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When it is not found, a full rebuild will be done. -config: 30335cb2e3d23d8b5f649a094b762bd1 +config: 75f8c34de863dec7a003520074cc7920 tags: 645f666f9bcd5a90fca523b33c5a78b7 diff --git a/01-IntroductionToPython.html b/01-IntroductionToPython.html index 89b9920..f366a66 100644 --- a/01-IntroductionToPython.html +++ b/01-IntroductionToPython.html @@ -1,298 +1,380 @@ - - + + + + + Introduction to Python — Python Companion to Statistical Thinking in the 21st Century - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - - + + + + +
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Introduction to Python

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Introduction to Python#

Contributors: John Butler, Russell Poldrack

In this chapter we will give you an overview of the basic features of the Python programming language. This introduction won’t make you an ace programmer — only practice will do that. However, it will give you an introduction to some of the most important features of the language.

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Why programming is hard to learn

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Why programming is hard to learn#

Programming a computer is a skill, just like playing a musical instrument or speaking a second language. And just like those skills, it takes a lot of work to get good at it — the only way to acquire a skill is through practice. There is nothing special or magical about people who are experts, other than the quality and quantity of their experience! However, not all practice is equally effective. A large amount of psychological research has shown that practice needs to be deliberate, meaning that it focuses on developing the specific skills that one needs to perform the skill, at a level that is always pushing one’s ability.

If you have never programmed before, then it’s going to seem hard, just as it would seem hard for a native English speaker to start speaking Mandarin. However, just as a beginning guitarist needs to learn to play their scales, we will teach you how to perform the basics of programming, which you can then use to do more powerful things.

One of the most important aspects of computer programming is that you can try things to your heart’s content; the worst thing that can happen is that the program will crash. Trying new things and making mistakes is one of the keys to learning.

The hardest part of programming is figuring out why something didn’t work, which we call debugging. In programming, things are going to go wrong in ways that are often confusing and opaque. Every programmer has a story about spending hours trying to figure out why something didn’t work, only to realize that the problem was completely obvious in hindsight. The more practice you get, the better you will get at figuring out how to fix these errors. But there are a few strategies that can be helpful.

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Use the web

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Use the web#

In particular, you should take advantage of the fact that there are millions of people programming in Python around the world, so nearly any error message you see has already been seen by someone else. Whenever I experience an error that I don’t understand, the first thing that I do is to copy and paste the error message into a search engine. Often this will provide several pages discussing the problem and the ways that people have solved it.

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Rubber duck debugging

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Rubber duck debugging#

The idea behind rubber duck debugging is to pretend that you are trying to explain what your code is doing to an inanimate object, like a rubber duck. Often, the process of explaning it aloud is enough to help you find the problem.

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Getting access to Python

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Getting access to Python#

If you want to try out Python without having to install it on your own computer (or if you have a computer that doesn’t support it, such as a Chromebook), the best way to try it out is using a web platform that supports notebooks, which are documents that allow you to combine text and code; this book was actually written using such notebooks. Two good options are Google Colab and Kaggle Kernels. TBD: MAKE ALL CHAPTERS AVAILABLE ON BOTH

If you do want to install it on your own computer, we recommend installing the Anaconda software package, which will provide you with Python as well as many related packages.

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Getting started with Python

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Getting started with Python#

When we work with Python, we can do this at the command line in a terminal or (as we will do) using a Jupyter notebook. Notebooks are composed of cells, each of which can contain either Python code or text.

Jupyter notebooks use a special kind of text known as Markdown, which allows formatting (such as headings or text styles like bold and italics). If you were to double-click on this cell in Jupyter, you would be able to edit the text. Once you are done editing, hit the Run button (the triangular arrow button above), or press Shift+Enter, and the text will be shown normally.

@@ -301,12 +383,12 @@

Getting started with Python

+
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Variables#

A variable is a symbol that stands for another value (just like “X” in algebra). We can create a variable by assigning a value to it using the = operator. If we then type the name of the variable, Python will print out its value — as long as that variable name is the last entry in the cell.

+
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Libraries#

Although Python has many useful features, many of the features we will need are not contained in the primary Python library but instead come from open source libraries that have been developed by various members of the python community.

Two packages that we will use extensively are NumPy and Pandas. These libraries are part of the SciPy stack, a group of Python libraries used for scientific computing.

In Python, to work with a library, we first have to import it and specify how we are going to call on the functions of that library (functions will be explained in more detail below).

Here is how we will import NumPy:

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Functions#

A function is an operator that takes some input and gives an output based on the input. For example, let’s say that we have a number, and we want to determine its absolute value. NumPy has a function called abs() that takes in a number and outputs its absolute value:

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Arrays#

You may have noticed that the my_random_numbers created above wasn’t like the variables that we had seen before — it contained several values in it. We refer to this kind of variable as an array.

If you want to create your own new array, you can do that using the np.array() function:

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Math with vectors and matrices#

You can apply mathematical operations to the elements of an array in the same way that you apply them to regular variables. Let’s say that we want to multiply each element in the array by the number 5:

+
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Dictionaries#

There is another kind of variable in Python that is very useful, which is known as a dictionary. A dictionary is like a container that stores values that are associated with particular keys. A dictionary is created using squiggly brackets; each entry must include a key and a value (which can be any kind of variable, including another dictionary), separated by a color. For example, let’s say that we wanted to store the ages of three people:

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ages = {'Lisa': 23, 'Angela': 25, 'Monique': 27}
+
ages = {'Lisa': 23, 'Angela': 25, 'Monique': 27}
 ages
 
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{'Lisa': 23, 'Angela': 25, 'Monique': 27}
+
{'Lisa': 23, 'Angela': 25, 'Monique': 27}
 
@@ -826,25 +915,25 @@

Dictionaries

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Data Frames#

Often in a dataset we will have a number of different variables that we want to work with. Instead of having a different named variable that stores each one, it is often useful to combine all of the separate variables into a single package, which is referred to as a data frame.

If you are familiar with a spreadsheet (say from Microsoft Excel) then you already have a basic understanding of a data frame. Let’s say that we have values of price and mileage for three different types of cars. We could start by creating a variable for each one, making sure that the three cars are in the same order for each of the variables:

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car_model = ("Ford Fusion", "Hyundai Accent", "Toyota Corolla")
+
car_model = ("Ford Fusion", "Hyundai Accent", "Toyota Corolla")
 car_price = np.array([25000, 16000, 18000])
 car_mileage = np.array([27, 36, 32])
 
@@ -854,7 +943,7 @@

Data Frames
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data = {'Price': car_price,
+
data = {'Price': car_price,
         'Mileage': car_mileage}
 cars_df2 = pd.DataFrame(data, index=car_model)
 cars_df2
@@ -910,12 +999,12 @@ 

Data Frames
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cars_df2['Mileage']
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cars_df2['Mileage']
 
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Ford Fusion       27
+
Ford Fusion       27
 Hyundai Accent    36
 Toyota Corolla    32
 Name: Mileage, dtype: int64
@@ -926,12 +1015,16 @@ 

