diff --git a/assignment5.Rmd b/assignment5.Rmd index 288bcb3..b705974 100644 --- a/assignment5.Rmd +++ b/assignment5.Rmd @@ -16,7 +16,8 @@ The data you will be using comes from the Assistments online intelligent tutorin ## Start by uploading the data ```{r} -D1 <- +D1 <- read.csv("Assistments-confidence.csv", header = TRUE) +library(tidyverse) ``` @@ -27,6 +28,7 @@ D1 <- library(ggplot2) library(GGally) +library(corrplot) ggpairs(D1, 2:8, progress = FALSE) #ggpairs() draws a correlation plot between all the columns you identify by number (second option, you don't need the first column as it is the student ID) and progress = FALSE stops a progress bar appearing as it renders your plot @@ -38,7 +40,7 @@ ggcorr(D1[,-1], method = c("everything", "pearson")) #ggcorr() doesn't have an e ## Create a new data frame with the mean_correct variable removed, we want to keep that variable intact. The other variables will be included in our PCA. ```{r} -D2 <- +D2 <- as_tibble(D1) %>% select(-1,-5)#removed the first ID column ``` @@ -67,20 +69,24 @@ plot(pca, type = "lines") ``` ## Decide which components you would drop and remove them from your data set. +I will select components 1-4 since they account for about 80% of the variance. ## Part II ```{r} #Now, create a data frame of the transformed data from your pca. +library(broom) -D3 <- #Attach the variable "mean_correct" from your original data frame to D3. - +D3 <- augment(pca, data = D1) %>% select(mean_correct,.fittedPC1:.fittedPC4) #Now re-run your correlation plots between the transformed data and mean_correct. If you had dropped some components would you have lost important infomation about mean_correct? +ggpairs(D3,progress = FALSE) + +ggcorr(D3, method = c("everything", "pearson")) ``` @@ -91,9 +97,11 @@ pca$rotation #Examine the eigenvectors, notice that they are a little difficult to interpret. It is much easier to make sense of them if we make them proportional within each component -loadings <- abs(pca$rotation) #abs() will make all eigenvectors positive +loadings <- abs(pca$rotation)#abs() will make all eigenvectors positive +loadings #Now examine your components and try to come up with substantive descriptions of what some might represent? +# The first PC is mostly made up of "mean hint" which according to the first correlations was one of the few things that were correlated to "Mean Correct". Similarly, PC 2 is mostly made up of "prior percent" correct (.82) as it makes sense that a student will be able to correctly answer questions correct if they had have the previous knowledge needed. It is interesting that PC6 had the highest weight of "mean hint" and a highest weight of "prior percent" than PC6 so that is what we loose by removing some variance and one of the components. I also think that we started with really low correlations so removing anything, removes information needed. If we had variables that were highly correlated to start, it would be easier to remove components. #You can generate a biplot to help you, though these can be a bit confusing. They plot the transformed data by the first two components. Therefore, the axes represent the direction of maximum variance accounted for. Then mapped onto this point cloud are the original directions of the variables, depicted as red arrows. It is supposed to provide a visualization of which variables "go together". Variables that possibly represent the same underlying construct point in the same direction. @@ -102,13 +110,32 @@ biplot(pca) ``` # Part III -Also in this repository is a data set collected from TC students (tc-program-combos.csv) that shows how many students thought that a TC program was related to andother TC program. Students were shown three program names at a time and were asked which two of the three were most similar. Use PCA to look for components that represent related programs. Explain why you think there are relationships between these programs. +Also in this repository is a data set collected from TC students (tc-program-combos.csv) that shows how many students thought that a TC program was related to another TC program. Students were shown three program names at a time and were asked which two of the three were most similar. Use PCA to look for components that represent related programs. Explain why you think there are relationships between these programs. -```{r} + +Load and clean the data +```{r} +P1 <- read.csv("tc-program-combos.csv", header = TRUE) %>% select (-1) ``` +PCA +```{r} +PCA_T <- prcomp(P1, scale. = TRUE) +``` +Examining the Components +```{r} +PCA_T$sdev +PCA_T$sdev^2 +summary(PCA_T) +plot(PCA_T, type="lines") +#PCA_T$rotation +PCAT_loadings <- abs(PCA_T$rotation) +biplot(PCA_T) + +#from the biplot you can see how students identify similar programs as "similar" programs point to the same direction. For example, you can see Kinesiology, Health Education and Nutrition on the top left, and Psychology programs on the top left. +``` diff --git a/assignment5.html b/assignment5.html new file mode 100644 index 0000000..928609d --- /dev/null +++ b/assignment5.html @@ -0,0 +1,644 @@ + + + + + + + + + + + + + +Principle Component Aanalysis + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + +
+

