741935,0.897597765,0.152627586,0.538589672,Sight
+muffled,5.111319444,1.142815053,0.078992908,0.560651925,Sound
+muggy,6.271428571,1.249559776,0.616875,0.585261777,Sight
+mumbling,5.153846154,0.782781006,0.177416667,0.607583333,Sound
+murky,5.51844898,0.988750067,0.159689362,0.596984783,Sight
+murmuring,5.677307692,1.067543962,0.42872,0.533206909,Sound
+mushroomy,5.5,1.013691843,0.0712,0.244796859,Taste
+mushy,5.556268657,0.867003202,0.128294118,0.555443486,Touch
+musky,5.776756757,1.327421682,0.542472222,0.872534757,Smell
+musty,5.468288288,0.919022527,0.176432692,0.589609997,Smell
+mute,5.380520231,1.216780528,-0.067005814,0.617864271,Sound
+muttering,6.056666667,0.992819739,0.462,0.628507852,Sound
+narrow,5.395671141,0.7570011,0.089992916,0.577221873,Sight
+noisy,5.364528302,1.057078363,0.130005525,0.583427757,Sound
+noxious,4.849032258,1.110612134,0.002,0.677447373,Smell
+nutty,6.013333333,1.128756244,0.327776699,0.626130025,Taste
+odorous,5.362857143,1.060769194,0.153619048,0.520100473,Smell
+oily,5.468781726,1.00021056,0.278619792,0.576499208,Sight
+oniony,5.8725,1.495576536,0.463125,1.133378926,Taste
+open,5.32737931,0.951692065,0.090132427,0.639734764,Sight
+orange,5.591410959,0.878637132,0.354780521,0.620530353,Sight
+oval,5.605567568,0.835400833,0.309716578,0.562866925,Sight
+painful,5.002169391,1.153607197,-0.041280125,0.654378152,Touch
+palatable,5.721764706,0.975338843,-0.117984375,0.461166968,Taste
+pale,5.593280632,0.986369369,0.292592742,0.559286754,Sight
+patterned,5.686810345,0.771969042,0.33604386,0.573558417,Sight
+peachy,6.249565217,1.18571829,0.63812,0.711728166,Taste
+peppery,5.613488372,1.088246563,0.64148,0.880566909,Taste
+perfumed,5.936666667,1.217066264,0.560642857,0.646564321,Smell
+petite,5.988045977,1.120255814,0.354597938,0.654620985,Sight
+pink,5.613787529,0.924535048,0.346882151,0.622891435,Sight
+plain,5.499516729,1.15646522,0.069840074,0.699625298,Sight
+polished,5.67347541,0.895167319,0.154309764,0.668340137,Sight
+popping,5.0285,0.6495,0.234,0.570844585,Sound
+portly,5.631935484,0.881438782,-0.034467742,0.492371474,Sight
+prickly,5.739300699,1.112667695,0.201934783,0.505675051,Touch
+puffy,5.876375,1.049182151,0.359865854,0.52331024,Sight
+pulsing,5.612375,1.046893883,0.239727273,0.510068402,Touch
+pungent,5.657480916,1.064579769,0.345969231,0.791636567,Smell
+puny,5.683370787,1.076620162,0.012376344,0.586010922,Sight
+purple,5.5071777,0.894631129,0.32635163,0.63989033,Sight
+purring,5.850555556,1.124401365,0.185722222,0.515612856,Sound
+putrid,5.175081967,1.324022584,-0.031576271,0.739303754,Smell
+quiet,5.609460133,1.152530329,0.136330465,0.669146768,Sound
+radiant,6.145970874,1.30921152,0.417542289,0.625538354,Sight
+rancid,5.129649123,1.254790728,0.064018519,0.766541108,Smell
+raspy,5.282380952,1.064505347,0.12175,0.502524155,Sound
+raucous,5.646625,1.163470796,0.173163522,0.656211201,Sound
+rectangular,5.426,0.709204217,0.173286996,0.473801093,Sight
+red,5.414637416,0.88615028,0.235629563,0.642456415,Sight
+reddish,5.48795082,0.877193882,0.230172131,0.554633979,Sight
+reeking,4.63047619,1.290073371,-0.255,0.776364289,Smell
+resounding,5.319145299,1.107080676,-0.053423423,0.512518986,Sound
+reverberating,5.211578947,0.996639635,0.190944444,0.6600573,Sound
+rhythmic,5.621275168,0.972054313,0.105539474,0.598217415,Sound
+ripe,5.779674797,1.190499841,0.360181818,0.80877701,Taste
+rippled,5.636756757,0.706797856,0.303235294,0.480770248,Sight
+rippling,5.553186813,0.999720682,0.272690476,0.599385813,Sight
+roasted,5.898198198,1.000643968,0.59590678,0.769676145,Taste
+roasting,5.4925,0.788801418,0.350222222,0.521932324,Sight
+rotten,5.583674912,1.151858843,0.0804947,0.662255249,Sight
+rotund,5.673243243,0.906402929,0.053848485,0.439544979,Sight