Data Frames
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cars_df2['Mileage'][1]
+
cars_df2['Mileage'][1]
 

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1.10 Working with data files#

When we are doing statistics, we often need to load in the data that we will analyze. Those data will live in a file on one’s computer or on the internet. For this example, let’s use a file that is hosted on the internet, which contains the gross domestic product (GDP) values for a number of countries around the world. This file is stored as a comma-separated value (or CSV) file, meaning that the values for each of the variables in the dataset are separated by commas. There are three variables: the relative rank of the countries, the name of the country, and its GDP value. Here is what the first few lines of the file look like: We can load the file using the pd.read_csv() function:

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data_url = 'https://raw.githubusercontent.com/psych10/psych10/master/notebooks/Session03-IntroToR/gdp.csv'
+
data_url = 'https://raw.githubusercontent.com/psych10/psych10/master/notebooks/Session03-IntroToR/gdp.csv'
 gdp_data = pd.read_csv(data_url)
 
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1.10 Working with data files.head() which will show us the top 5 rows of the data frame:

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gdp_data.head()
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gdp_data.head()
 
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1.10 Working with data filesdir() function:

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Summarizing Data

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Summarizing Data#

This chapter will introduce you to how to summarize data using data frames in Pandas.

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Working with data frames

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Working with data frames#

In the last chapter you were introduced to the concept of a data frame, which we will use throughout much of this book.
In particular, we will use a dataset known as NHANES for several of our examples, so let’s load the library that provides us access to the data. This is a large dataset collected from a sample of individuals in the United States every two years, which measures many different aspects of their health and lifestyles. To access the data, we will use a Python package called nhanes that contains a function to load a cleaned-up version of the dataset.

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from nhanes.load import load_NHANES_data
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from nhanes.load import load_NHANES_data
 nhanes_data = load_NHANES_data()
 
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Working with data frames.head() method to look inside the data frame:

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nhanes_data.head()
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nhanes_data.head()
 
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Working with data frames.shape method:

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nhanes_data.shape
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nhanes_data.shape
 
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(8366, 197)
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(8366, 197)
 
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Working with data frames.index operator:

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nhanes_data.index
+
nhanes_data.index
 
-
Float64Index([ 93703.0,  93704.0,  93705.0,  93706.0,  93707.0,  93708.0,
-               93709.0,  93711.0,  93712.0,  93713.0,
-              ...
-              102946.0, 102947.0, 102948.0, 102949.0, 102951.0, 102952.0,
-              102953.0, 102954.0, 102955.0, 102956.0],
-             dtype='float64', name='SEQN', length=8366)
+
Index([ 93703.0,  93704.0,  93705.0,  93706.0,  93707.0,  93708.0,  93709.0,
+        93711.0,  93712.0,  93713.0,
+       ...
+       102946.0, 102947.0, 102948.0, 102949.0, 102951.0, 102952.0, 102953.0,
+       102954.0, 102955.0, 102956.0],
+      dtype='float64', name='SEQN', length=8366)
 
@@ -497,12 +599,12 @@

Working with data frames.columns operator:

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nhanes_data.columns
+
nhanes_data.columns
 
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Index(['GeneralHealthCondition', 'EverBreastfedOrFedBreastmilk',
+
Index(['GeneralHealthCondition', 'EverBreastfedOrFedBreastmilk',
        'AgeStoppedBreastfeedingdays', 'AgeFirstFedFormuladays',
        'AgeStoppedReceivingFormuladays', 'AgeStartedOtherFoodbeverage',
        'AgeFirstFedMilkdays', 'TypeOfMilkFirstFedWholeMilk',
@@ -520,17 +622,17 @@ 

Working with data frames -

Selecting rows (observations) and columns (variables)

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Selecting rows (observations) and columns (variables)#

We often want to select a subset of rows from a data frame. You saw in the last chapter how we can access specific rows using the .loc operator. This operator requires us to refer to the row names (that is, the index) and column names. For example, if we wanted to know the value of the GeneralHealthCondition variable for the indivdiual labeled 93707, we could use the following:

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Missing values#

You will notice that the GeneralHealthCondition variable for the first individual in the previous cell contained NaN, which stands for “not a number”. This is generally used to denote that the data are missing for this particular observation; perhaps they declined to answer the question, or the interviewer failed to properly record the answer. Missing data are common when we are working with real data. There are many sophsticated ways to deal with missing data in statistics, but for the moment we will just remove observations that have a missing data on one of our variables of interest, which we can do using the .dropna() operator:

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my_subset = nhanes_data.loc[[102951, 102955, 93707], ['GeneralHealthCondition', 'Gender', 'AgeInYearsAtScreening']]
+
my_subset = nhanes_data.loc[[102951, 102955, 93707], ['GeneralHealthCondition', 'Gender', 'AgeInYearsAtScreening']]
 my_subset.dropna()
 
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Missing values -

Selecting rows by value

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Selecting rows by value#

Let’s say that we want to analyze NHANES data, but only for those individuals who are over 50 years of age. We can use the query operator on a data frame to find rows that match particular conditions:

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Creating new variables#

We can add a new variable to a data frame easily, by simply giving it a new name. Let’s say that we wanted to convert the weight value in NHANES (stored in the WeightKg variable) from kilograms to pounds.

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nhanes_data['WeightLbs'] = nhanes_data['WeightKg'] * 2.205
+
nhanes_data['WeightLbs'] = nhanes_data['WeightKg'] * 2.205
 
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Creating new variablesThis shows another way to refer to a particular variable in a dataframe: simply put its name in square brackets. Pandas also has the ability to replace particular values in a variable. First, let’s look at the values of the Gender variable in the data frame, to see what values it takes. We can see all of the unique values of a variable using the .unique() operator:

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nhanes_data['Gender'].unique()
+
nhanes_data['Gender'].unique()
 
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array(['Female', 'Male'], dtype=object)
+
array(['Female', 'Male'], dtype=object)
 
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Creating new variablesNow let’s say that we wanted to recode the Gender variable so that instead of “Female” and “Male” its values were “F” and “M”. One way to do this would be to use the .rename() operator on the data frame:

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nhanes_data['GenderMF'] = nhanes_data['Gender'].replace({'Female': 'F', 'Male': 'M'})
+
nhanes_data['GenderMF'] = nhanes_data['Gender'].replace({'Female': 'F', 'Male': 'M'})
 nhanes_data['GenderMF'].unique()
 
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array(['F', 'M'], dtype=object)
+
array(['F', 'M'], dtype=object)
 
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Understanding your data

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Understanding your data#

Let’s say that we want to learn more about the variable labeled GeneralHealthCondition in the dataset. We can load some information about that variable using the open_variable_page() function from the nhanes package:

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from nhanes.load import open_variable_page
+
from nhanes.load import open_variable_page
 open_variable_page('GeneralHealthCondition')
 

This shows us the question that was asked (“Would you say {your/SP’s} health in general is…”).