Data

+

The data you will be using comes from the Assistments online intelligent tutoring system (https://www.assistments.org/). It describes students working through online math problems. Each student has the following data associated with them:

+ +
+
+

Start by uploading the data

+
D1 <- read.csv("Assistments-confidence.csv", header = TRUE)
+library(tidyverse)
+
## -- Attaching packages --------------------------------------- tidyverse 1.3.0 --
+
## v ggplot2 3.3.2     v purrr   0.3.4
+## v tibble  3.0.4     v dplyr   1.0.2
+## v tidyr   1.1.2     v stringr 1.4.0
+## v readr   1.4.0     v forcats 0.5.0
+
## Warning: package 'tibble' was built under R version 4.0.3
+
## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
+## x dplyr::filter() masks stats::filter()
+## x dplyr::lag()    masks stats::lag()
+
+
+

Create a correlation matrix of the relationships between the variables, including correlation coefficients for each pair of variables/features.

+
#You can install the corrplot package to plot some pretty correlation matrices (sometimes called correlograms)
+
+library(ggplot2)
+library(GGally)
+
## Warning: package 'GGally' was built under R version 4.0.3
+
## Registered S3 method overwritten by 'GGally':
+##   method from   
+##   +.gg   ggplot2
+
library(corrplot)
+
## Warning: package 'corrplot' was built under R version 4.0.3
+
## corrplot 0.84 loaded
+
ggpairs(D1, 2:8, progress = FALSE) #ggpairs() draws a correlation plot between all the columns you identify by number (second option, you don't need the first column as it is the student ID) and progress = FALSE stops a progress bar appearing as it renders your plot
+

+
ggcorr(D1[,-1], method = c("everything", "pearson")) #ggcorr() doesn't have an explicit option to choose variables so we need to use matrix notation to drop the id variable. We then need to choose a "method" which determines how to treat missing values (here we choose to keep everything, and then which kind of correlation calculation to use, here we are using Pearson correlation, the other options are "kendall" or "spearman")
+

+
#Study your correlogram images and save them, you will need them later. Take note of what is strongly related to the outcome variable of interest, mean_correct.
+
+
+

Create a new data frame with the mean_correct variable removed, we want to keep that variable intact. The other variables will be included in our PCA.

+
D2 <- as_tibble(D1) %>%  select(-1,-5)#removed the first ID column
+
+
+

Now run the PCA on the new data frame

+
pca <- prcomp(D2, scale. = TRUE)
+
+
+

Although princomp does not generate the eigenvalues directly for us, we can print a list of the standard deviation of the variance accounted for by each component.

+
pca$sdev
+
## [1] 1.2825140 1.0543565 1.0245688 0.9621486 0.8556715 0.7320146
+
#To convert this into variance accounted for we can square it, these numbers are proportional to the eigenvalue
+
+pca$sdev^2
+
## [1] 1.6448423 1.1116675 1.0497412 0.9257299 0.7321737 0.5358454
+
#A summary of our pca will give us the proportion of variance accounted for by each component
+
+summary(pca)
+
## Importance of components:
+##                           PC1    PC2    PC3    PC4    PC5     PC6
+## Standard deviation     1.2825 1.0544 1.0246 0.9621 0.8557 0.73201
+## Proportion of Variance 0.2741 0.1853 0.1750 0.1543 0.1220 0.08931
+## Cumulative Proportion  0.2741 0.4594 0.6344 0.7887 0.9107 1.00000
+
#We can look at this to get an idea of which components we should keep and which we should drop
+
+plot(pca, type = "lines")
+

+
+
+

Decide which components you would drop and remove them from your data set.