+rough,5.512853224,0.901873678,0.125721992,0.546612152,Touch
+round,5.523026052,0.835260955,0.227340771,0.509112878,Sight
+rubbery,5.532705882,0.883447663,0.201414634,0.500146724,Touch
+rumbling,5.144545455,1.049678111,0.257403509,0.475266188,Sound
+rustling,5.909444444,1.336196151,0.742411765,0.683719738,Sound
+rusty,5.322663934,0.722808801,0.19217094,0.507610193,Sight
+salty,5.693988764,1.125207068,0.366546961,0.661932775,Taste
+savory,5.915,1.112931422,0.554269231,0.767600039,Taste
+scaly,5.411454545,0.89876902,0.064358491,0.636096414,Touch
+scented,5.828333333,1.142618569,0.429034483,0.719776793,Smell
+scentless,5.755,0.691153072,0.2435,0.7045,Smell
+scratchy,5.722638889,0.909796493,0.266861111,0.579059918,Touch
+scrawny,5.616933333,1.007302182,0.0185,0.428858169,Sight
+screaming,5.282484472,1.21198739,0.054046358,0.50125176,Sound
+screeching,5.503548387,1.158163597,0.082896552,0.375656797,Sound
+searing,5.215639535,1.17758033,0.057058824,0.678516539,Sight
+shadowy,5.466477273,0.904985085,0.140469636,0.536082943,Sight
+shaggy,5.620693069,0.872337751,0.123040816,0.467633935,Sight
+shallow,5.515985222,0.868690464,0.117414392,0.619009821,Sight
+sharp,5.202479241,0.992634241,0.006454545,0.609432201,Touch
+sheer,5.299710366,1.319592897,-0.011308442,0.801039573,Sight
+shimmering,5.891686747,0.961673846,0.436939759,0.61227206,Sight
+shiny,5.413979058,0.765299716,0.233790761,0.528342746,Sight
+short,5.433436929,0.843992132,0.113323801,0.550142679,Sight
+shrieking,5.240377358,1.14245264,0.067137255,0.508392465,Sound
+shrill,5.087352941,1.123069427,-0.042845361,0.625938954,Sound
+silent,5.331201672,1.142956937,0.085399784,0.614644774,Sound
+silky,5.740194805,0.893441036,0.479698718,0.637077156,Touch
+silver,5.784931507,0.927533833,0.314506849,0.518128665,Sight
+skinny,5.533890909,0.945719894,0.141282528,0.537469978,Sight
+slick,5.504644068,0.871806749,0.158537162,0.547499593,Sight
+slimy,5.19446281,0.995778632,0.097826087,0.534714272,Touch
+slippery,5.446836364,0.802833288,0.134652985,0.616811342,Touch
+slushy,5.309310345,0.923288884,0.241576923,0.477427909,Sight
+small,5.314277127,0.931466118,0.137467971,0.669430168,Sight
+smelly,5.176338028,0.971852545,0.100293233,0.523363618,Smell
+smoky,5.538823529,1.023393333,0.366246753,0.737585865,Smell
+smooth,5.540558003,0.844622963,0.255969343,0.607529147,Touch
+snarling,5.163188406,1.083689887,-0.100161765,0.565746767,Sound
+snorting,5.395,1.555384357,-0.012818182,0.347546882,Sound
+soapy,5.916444444,1.197076424,0.355914894,0.616217778,Sight
+sodden,5.409310345,0.894654825,0.126973214,0.477117754,Sight
+soft,5.454473925,0.931899914,0.150204896,0.618303033,Touch
+soggy,5.44441989,0.952430661,0.186431034,0.553433561,Touch
+solid,5.527763272,0.878413683,0.061660494,0.549814071,Touch
+sonorous,5.874878049,0.965215288,0.131911111,0.602532984,Sound
+sore,5.056132075,0.855290144,-0.166278846,0.614837746,Touch
+soundless,5.229795918,1.007393798,0.043431818,0.661751428,Sound
+sour,5.491092437,1.20917879,0.140840164,0.693063406,Taste
+sparkly,5.738867925,0.877604743,0.201641509,0.493681319,Sight
+speckled,5.589193548,0.867456226,0.334540984,0.519923954,Sight
+spicy,5.843225806,1.037600841,0.680765854,0.838365366,Taste
+spiky,5.707368421,0.977206057,0.35095082,0.62302077,Touch
+spilling,5.7,0.984065602,0.388857143,0.475149587,Sight
+spotless,5.9184,0.971784089,0.386662162,0.573155439,Sight
+spotted,5.405044248,1.033491877,0.159948276,0.53057059,Sight
+square,5.54826087,0.838554082,0.20334058,0.499822406,Sight
+squeaking,5.32,0.710180677,0.3355,0.572130889,Sound
+squealing,5.372142857,1.165439505,0.10156,0.308041884,Sound
+stagnant,5.371034483,0.950353394,0.120012195,0.594745434,Sight
+stale,5.518959538,1.181044654,0.208488506,0.656731871,Taste
+steep,5.228913043,0.811227243,0.168665198,0.595416103,Sight