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Summarizing data using a frequency distribution

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Summarizing data using a frequency distribution#

Let’s say that we want to know the frequencies of all of the different answers to the GeneralHealthCondition question. We can do this using the .value_counts() method of the data frame:

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nhanes_data['GeneralHealthCondition'].value_counts()
+
nhanes_data['GeneralHealthCondition'].value_counts()
 
-
Good         2454
+
GeneralHealthCondition
+Good         2454
 Very good    1544
 Fair or      1175
 Excellent     619
 Poor?         172
-Name: GeneralHealthCondition, dtype: int64
+Name: count, dtype: int64
 
@@ -777,34 +880,35 @@

Summarizing data using a frequency distribution.sum() operator:

-
GeneralHealthCondition_frequency_dist = nhanes_data['GeneralHealthCondition'].value_counts()
+
GeneralHealthCondition_frequency_dist = nhanes_data['GeneralHealthCondition'].value_counts()
 GeneralHealthCondition_frequency_dist / GeneralHealthCondition_frequency_dist.sum()
 
-
Good         0.411469
+
GeneralHealthCondition
+Good         0.411469
 Very good    0.258887
 Fair or      0.197015
 Excellent    0.103789
 Poor?        0.028840
-Name: GeneralHealthCondition, dtype: float64
+Name: count, dtype: float64
 
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Data Cleaning

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Data Cleaning#

When we work with real data, they often have problems that we have to fix before we can analyze them properly. An example is the GeneralHealthCondition variable that we worked with in the previous example. You may have noticed that the values of the variable had some extranous information, which were held over from the way that the question is worded (including “Fair or” and “Poor?”). We can clean these up by replacing the problematic values using the .replace() method:

-
nhanes_data['GeneralHealthConditionFixed'] = nhanes_data['GeneralHealthCondition'].replace({'Fair or': 'Fair', 'Poor?': 'Poor'})
+
nhanes_data['GeneralHealthConditionFixed'] = nhanes_data['GeneralHealthCondition'].replace({'Fair or': 'Fair', 'Poor?': 'Poor'})
 nhanes_data['GeneralHealthConditionFixed'].unique()
 
-
array([nan, 'Good', 'Very good', 'Fair', 'Excellent', 'Poor'],
+
array([nan, 'Good', 'Very good', 'Fair', 'Excellent', 'Poor'],
       dtype=object)
 
@@ -813,29 +917,31 @@

Data Cleaningdetails of the smoking questionnaire, we will see that not all individuals got the same questions; for example, if a person said that they had not smoked more than 100 cigarettes in their life (recorded in the SmokedAtLeast100CigarettesInLife variable), then they were not asked the question about whether they currently smoked cigarettes (stored as DoYouNowSmokeCigarettes). We can see this in the number of respondents to each question:

-
print(nhanes_data['SmokedAtLeast100CigarettesInLife'].value_counts())
+
print(nhanes_data['SmokedAtLeast100CigarettesInLife'].value_counts())
 
-
0.0    3301
+
SmokedAtLeast100CigarettesInLife
+0.0    3301
 1.0    2232
-Name: SmokedAtLeast100CigarettesInLife, dtype: int64
+Name: count, dtype: int64
 
-
nhanes_data['DoYouNowSmokeCigarettes'].value_counts()
+
nhanes_data['DoYouNowSmokeCigarettes'].value_counts()
 
-
Not at all    1260
+
DoYouNowSmokeCigarettes
+Not at all    1260
 Every day      763
 Some days      209
-Name: DoYouNowSmokeCigarettes, dtype: int64
+Name: count, dtype: int64
 
@@ -843,7 +949,7 @@

Data Cleaning
-
nhanes_data[['SmokedAtLeast100CigarettesInLife', 'DoYouNowSmokeCigarettes']].head(10)
+
nhanes_data[['SmokedAtLeast100CigarettesInLife', 'DoYouNowSmokeCigarettes']].head(10)
 
@@ -933,13 +1039,13 @@

Data Cleaning

+

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\ No newline at end of file diff --git a/03-DataVisualization.html b/03-DataVisualization.html index 928be90..0c295b3 100644 --- a/03-DataVisualization.html +++ b/03-DataVisualization.html @@ -1,245 +1,355 @@ - - + + + + + Data Visualization — Python Companion to Statistical Thinking in the 21st Century - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - - + + + + +
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Data Visualization

+ +
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Data Visualization#

There are two main packages that we will use for visualization in Python: matplotlib and seaborn, which is based on matplotlib. First, let’s import these. It is customary to import the pyplot module from matplotlib, since it contains most of the important plotting functions:

-
import matplotlib.pyplot as plt
+
import matplotlib.pyplot as plt
 import seaborn as sns
 import numpy as np
 import pandas as pd
@@ -247,8 +357,8 @@ 

Data Visualization -

Let’s think through a visualization

+
+

Let’s think through a visualization#

Principles we want to keep in mind:

+
+

Plotting the distribution of a single variable#

One of the most common uses of plotting is to plot the distribution of the data — which you can think of as the shape of the data. There are various ways to do this, but one of the most common is known as a histogram, which plots the number of observations that fall into specific bins. We can plot a histogram using the plt.hist() function from matplotlib. As an example, let’s look at the distribution of ages in the NHANES dataset. First we need to load the data:

-
from nhanes.load import load_NHANES_data
+
from nhanes.load import load_NHANES_data
 nhanes_data = load_NHANES_data()
 
@@ -276,78 +386,91 @@

Plotting the distribution of a single variable
-
age_histogram = plt.hist(nhanes_data['AgeInYearsAtScreening'])
+
age_histogram = plt.hist(nhanes_data['AgeInYearsAtScreening'])
 
-_images/03-DataVisualization_7_0.png +_images/009548cab95ffa8a7e6e72674a984879b409cf5b162b4f47c3c9743f353493b2.png

You can see from this plot that the plt.hist() function has binned together individuals across several years; That’s because we let it automatically determine the size of the bins. Let’s say that instead we want to bin each year separately. We can do this using the bins argument to plt.hist. Because this argument takes a list of bins, we need to create a list that spans from the youngest to the oldest age. We can do this using the np.arange() function from numpy, which generates a list of numbers that span a particular range. In this case, we need to span from the youngest to the oldest value, which are equivalent to the minimum and maximum values which we can obtain using the .min() and .max() operators; because Python starts at zero, we need to add one to the maximum in order to get the bins to cover the entire range:

-
bins = np.arange(nhanes_data['AgeInYearsAtScreening'].min(), nhanes_data['AgeInYearsAtScreening'].max() + 1)
+
bins = np.arange(nhanes_data['AgeInYearsAtScreening'].min(), nhanes_data['AgeInYearsAtScreening'].max() + 1)
 age_histogram_1year_bins = plt.hist(nhanes_data['AgeInYearsAtScreening'], bins=bins)
 
-_images/03-DataVisualization_9_0.png +_images/d5a215e21518ff12eeed427ee06627e2ce32b3c90fcbb06fcb0b625a4f27830c.png

Sometimes it’s more useful to look at the density rather than the counts, which we can do by setting density=True in our call to the histogram function:

-
age_density_1year_bins = plt.hist(nhanes_data['AgeInYearsAtScreening'], bins=bins, density=True)
+
age_density_1year_bins = plt.hist(nhanes_data['AgeInYearsAtScreening'], bins=bins, density=True)
 
-_images/03-DataVisualization_11_0.png +_images/89322a4dd57c9d3d1a7bad9acee2f404fb77c1300ad0573d470f2a3e1e211813.png

Now we see the proportion of individuals that fall into each age bin. Why do you think there are so many eighty-year-olds in the dataset? Have a look at the documentation for the Age question and see if you can figure it out.