+

I will select components 1-4 since they account for about 80% of the variance.

+
+
+

Part II

+
#Now, create a data frame of the transformed data from your pca.
+library(broom)
+
## Warning: package 'broom' was built under R version 4.0.3
+
#Attach the variable "mean_correct" from your original data frame to D3.
+
+D3 <- augment(pca, data = D1) %>% select(mean_correct,.fittedPC1:.fittedPC4)
+
+#Now re-run your correlation plots between the transformed data and mean_correct. If you had dropped some components would you have lost important infomation about mean_correct?
+
+ggpairs(D3,progress = FALSE)
+

+
ggcorr(D3, method = c("everything", "pearson"))
+

## Now print out the loadings for the components you generated:

+
pca$rotation
+
##                               PC1         PC2         PC3        PC4
+## prior_prob_count      -0.26034140  0.45818753 -0.40090679 -0.6897642
+## prior_percent_correct  0.16840319  0.81617867  0.09267306  0.2640040
+## problems_attempted    -0.45568733  0.31685183  0.36387724  0.3168141
+## mean_hint             -0.63337594 -0.12501620 -0.08008842 -0.1122586
+## mean_attempt          -0.54200011 -0.08510858 -0.04585364  0.3108682
+## mean_confidence        0.03581325  0.02547483 -0.83051917  0.4948890
+##                                PC5         PC6
+## prior_prob_count      -0.007142834 -0.29280482
+## prior_percent_correct  0.298843852  0.37134715
+## problems_attempted    -0.592336569 -0.32911025
+## mean_hint             -0.102302115  0.74412634
+## mean_attempt           0.697232132 -0.33781385
+## mean_confidence       -0.251357022 -0.01452143
+
#Examine the eigenvectors, notice that they are a little difficult to interpret. It is much easier to make sense of them if we make them proportional within each component
+
+loadings <- abs(pca$rotation)#abs() will make all eigenvectors positive
+loadings
+
##                              PC1        PC2        PC3       PC4         PC5
+## prior_prob_count      0.26034140 0.45818753 0.40090679 0.6897642 0.007142834
+## prior_percent_correct 0.16840319 0.81617867 0.09267306 0.2640040 0.298843852
+## problems_attempted    0.45568733 0.31685183 0.36387724 0.3168141 0.592336569
+## mean_hint             0.63337594 0.12501620 0.08008842 0.1122586 0.102302115
+## mean_attempt          0.54200011 0.08510858 0.04585364 0.3108682 0.697232132
+## mean_confidence       0.03581325 0.02547483 0.83051917 0.4948890 0.251357022
+##                              PC6
+## prior_prob_count      0.29280482
+## prior_percent_correct 0.37134715
+## problems_attempted    0.32911025
+## mean_hint             0.74412634
+## mean_attempt          0.33781385
+## mean_confidence       0.01452143
+
#Now examine your components and try to come up with substantive descriptions of what some might represent?
+# The first PC is mostly made up of "mean hint" which according to the first correlations was one of the few things that were correlated to "Mean Correct". Similarly, PC 2 is mostly made up of "prior percent" correct (.82) as it makes sense that a student will be able to correctly answer questions correct if they had have the previous knowledge needed. It is interesting that PC6 had the highest weight of "mean hint" and a highest weight of "prior percent" than PC6 so that is what we loose by removing some variance and one of the components. I also think that we started with really low correlations so removing anything, removes information needed. If we had variables that were highly correlated to start, it would be easier to remove components. 
+
+#You can generate a biplot to help you, though these can be a bit confusing. They plot the transformed data by the first two components. Therefore, the axes represent the direction of maximum variance accounted for. Then mapped onto this point cloud are the original directions of the variables, depicted as red arrows. It is supposed to provide a visualization of which variables "go together". Variables that possibly represent the same underlying construct point in the same direction.  
+
+biplot(pca)
+

# Part III
+Also in this repository is a data set collected from TC students (tc-program-combos.csv) that shows how many students thought that a TC program was related to another TC program. Students were shown three program names at a time and were asked which two of the three were most similar. Use PCA to look for components that represent related programs. Explain why you think there are relationships between these programs.