+stenchy,5.62,0.556153072,5,4.936015703,Smell
+sticky,5.337,0.882716138,0.106022508,0.606300197,Touch
+stinging,4.847368421,1.156811761,0.063704698,0.672684037,Touch
+stinky,5.485862069,1.090875091,0.092655172,0.603621231,Smell
+stormy,5.501052632,1.272739351,0.173115044,0.524603576,Sight
+straight,5.43860735,0.919301883,-0.047220307,0.578728983,Sight
+strange,5.480895296,1.077857751,0.074687943,0.612033559,Sight
+striped,5.579267016,0.807011278,0.350292308,0.623508926,Sight
+strong,5.430486233,0.952014954,0.005910628,0.640922269,Sight
+sturdy,5.644059946,0.819068987,0.160066852,0.493155434,Touch
+sunny,5.931320132,1.069463838,0.366020134,0.571214588,Sight
+sweaty,5.453203883,1.09066436,0.115298507,0.527757547,Smell
+sweet,5.681452671,1.170262758,0.237406452,0.647960889,Taste
+swift,5.248944282,1.059052399,0.012349112,0.605774776,Sight
+swinging,5.463868613,0.820224271,0.223940741,0.442790187,Sight
+tall,5.5425,0.852785801,0.163086614,0.55195898,Sight
+tangerine,5.902564103,1.116705658,0.577333333,0.687245573,Sight
+tangy,5.876625,1.07847067,0.563788235,0.830056111,Taste
+tapering,5.485172414,0.60896517,0.124241379,0.419897093,Sight
+tart,5.990337079,1.209947611,0.428849462,0.709952146,Taste
+tasteless,5.661060606,1.089008972,0.015242424,0.750545455,Taste
+tasty,5.894336735,1.117012917,0.481465347,0.71510554,Taste
+tender,5.712243437,1.13440756,0.289879012,0.682746506,Touch
+tepid,5.940707071,1.267179042,0.35543299,0.741623899,Touch
+thorny,5.458902439,0.98477452,0.037209877,0.556272574,Touch
+thudding,4.881333333,1.362410205,0.253357143,0.542930815,Sound
+thumping,5.002413793,1.092095357,0.185733333,0.649404188,Sound
+thunderous,5.203923077,1.123603455,0.106540323,0.603171128,Sound
+ticklish,5.367419355,0.822332132,-0.0604375,0.474496074,Touch
+tight,5.378703704,0.75882593,0.055315589,0.531305436,Touch
+tinkling,6.292962963,1.515611876,0.549466667,0.600540662,Sound
+tiny,5.37049417,0.901232817,0.204778036,0.60715513,Sight
+tough,5.355045872,0.932766152,-0.023501699,0.545038517,Touch
+translucent,5.597755102,0.904458043,0.322663102,0.583639639,Sight
+transparent,5.49226506,0.862375933,0.064519802,0.564797185,Sight
+triangular,5.473554502,0.739806418,0.136162437,0.456692348,Sight
+ugly,5.146509972,1.082317404,-0.01544919,0.565952247,Sight
+uneven,5.636811594,0.872340733,0.100157895,0.534096665,Sight
+unpalatable,5.535185185,1.118233188,-0.113962963,0.444034129,Taste
+unripe,6.737,1.673153072,0.767166667,1.074841185,Sight
+vegetal,5.915652174,0.995752004,0.746043478,0.867964032,Sight
+vinegary,6.248095238,1.53904714,0.434545455,0.680369347,Taste
+vivid,5.823141892,1.066446988,0.056786942,0.596107231,Sight
+wailing,5.548043478,1.300350935,0.074711111,0.563155207,Sound
+warbling,5.17,0.893286643,0.348083333,0.557585951,Sound
+warm,5.774844398,1.061902707,0.375366701,0.675182277,Touch
+waxy,5.73527027,0.987639831,0.389679487,0.567326553,Touch
+weak,5.453759178,0.930084439,-0.045140762,0.630720118,Sight
+weightless,5.756875,1.074879469,0.471691176,0.780638819,Touch
+wet,5.415122898,0.894773336,0.199112113,0.566535373,Touch
+whimpering,4.9045,1.5575,-0.352045455,0.615036889,Sound
+whining,5.29097561,1.009455702,-0.056097561,0.616146724,Sound
+whispering,5.8955,1.115191843,0.033121951,0.635998851,Sound
+whistling,5.827142857,1.086373184,0.235384615,0.588152638,Sound
+white,5.353693765,0.938191723,0.131610348,0.656168893,Sight
+wide,5.450415677,0.824700038,0.096359509,0.588883461,Sight
+wiry,5.665333333,0.832443997,0.078242424,0.542486911,Sight
+wispy,5.67872093,0.981546992,0.135094118,0.628858639,Sight
+wizened,5.733333333,1.133247399,0.099627907,0.568882625,Sight
+woolly,5.503770492,0.857805458,0.267328358,0.485987653,Touch
+yellow,5.451604376,0.891756232,0.252861261,0.639369331,Sight
\ No newline at end of file