-
-

Bar vs. line plots

+
+

Bar vs. line plots#

The histograms above are an example of bar plots where each number is represented by a bar. We could also plot the distribution using a line instead. One reason to do this is that we can make the line a bit smoother than the actual data. For example, here are the histogram data from above, plotted as a line:

-
plt.plot(age_density_1year_bins[1][1:], age_density_1year_bins[0])
+
plt.plot(age_density_1year_bins[1][1:], age_density_1year_bins[0])
 
-
[<matplotlib.lines.Line2D at 0x7fd04ebd0340>]
+
[<matplotlib.lines.Line2D at 0x7fecc7e126f0>]
 
-_images/03-DataVisualization_14_1.png +_images/90f4869aee8f2f3f01101280efc2010e41175b9237e9445902fd100dd33f0ccb.png

Here we have taken advantage of the fact that the output of our histogram command above contains both the bins (in its [1] position) and the histogram values (in its [0]) position. Why do we include [1:] after the bins variable? This is because the bins include both the upper and lower edges of the bin, which means that there is one more bin value than there are average values. Adding [1:] is equivalent to saying “start with the second bin” which is equivalent to using the top edges of each bin for our X axis.

Now let’s plot a smoothed version of the histogram, using the sns.distplot() function from the seaborn library.

-
sns.distplot(nhanes_data['AgeInYearsAtScreening'], bins=bins)
+
sns.distplot(nhanes_data['AgeInYearsAtScreening'], bins=bins)
 
-
<matplotlib.axes._subplots.AxesSubplot at 0x7fd04eb7d160>
+
/tmp/ipykernel_161/3685571357.py:1: UserWarning: 
+
+`distplot` is a deprecated function and will be removed in seaborn v0.14.0.
+
+Please adapt your code to use either `displot` (a figure-level function with
+similar flexibility) or `histplot` (an axes-level function for histograms).
+
+For a guide to updating your code to use the new functions, please see
+https://gist.github.com/mwaskom/de44147ed2974457ad6372750bbe5751
+
+  sns.distplot(nhanes_data['AgeInYearsAtScreening'], bins=bins)
 
-_images/03-DataVisualization_16_1.png +
<Axes: xlabel='AgeInYearsAtScreening', ylabel='Density'>
+
+_images/4829581befa89ac7cab1c71a89b2313d7f79284539d1170582309e29119fe8c8.png
-

You can see that the line is now much smoother (less bumpy) than the one above. It generally follows the overall shape of the data pretty closely, but you can also see that it mostly hides the large bump at 80 years. It’s always important to keep in mind that anything we do to the data has the potential to distort their message.

-
-
-

Plots with two variables

+

You can see that the line is now much smoother (less bumpy) than the one above. It generally follows the overall shape of the data pretty closely, but you can also see that it mostly hides the large bump at 80 years. It’s always important to keep in mind that anything we do to the data has the potential to distort their message.

+
+

+
+

Plots with two variables#

Another common use of visualization is to examine the relationship betwen two variables. For example, let’s say that we wanted to plot average height as a function of age in the NHANES dataset. We would first summarize the data to obtain the average height for each age:

-
mean_height_by_age = nhanes_data.groupby('AgeInYearsAtScreening')['StandingHeightCm'].mean()
+
mean_height_by_age = nhanes_data.groupby('AgeInYearsAtScreening')['StandingHeightCm'].mean()
 
@@ -355,75 +478,74 @@

Plots with two variables.groupby() along with a builtin in method for computing the average of a variable in a dataframe (.mean()). This returns a single average height value for all of the individuals in each age group, which we can then plot. While we are at it, we will add descriptive labels to the X and Y axes, which is always a good idea:

+
+

Plotting dispersion#

An important job of statistical visualization is to show us the variability, or dispersion, of our data. We have already see how to do this using histograms; now let’s look at how we can compare distributions.

Let’s start with a simple example: Comparing the height of adult men and women in the NHANES sample. One commonly used plot is the box plot (sometimes known as a box and whiskers plot). This form of plot uses quartiles to give us a sense of spread. The thickest line, somewhere inside the box, represents the median. The upper and lower bounds of the box (the hinges) are the first and third quartiles (can you use them to approximate the interquartile range?). The lines extending from the hinges are the remaining data points, excluding outliers, which are plotted as individual points.

-
adult_nhanes_data = nhanes_data.query('AgeInYearsAtScreening > 17')
+
adult_nhanes_data = nhanes_data.query('AgeInYearsAtScreening > 17')
 sns.boxplot(x='Gender', y='StandingHeightCm', data=adult_nhanes_data)
 
-
<matplotlib.axes._subplots.AxesSubplot at 0x7fd04e8c2190>
+
<Axes: xlabel='Gender', ylabel='StandingHeightCm'>
 
-_images/03-DataVisualization_26_1.png +_images/faad66bcbd88fbcc81b22682b835a464d181c2a1a7b93b22313d310304cc6fe6.png

This tells us that the median male is taller than 75% of all of the females in the sample.

Another type of plot that is commonly used is the violin plot, which shows the shape of the entire distribution:

-

-sns.violinplot(x='Gender', y='StandingHeightCm', data=adult_nhanes_data)
+
sns.violinplot(x='Gender', y='StandingHeightCm', data=adult_nhanes_data)
 
-
<matplotlib.axes._subplots.AxesSubplot at 0x7fd047fd25e0>
+
<Axes: xlabel='Gender', ylabel='StandingHeightCm'>
 
-_images/03-DataVisualization_28_1.png +_images/54dc2a5ae8fba2c89395490b69cf96c3aefab9abd7ce9143bab168a427fcbaad.png
-
-

Scatter plot

+
+

Scatter plot#

When we have multiple continuous variables, we can use points to plot each variable on an axis. This is known as a scatter plot. As an example, let’s look at the blood pressure readings taken in the NHANES study. Each individual has their blood pressure taken three times. Here we will plot the first reading against the second reading, using a scatter plot. We will also add a line showing where the x axis is equal to the y axis, which makes it easier to see how the two variables are related to each other.

-
sns.scatterplot(x='SystolicBloodPres1StRdgMmHg',
+
sns.scatterplot(x='SystolicBloodPres1StRdgMmHg',
                 y='SystolicBloodPres2NdRdgMmHg',
                 data=adult_nhanes_data)
 plt.plot([adult_nhanes_data['SystolicBloodPres1StRdgMmHg'].min(),
@@ -437,18 +559,17 @@ 

Scatter plot -
Text(0, 0.5, 'Systolic BP - Second reading')
+
Text(0, 0.5, 'Systolic BP - Second reading')
 
-_images/03-DataVisualization_30_1.png +_images/28af6734dd1ae9f11e4a0102ba2d6ba12853ae81a263fba68d883ce266dd3206.png

Here we can see that the two variables are closely related to one another. We can also see that most of the blue points fall below the black line, which tells us that the second reading is generally somewhat lower than the first reading.

In this section we will recreate a version of Figure 4.2 from Statistical Thinking in the 21st Century.

-

-oring_data = pd.read_csv('https://raw.githubusercontent.com/statsthinking21/statsthinking21-python/master/notebooks/data/orings.csv', index_col=0)
+
oring_data = pd.read_csv('https://raw.githubusercontent.com/statsthinking21/statsthinking21-python/master/notebooks/data/orings.csv', index_col=0)
 
 ax = sns.lineplot(x='Temperature', y='DamageIndex', data=oring_data, ci=None)
 plt.xlabel('Temperature at time of launch')
@@ -461,43 +582,146 @@ 

Scatter plot -_images/03-DataVisualization_33_0.png -

+
/tmp/ipykernel_161/488923491.py:3: FutureWarning: 
+
+The `ci` parameter is deprecated. Use `errorbar=None` for the same effect.
+
+  ax = sns.lineplot(x='Temperature', y='DamageIndex', data=oring_data, ci=None)
+
+_images/4bbd8449718db0a3ae3bf9eef978713e8c7156b4e5bbe581682d7c11855e6a3d.png
+

+
+

+ + + + +
+ + + + + + + + +
+ + + +
-
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+ +
+ - © Copyright 2020.
-

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+
\ No newline at end of file diff --git a/04-FittingSimpleModels.html b/04-FittingSimpleModels.html index 6179ce2..20bb753 100644 --- a/04-FittingSimpleModels.html +++ b/04-FittingSimpleModels.html @@ -1,241 +1,353 @@ - - + + + + + Fitting simple models — Python Companion to Statistical Thinking in the 21st Century - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - - + + + + +
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Fitting simple models

-
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- On this page -
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- -
+ + -
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Fitting simple models

+ +
+ +
+

Fitting simple models#

In this chapter we will focus on how to compute the measures of central tendency and variability that were covered in the previous chapter. Most of these can be computed using a built-in Python function, but we will show how to do them manually in order to give some intuition about how they work. First let’s load the NHANES data that we will use for our examples.

-
from nhanes.load import load_NHANES_data
+
from nhanes.load import load_NHANES_data
 nhanes_data = load_NHANES_data()
 adult_nhanes_data = nhanes_data.query('AgeInYearsAtScreening > 17')
 
@@ -245,26 +357,26 @@

Fitting simple modelsSince we will be analyzing the StandingHeightCm variable, we should exclude any observations that are missing this measurement. We will also recode the variable to be called Height in order to simplify the coding later.

-
adult_nhanes_data = adult_nhanes_data.dropna(subset=['StandingHeightCm']).rename(columns={'StandingHeightCm': 'Height'})
+
adult_nhanes_data = adult_nhanes_data.dropna(subset=['StandingHeightCm']).rename(columns={'StandingHeightCm': 'Height'})
 
-
-

Mean

+
+

Mean#

The mean is defined as the sum of values divided by the number of values being summed: -\($ +$\( \bar{X} = \frac{\sum_{i=1}^{n}x_i}{n} -$\) +\)$ Let’s say that we want to obtain the mean height for adults in the NHANES database (contained in the data Height that we generated above). We would sum the individual heights (using the .sum() operator) and then divide by the number of values:

-
adult_nhanes_data['Height'].sum() / adult_nhanes_data['Height'].shape[0]
+
adult_nhanes_data['Height'].sum() / adult_nhanes_data['Height'].shape[0]
 
+
+

Median#

The median is the middle value after sorting the entire set of values. First we sort the data in order of their values:

+
+

Mode#

The mode is the most frequent value that occurs in a variable. For example, let’s say that we had the following data:

+
+

Creating functions#

It is often useful to create our own custom function in order to perform a particular action. Let’s do that for our mode function:

+
+

Variability#

Let’s first compute the variance, which is the average squared difference between each value and the mean. Let’s do this with our cleaned-up version of the height data, but instead of working with the entire dataset, let’s take a random sample of 150 individuals:

+
+

Z-scores#

A Z-score is obtained by first subtracting the mean and then dividing by the standard deviation of a distribution. Let’s do this for the height_sample data.

+

+ + + + +
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Probability

+ +
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Probability#

In this chapter we will go over how to perform probability computations in Python.

-
-

Basic probability calculations

+
+

Basic probability calculations#

Let’s create a vector of outcomes from one to 6, using the np.arange() function to create such a sequence. This function takes the minimum and maximum values as its inputs, but note that the maximum is not included in the sequence; that is, the sequence goes up to but not including the maximum. Thus, we would have to give 1 and 7 as the minimum and maximum in order to get a sequence of numbers from 1 to 6:

+
+

Empirical frequency#

Let’s walk through how we computed empirical frequency of rain in San Francisco.

First we load the data:

+
+

Conditional probability#

Let’s determine the conditional probability of someone having hearing problems, given that they are over 70 years of age, using the NHANES dataset. First, let’s create a new variable called Over70 that denotes whether each individual is over 70 or not.

-
from nhanes.load import load_NHANES_data
+
from nhanes.load import load_NHANES_data
 nhanes_data = load_NHANES_data()
 
 nhanes_data['Over70'] = nhanes_data['AgeInYearsAtScreening'] > 70
@@ -502,7 +624,7 @@ 

Conditional probabilityNow let’s create a cleaned-up dataset that only includes the over70 variable along with the variable called HaveSeriousDifficultyHearing that denotes whether a person reports having serious hearing difficulty (coded as 1 for “yes” and 0 for “no”).

-
hearing_data = nhanes_data[['Over70', 'HaveSeriousDifficultyHearing']].dropna()
+
hearing_data = nhanes_data[['Over70', 'HaveSeriousDifficultyHearing']].dropna()
 hearing_data
 
@@ -599,13 +721,13 @@

Conditional probabilityFirst, what’s the probability of being over 70?

-
p_over_70 = hearing_data['Over70'].mean()
+
p_over_70 = hearing_data['Over70'].mean()
 p_over_70
 
-
0.10531978481769277
+
0.10531978481769277
 
@@ -613,13 +735,13 @@

Conditional probabilitySecond, what’s the probability of having hearing problems?

-
p_hearing_problem = hearing_data['HaveSeriousDifficultyHearing'].mean()
+
p_hearing_problem = hearing_data['HaveSeriousDifficultyHearing'].mean()
 p_hearing_problem
 
-
0.06276150627615062
+
0.06276150627615062
 
@@ -627,7 +749,7 @@

Conditional probabilityWhat’s the probability for each combination of hearing problems/no problems and over 70/ not? We can create a table that finds the joint probability for each combination, using the pd.crosstab() function:

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Sampling

-
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Sampling

+ +
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+

Sampling#

In this chapter we will learn how to use Python to understand sampling and sampling error.