From 0d22dbb79df67b0f4d7e68a92521137797d68317 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Fri, 1 Nov 2024 11:40:22 +0530
Subject: [PATCH 50/55] predict()

Mainly added predict function

Also, included relationship between r and R^2
---
 Module 6/Regression.Rmd | 39 ++++++++++++++++++++++++++++++++++++++-
 1 file changed, 38 insertions(+), 1 deletion(-)

diff --git a/Module 6/Regression.Rmd b/Module 6/Regression.Rmd
index 5f6d2fb5..f2c68e87 100644
--- a/Module 6/Regression.Rmd	
+++ b/Module 6/Regression.Rmd	
@@ -37,6 +37,12 @@ summary(regression.1)
 cor.test( x = parenthood$dan.sleep, y = parenthood$dan.grump )
 ```
 
+```{r}
+R <-0.903^2
+R
+```
+
+
 
 ## Multiple linear regression
 dan.grump ~ dan.sleep + baby.sleep (Is Dan's sleep and the baby's sleep both together leading to grumpiness the next day?)
@@ -50,6 +56,8 @@ print( regression.2 )
 summary(regression.2)
 ```
 
+
+
 ```{r}
 library(lsr)
 correlate(parenthood, test=TRUE)
@@ -141,6 +149,7 @@ plot(x = regression.2, which = 1)
 
 
 ```{r}
+library(carData)
 residualPlots( model = regression.2 ) 
 ```
 
@@ -164,7 +173,7 @@ vif( mod = regression.2 )
 ```
 
 
-
+MODEL SELECTION
 
 Backward elimination
 
@@ -202,6 +211,32 @@ anova( M0, M1 )
 ```
 
 
+Using predict() function to interpret the model
+```{r}
+regression.1 <- lm( formula = dan.grump ~ dan.sleep,  
+                    data = parenthood )
+```
+
+
+
+```{r}
+xvals <- seq(from = 0, to = 10, by = 1)
+mypreds <- tibble(dan.sleep = xvals)
+```
+
+
+```{r}
+mypreds$fit <- predict(regression.1, newdata = mypreds)
+mypreds
+```
+
+
+
+
+
+
+
+
 Using the performance package from easystats for model checking
 
 ```{r}
@@ -213,6 +248,8 @@ model_performance(M1)
 ```
 
 
+
+
 ```{r}
 # checking model assumptions
 check_model(M1)

From a58b6273c1da947640a7c23bf7986751d6668cc2 Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Fri, 1 Nov 2024 12:25:56 +0530
Subject: [PATCH 51/55] Assumptions checks modified