-
-

Sampling error

+
+

Sampling error#

Here we will repeatedly sample from the NHANES Height variable in order to obtain the sampling distribution of the mean. First let’s load the data and clean them up.

+
+

Central limit theorem#

The central limit theorem tells us that the sampling distribution of the mean becomes normal as the sample size grows. Let’s test this by sampling a clearly non-normal variable and look at the normality of the results using a Q-Q plot. For example, let’s look at the variable that represents annual family income. This variable is oddly distributed:

-
plt.hist(adult_nhanes_data['AnnualFamilyIncome'])
+
plt.hist(adult_nhanes_data['AnnualFamilyIncome'])
 
-
-
/opt/conda/lib/python3.8/site-packages/numpy/lib/histograms.py:839: RuntimeWarning: invalid value encountered in greater_equal
-  keep = (tmp_a >= first_edge)
-/opt/conda/lib/python3.8/site-packages/numpy/lib/histograms.py:840: RuntimeWarning: invalid value encountered in less_equal
-  keep &= (tmp_a <= last_edge)
-
-
-
(array([369., 626., 925., 541., 357., 302., 241.,   0., 463., 902.]),
+
(array([369., 626., 925., 541., 357., 302., 241.,   0., 463., 902.]),
  array([  2000.,  11800.,  21600.,  31400.,  41200.,  51000.,  60800.,
          70600.,  80400.,  90200., 100000.]),
- <a list of 10 Patch objects>)
+ <BarContainer object of 10 artists>)
 
-_images/06-Sampling_7_2.png +_images/ad5ee260a4412336e0b7fdb5e3ef108e2bd4eaa932b70d92084b08f6796d4b7d.png

This odd distribution comes in part from the how the variable is coded, as shown here. Let’s resample this variable 5000 times, compute the mean, and examine the distribution. To do this, we will create a function that resamples and returns the mean:

-
def sample_and_return_mean(df, variable_name, 
+
+
+

Confidence intervals#

Remember that confidence intervals are intervals that will contain the population parameter in a certain proportion of samples from the population. In this example we will walk through the simulation that was presented in the book to show that this actually works properly. To do this, let’s create a function that takes a sample from the NHANES population and returns the confidence interval for the mean of the Height variable within that sample. We will use the t distribution to obtain our confidence intervals.

-
def get_confidence_interval(df, variable_name, 
+
def get_confidence_interval(df, variable_name, 
                             ci_percent=95,
                             sample_size=50):
     sample_df = df.sample(sample_size)
@@ -408,8 +521,7 @@ 

Confidence intervalsUsing this function, let’s resample the data 1000 times and look how often the resulting interval contains the population mean.

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Resampling and simulation

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Generating random samples

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Resampling and simulation#

+
+

Generating random samples#

Here we will generate random samples from a number of different distributions and plot their histograms. We could write out separate commands to plot each of our functions of interest, but that would involve repeating a lot of code, so instead we will take advantage of the fact that Python allows us to treat modules as variables. We will specify the module that creates each distribution, and then loop through them, each time incrementing the panel number. Some distributions also take specific parameters; for example, the Chi-squared distribution requires specifying the degrees of freedom. We will store those in a separate dictionary and use them as needed.

-
import scipy.stats
+
import scipy.stats
 import matplotlib.pyplot as plt
 
 num_samples = 10000
@@ -254,16 +376,16 @@ 

Generating random samples -_images/07-ResamplingAndSimulation_1_0.png +_images/e0115ad0320e66a7e8511052792c5cc021ad30a05eff63158bb83575a943b60f.png

-
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Simulating the maximum finishing time#

Let’s simulate 5000 samples of 150 observations, collecting the maximum value from each sample, and then plotting the distribution of maxima.

+
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The bootstrap#

The bootstrap is useful for creating confidence intervals in cases where we don’t have a parametric distribution. One example is for the median; let’s look at how that works. We will start by implementing it by hand, to see more closely how it works. We will start by collecting a sample of individuals from the NHANES dataset, and the using the bootstrap to obtain confidence intervals on the median for the Height variable.

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Hypothesis testing in Python

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Hypothesis testing in Python

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Hypothesis testing in Python#

In this chapter we will present several examples of using Python to perform hypothesis testing.

-
-

Simple example: Coin-flipping

+
+

Simple example: Coin-flipping#

Let’s say that we flipped 100 coins and observed 70 heads. We would like to use these data to test the hypothesis that the true probability is 0.5. First let’s generate our data, simulating 100,000 sets of 100 flips. We use such a large number because it turns out that it’s very rare to get 70 heads, so we need many attempts in order to get a reliable estimate of these probabilties. This will take a couple of minutes to complete.

+
+

Simulating p-values#

In this exercise we will perform hypothesis testing many times in order to test whether the p-values provided by our statistical test are valid. We will sample data from a normal distribution with a mean of zero, and for each sample perform a t-test to determine whether the mean is different from zero. We will then count how often we reject the null hypothesis; since we know that the true mean is zero, these are by definition Type I errors.

-

-num_runs = 5000
+
num_runs = 5000
 
 
 # create a function that will take a sample
 # and perform a one-sample t-test
 def sample_ttest(sampSize=32):
-    """
+    """
     perform a ttest on random data of n=sampSize
     """
 
@@ -326,45 +450,133 @@ 

Simulating p-values -
0.0516
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0.0512
 

We should see that the proportion of samples with p < .05 is about 5%.

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Statistical Power Analysis in Python

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Statistical Power Analysis in Python

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Statistical Power Analysis in Python#

In this chapter we focus specifically on statistical power. We will use the NHANES dataset, so let’s first set that up.

-

-import numpy as np
+
import numpy as np
 import pandas as pd
 
 np.random.seed(12345) 
@@ -235,13 +356,12 @@ 

Statistical Power Analysis in Python -

Power analysis

+
+

Power analysis#

We can compute a power analysis using functions from the statsmodels.stats.power package. Let’s focus on the power for an independent samples t-test in order to determine a difference in the mean between two groups. Let’s say that we think that an effect size of Cohen’s d=0.5 is realistic for the study in question (based on previous research) and would be of scientific interest. We wish to have 80% power to find the effect if it exists. We can compute the sample size needed for adequate power using the TTestIndPower() function:

+
+

Power curves#

We can also create plots that can show us how the power to find an effect varies as a function of effect size and sample size, at the alpha specified in the power analysis. We will use the plot_power() function. The x-axis is defined by the dep_var argument, while sample sizes (nobs) and effect sizes (effect_size) are provided as arrays.

-
#+
+
#+
 effect_sizes = np.array([0.2, 0.5, 0.8])
 sample_sizes = np.array(range(10, 500, 10))
 
@@ -280,18 +408,15 @@ 

Power curves -_images/09-StatisticalPower_5_0.png -

-
-
+
+

Simulating statistical power#

We can also simulate data to see whether the power analysis actually gives the right answer. We will sample data for two groups, with a difference of 0.5 standard deviations between their underlying distributions and a sample size based on power analysis, and we will then look at how often we reject the null hypothesis.