Added Normality check.
Loaded Car and CatData for linearity check
---
 Module 6/Regression.Rmd | 22 ++++++++++++++++++++++
 1 file changed, 22 insertions(+)

diff --git a/Module 6/Regression.Rmd b/Module 6/Regression.Rmd
index f2c68e87..aa95b841 100644
--- a/Module 6/Regression.Rmd	
+++ b/Module 6/Regression.Rmd	
@@ -97,6 +97,10 @@ ckd <- cooks.distance( model = regression.2 )
 ckd
 ```
 
+
+
+
+
 Directly plotting Cook's distance using plot from car package
 ```{r}
 plot(regression.2,which=4)
@@ -130,6 +134,17 @@ Directly make the QQ plot of residuals
 plot( x = regression.2, which = 2 )
 ```
 
+Normality Test
+```{r}
+shapiro.test(residuals(regression.2))
+```
+
+
+
+
+
+
+
 Checking linearity of relationship
 ```{r}
 yhat.2 <- fitted.values( object = regression.2 )
@@ -148,11 +163,18 @@ plot(x = regression.2, which = 1)
 ```
 
 
+Linearity test
+
 ```{r}
+library(car)
 library(carData)
+```
+
+```{r}
 residualPlots( model = regression.2 ) 
 ```
 
+
 If the curvature is significant, then you might want to transform the predictor using Box Cox Transformation. Or use the powerTransform() in the car package.
 
 

From 021451766063df14ed654168f8a19d64a297f18c Mon Sep 17 00:00:00 2001
From: juneeybug 
Date: Fri, 1 Nov 2024 15:36:11 +0530
Subject: [PATCH 52/55] Dummy coding

Added description
---
 Module 6/CategoricalRegression.Rmd     |  8 ++++++++
 Module 6/CategoricalRegression.nb.html | 10 ++++++----
 2 files changed, 14 insertions(+), 4 deletions(-)

diff --git a/Module 6/CategoricalRegression.Rmd b/Module 6/CategoricalRegression.Rmd
index eec3f61f..0bc8f10d 100644
--- a/Module 6/CategoricalRegression.Rmd	
+++ b/Module 6/CategoricalRegression.Rmd	
@@ -44,6 +44,14 @@ If you are interested in choosing the right colors for your plots, then here's a
 https://ggplot2-book.org/scales-colour
 
 
+Coding the variables
+Referred to as Dummy Coding or Treatment Coding
+Smell - 0
+Taste - 1
+
+
+
+
 ```{r}
 chem_mdl <- lm(Val ~ DominantModality, data = chem)
 summary(chem_mdl)
diff --git a/Module 6/CategoricalRegression.nb.html b/Module 6/CategoricalRegression.nb.html
index c3d10c57..77393d21 100644
--- a/Module 6/CategoricalRegression.nb.html	
+++ b/Module 6/CategoricalRegression.nb.html	
@@ -1790,8 +1790,8 @@ 
    R Notebook
    
 
 
    senses <- read_csv('winter_2016_senses_valence.csv')
    
 
-
-
    Rows: 405 Columns: 6── Column specification ──────────────────────────────────────────────────────────────────────────────────────────
+
+
    Rows: 405 Columns: 6── Column specification ─────────────────────────────────────────────────────────────────────────────────────────
 Delimiter: ","
 chr (2): Word, DominantModality
 dbl (4): Val, AbsVal, Sent, AbsSent
@@ -1801,7 +1801,7 @@ 
    R Notebook
    
 
 
    senses
    
 
-
+
 
    
 
+
+
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+
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+
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+
    
+
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+
+
    This is an R Markdown
+Notebook. When you execute code within the notebook, the results appear
+beneath the code.
    
+
    Try executing this chunk by clicking the Run button within
+the chunk or by placing your cursor inside it and pressing
+Ctrl+Shift+Enter.
    
+
+
+
+
    library(lsr)
    
+
+
+
+
+
+
+
    library(magrittr)
+library(tidyverse)
+library(ggplot2)
    
+
+
+
+
    For linear mixed models, load these libraries
    
+
+
+
+
    library(lme4)
+library(lmerTest)
    
+
+
+
+
    Add a new chunk by clicking the Insert Chunk button on the
+toolbar or by pressing Ctrl+Alt+I.
    
+
    When you save the notebook, an HTML file containing the code and
+output will be saved alongside it (click the Preview button or
+press Ctrl+Shift+K to preview the HTML file).
    
+
    The preview shows you a rendered HTML copy of the contents of the
+editor. Consequently, unlike Knit, Preview does not
+run any R code chunks. Instead, the output of the chunk when it was last
+run in the editor is displayed.
    
+
+
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