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Bayesian Statistics in Python

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Bayesian Statistics in Python#

In this chapter we will introduce how to basic Bayesian computations using Python.

-
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Applying Bayes’ theorem: A simple example

+
+

Applying Bayes’ theorem: A simple example#

TBD: MOVE TO MULTIPLE TESTING EXAMPLE SO WE CAN USE BINOMIAL LIKELIHOOD A person has a cough and flu-like symptoms, and gets a PCR test for COVID-19, which comes back postiive. What is the likelihood that they actually have COVID-19, as opposed to a regular cold or flu? We can use Bayes’ theorem to compute this. Let’s say that the local rate of symptomatic individuals who actually are infected with COVID-19 is 7.4% (as reported on July 10, 2020 for San Francisco); thus, our prior probability that someone with symptoms actually has COVID-19 is .074. The RT-PCR test used to identify COVID-19 RNA is highly specific (that is, it very rarelly reports the presence of the virus when it is not present); for our example, we will say that the specificity is 99%. Its sensitivity is not known, but probably is no higher than 90%.
First let’s look at the probability of disease given a single positive test.

+
+

Estimating posterior distributions#

In this example we will look at how to estimate entire posterior distributions.
We will implement the drug testing example from the book. In that example, we administered a drug to 100 people, and found that 64 of them responded positively to the drug. What we want to estimate is the probability distribution for the proportion of responders, given the data. For simplicity we started with a uniform prior; that is, all proprtions of responding are equally likely to begin with. In addition, we will use a discrete probability distribution; that is, we will estimate the posterior probabiilty for each particular proportion of responders, in steps of 0.01. This greatly simplifies the math and still retains the main idea.

+
+

Bayes factors#

There are no convenient off-the-shelf tools for estimating Bayes factors using Python, so we will use the rpy2 package to access the BayesFactor library in R. Let’s compute a Bayes factor for a T-test comparing the amount of reported alcohol computing between smokers versus non-smokers. First, let’s set up the NHANES data and collect a sample of 150 smokers and 150 nonsmokers.

-
#+
+
- - - - - - - - - - - - - - - - - - - - - - - - - -
HadCancer01
SmokeNow
03836470
187681
-
+
---------------------------------------------------------------------------
+ModuleNotFoundError                       Traceback (most recent call last)
+Cell In[2], line 1
+----> 1 import statsmodels.api as sm
+      2 table = sm.stats.Table.from_data(categorical_df, shift_zeros=False)
+      3 table.table_orig
+
+ModuleNotFoundError: No module named 'statsmodels'
+
+
+

We can also see the predicted frequencies under the null hypothesis of independence, which are stored in the .fittedvalues element:

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table.fittedvalues
+
table.fittedvalues
 
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HadCancer01
SmokeNow
03855.191336450.808664
1856.808664100.191336
-

Using these, we can compute the chi-squared statistic:

-
import numpy as np
+
 

We can confirm this by comparing it to the result from the built-in function to compute the association:

chi2_result = table.test_nominal_association() @@ -369,87 +415,132 @@

The Pearson Chi-squared test
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table.standardized_resids
 
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HadCancer01
SmokeNow
0-2.2401852.240185
12.240185-2.240185
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This shows that there is an unexpectedly large number of people who smoke but don’t have cancer, and similarly an unexpectedly low number of smokers who report having had cancer before. Does this tell us that smoking results in lower rates of cancer?

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The General Linear Model

+ +
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The General Linear Model#

In this chapter we will explore how to fit general linear models in Python. We will focus on the tools provided by the statsmodels package.

-
from nhanes.load import load_NHANES_data
+
from nhanes.load import load_NHANES_data
 nhanes_data = load_NHANES_data()
 adult_nhanes_data = nhanes_data.query('AgeInYearsAtScreening > 17')
 
-
-

Linear regression

+
+

Linear regression#

To perform linear regression in Python, we use the OLS() function (which stands for ordinary least squares) from the statsmodels package. Let’s generate some simulated data and use this function to compute the linear regression solution.

-

-import numpy as np
+
import numpy as np
 import pandas as pd
 import matplotlib.pyplot as plt
 
@@ -250,7 +365,7 @@ 

Linear regressiondef generate_linear_data(slope, intercept, noise_sd=1, x=None, npoints=100, seed=None): - """ + """ generate data with a given slope and intercept and add normally distributed noise @@ -280,16 +395,16 @@

Linear regression -
<matplotlib.collections.PathCollection at 0x7febd5275be0>
+
<matplotlib.collections.PathCollection at 0x7ff27f1a7140>
 
-_images/13-GeneralLinearModel_3_1.png +_images/52b4a700144d39470dd1ef18cb796f381a9edfa8d0fac57763a542cb53d73248.png

We can then perform linear regression on these data using the ols function. This function doesn’t automatically include an intercept in its model, so we need to add one to the design. Fitting the model using this function is a two-step process. First, we set up the model and store it to a variable (which we will call ols_model). Then, we actually fit the model, which generates the results that we store to a different variable called ols_results, and view a summary using the .summary() method of the results variable.

-

We should see three things in these results:

+
+

Model criticism and diagnostics#

Once we have fitted the model, we want to look at some diagnostics to determine whether the model is actually fitting properly.
The first thing to examine is to make sure that the residuals are (at least roughly) normally distributed. We can do this using a Q-Q plot:

-
import seaborn as sns
+
import seaborn as sns
 import scipy.stats
 
 _ = scipy.stats.probplot(ols_result.resid, plot=sns.mpl.pyplot)
 
-
-_images/13-GeneralLinearModel_9_0.png -

This looks pretty good, in the sense that the residual data points fall very close to the unit line. This is not surprising, since we generated the data with normally distributed noise. We should also plot the predicted (or fitted) values against the residuals, to make sure that the model does work systematically better for some predicted values versus others.

-
plt.scatter(ols_result.fittedvalues, ols_result.resid)
+
plt.scatter(ols_result.fittedvalues, ols_result.resid)
 plt.xlabel('Fitted value')
 plt.ylabel('Residual')
 
-
-
Text(0, 0.5, 'Residual')
-
-
-_images/13-GeneralLinearModel_11_1.png -

As expected, we see no clear relationship.

-
-
-

Examples of problematic model fit

+
+
+

Examples of problematic model fit#

Let’s say that there was another variable at play in this dataset, which we were not aware of. This variable causes some of the cases to have much larger values than others, in a way that is unrelated to the X variable. We play a trick here using the seq() function to create a sequence from zero to one, and then threshold those 0.5 (in order to obtain half of the values as zero and the other half as one) and then multiply by the desired effect size:

-
simulated_data.loc[:, 'x2'] = (simulated_data.index < (simulated_data.shape[0] / 2)).astype('int')
+
simulated_data.loc[:, 'x2'] = (simulated_data.index < (simulated_data.shape[0] / 2)).astype('int')
 hidden_effect_size = 10
 simulated_data.loc[:, 'y2'] = simulated_data['y'] + simulated_data['x2'] * hidden_effect_size
 
@@ -423,7 +480,7 @@

Examples of problematic model fit
-
ols_model2 = ols(formula='y2 ~ x + 1', data=simulated_data)
+
 

The lack of normality is clear from the Q-Q plot, and we can also see that there is obvious structure in the residuals.

Let’s look at another potential problem, in which the y variable is nonlinearly related to the X variable. We can create these data by squaring the X variable when we generate the Y variable:

-
noise_sd = 0.1
+
noise_sd = 0.1
 simulated_data['y3'] = (simulated_data['x']**2) * slope + intercept + np.random.randn(simulated_data.shape[0]) * noise_sd
 
 plt.scatter(simulated_data['x'], simulated_data['y3'])
 
-
-
<matplotlib.collections.PathCollection at 0x7febccb2f850>
-
-
-_images/13-GeneralLinearModel_17_1.png -
-
ols_model3 = ols(formula='y3 ~ x + 1', data=simulated_data)
+
ols_model3 = ols(formula='y3 ~ x + 1', data=simulated_data)
 ols_result3 = ols_model3.fit()
 ols_result3.summary()
 
-
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
OLS Regression Results
Dep. Variable: y3 R-squared: 0.008
Model: OLS Adj. R-squared: -0.002
Method: Least Squares F-statistic: 0.8369
Date: Thu, 22 Dec 2022 Prob (F-statistic): 0.363
Time: 15:26:38 Log-Likelihood: -149.89
No. Observations: 100 AIC: 303.8
Df Residuals: 98 BIC: 309.0
Df Model: 1
Covariance Type: nonrobust
- - - - - - - - - - -
coef std err t P>|t| [0.025 0.975]
Intercept 10.7813 0.110 98.295 0.000 10.564 10.999
x 0.1131 0.124 0.915 0.363 -0.132 0.358
- - - - - - - - - - - - - -
Omnibus: 62.137 Durbin-Watson: 1.699
Prob(Omnibus): 0.000 Jarque-Bera (JB): 203.933
Skew: 2.285 Prob(JB): 5.21e-45
Kurtosis: 8.297 Cond. No. 1.15


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Now we see that there is no significant linear relationship between \(X^2\) and Y/ But if we look at the residuals the problem with the model becomes clear:

plt.figure(figsize=(12, 6)) @@ -539,75 +528,18 @@

Examples of problematic model fitx**2 in the model:

-
simulated_data.loc[:, 'x_squared'] = simulated_data['x'] ** 2
+
simulated_data.loc[:, 'x_squared'] = simulated_data['x'] ** 2
 ols_model4 = ols(formula='y3 ~ x_squared + 1', data=simulated_data)
 ols_result4 = ols_model4.fit()
 ols_result4.summary()
 
-
-
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
OLS Regression Results
Dep. Variable: y3 R-squared: 0.992
Model: OLS Adj. R-squared: 0.992
Method: Least Squares F-statistic: 1.243e+04
Date: Thu, 22 Dec 2022 Prob (F-statistic): 4.87e-105
Time: 15:26:38 Log-Likelihood: 92.204
No. Observations: 100 AIC: -180.4
Df Residuals: 98 BIC: -175.2
Df Model: 1
Covariance Type: nonrobust
- - - - - - - - - - -
coef std err t P>|t| [0.025 0.975]
Intercept 10.0215 0.012 841.568 0.000 9.998 10.045
x_squared 0.9740 0.009 111.473 0.000 0.957 0.991
- - - - - - - - - - - - - -
Omnibus: 0.981 Durbin-Watson: 2.260
Prob(Omnibus): 0.612 Jarque-Bera (JB): 1.073
Skew: 0.216 Prob(JB): 0.585
Kurtosis: 2.735 Cond. No. 2.09


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Now we see that the effect of \(X^2\) is significant, and if we look at the residual plot we should see that things look much better:

+
+

Extending regression to binary outcomes.#

-
from statsmodels.formula.api import logit
+
from statsmodels.formula.api import logit
 
 diabetes_df = adult_nhanes_data.query(
     'DoctorToldYouHaveDiabetes != "Borderline"').dropna(
@@ -645,117 +571,29 @@ 

Extending regression to binary outcomes.logit() function:

-

-logitfit = logit(formula = 'Diabetes ~ Age + BMI', data = diabetes_df).fit(disp=0)
+
logitfit = logit(formula = 'Diabetes ~ Age + BMI', data = diabetes_df).fit(disp=0)
 
 logitfit.summary()
 
-
-
- - - - - - - - - - - - - - - - - - - - - - -
Logit Regression Results
Dep. Variable: Diabetes No. Observations: 5267
Model: Logit Df Residuals: 5264
Method: MLE Df Model: 2
Date: Thu, 22 Dec 2022 Pseudo R-squ.: 0.1661
Time: 15:26:38 Log-Likelihood: -1895.1
converged: True LL-Null: -2272.6
Covariance Type: nonrobust LLR p-value: 1.122e-164
- - - - - - - - - - - - - -
coef std err z P>|z| [0.025 0.975]
Intercept -7.3101 0.276 -26.478 0.000 -7.851 -6.769
Age 0.0622 0.003 21.763 0.000 0.057 0.068
BMI 0.0693 0.005 12.769 0.000 0.059 0.080

This looks very similar to the output from the ols() function, and it shows us that there is a significant relationship between the age, weight, and diabetes. The model provides us with a predicted probability that each individual will have diabetes; if this is greater than 0.5, then that means that the model predicts that the individual is more likely than not to have diabetes.
We can start by simply comparing those predictions to the actual outcomes.

-

-diabetes_df.loc[:, 'LogitPrediction'] = (logitfit.predict() > 0.5).astype('int')
+
diabetes_df.loc[:, 'LogitPrediction'] = (logitfit.predict() > 0.5).astype('int')
 
 pd.crosstab(diabetes_df['Diabetes'], diabetes_df['LogitPrediction'])
 
-
-
- - - - - - - - - - - - - - - - - - - - - - - - - - -
LogitPrediction01
Diabetes
0440050
175661
-

This table shows that the model did somewhat well, in that it labeled most non-diabetic people as non-diabetic, and most diabetic people as diabetic. However, it also made a lot of mistakes, mislabeling nearly half of all diabetic people as non-diabetic.

We would often like a single number that tells us how good our prediction is. We could simply ask how many of our predictions are correct on average:

-
(diabetes_df['LogitPrediction'] == diabetes_df['Diabetes']).mean()
-
-
-
-

+
+

Cross-validation#

Cross-validation is a powerful technique that allows us to estimate how well our results will generalize to a new dataset. Here we will build our own crossvalidation code to see how it works, continuing the logistic regression example from the previous section. In cross-validation, we want to split the data into several subsets and then iteratively train the model while leaving out each subset (which we usually call folds) and then test the model on that held-out fold. We can use one of the tools from the scikit-learn package to create our cross-validation folds for us. Let’s start by using 10-fold crossvalidation, in which we split the data into 10 parts, and the fit the model while holding out one of those parts and then testing it on the held-out data.

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