+

Load and clean the data

+
P1 <- read.csv("tc-program-combos.csv", header = TRUE) %>% select (-1)
+

PCA

+
PCA_T <- prcomp(P1, scale. = TRUE)
+

Examining the Components

+
PCA_T$sdev
+
##  [1] 2.66699514 2.33303087 2.03824332 1.80893489 1.71451092 1.60411744
+##  [7] 1.58798960 1.49222150 1.46424566 1.39138869 1.33520786 1.32516917
+## [13] 1.31212863 1.26312253 1.25366254 1.22338737 1.21896028 1.18649390
+## [19] 1.13127469 1.12814038 1.10432926 1.06319093 1.01168384 0.99665812
+## [25] 0.96528383 0.95048841 0.93256896 0.90507518 0.85160824 0.83479067
+## [31] 0.81879538 0.78538963 0.76079365 0.73350908 0.72278124 0.67319166
+## [37] 0.66343310 0.64839067 0.62448974 0.60331242 0.56846989 0.55769066
+## [43] 0.51031628 0.49442626 0.47128286 0.44551299 0.43288829 0.41344476
+## [49] 0.37259965 0.36653720 0.35015529 0.33278376 0.32799766 0.30414406
+## [55] 0.28040415 0.27066834 0.23729873 0.21156010 0.17616906 0.16541514
+## [61] 0.14778289 0.14204907 0.11092615 0.07054620 0.04430024 0.03588606
+## [67] 0.01241193
+
PCA_T$sdev^2
+
##  [1] 7.1128630864 5.4430330560 4.1544358466 3.2722454371 2.9395476996
+##  [6] 2.5731927459 2.5217109546 2.2267250142 2.1440153558 1.9359624893
+## [11] 1.7827800336 1.7560733413 1.7216815395 1.5954785267 1.5716697682
+## [16] 1.4966766496 1.4858641675 1.4077677636 1.2797824290 1.2727007063
+## [21] 1.2195431178 1.1303749458 1.0235041934 0.9933274149 0.9317728773
+## [26] 0.9034282205 0.8696848576 0.8191610815 0.7252366024 0.6968754570
+## [31] 0.6704258671 0.6168368672 0.5788069731 0.5380355701 0.5224127176
+## [36] 0.4531870174 0.4401434846 0.4204104623 0.3899874305 0.3639858772
+## [41] 0.3231580154 0.3110188729 0.2604227077 0.2444573271 0.2221075356
+## [46] 0.1984818264 0.1873922684 0.1709365720 0.1388304984 0.1343495215
+## [51] 0.1226087294 0.1107450296 0.1075824634 0.0925036076 0.0786264855
+## [56] 0.0732613524 0.0563106886 0.0447576745 0.0310355381 0.0273621679
+## [61] 0.0218397813 0.0201779378 0.0123046098 0.0049767662 0.0019625111
+## [66] 0.0012878092 0.0001540561
+
summary(PCA_T)
+
## Importance of components:
+##                           PC1     PC2     PC3     PC4     PC5     PC6     PC7
+## Standard deviation     2.6670 2.33303 2.03824 1.80893 1.71451 1.60412 1.58799
+## Proportion of Variance 0.1062 0.08124 0.06201 0.04884 0.04387 0.03841 0.03764
+## Cumulative Proportion  0.1062 0.18740 0.24941 0.29825 0.34212 0.38053 0.41816
+##                            PC8    PC9    PC10    PC11    PC12   PC13    PC14
+## Standard deviation     1.49222 1.4642 1.39139 1.33521 1.32517 1.3121 1.26312
+## Proportion of Variance 0.03323 0.0320 0.02889 0.02661 0.02621 0.0257 0.02381
+## Cumulative Proportion  0.45140 0.4834 0.51229 0.53890 0.56511 0.5908 0.61462
+##                           PC15    PC16    PC17    PC18   PC19   PC20   PC21
+## Standard deviation     1.25366 1.22339 1.21896 1.18649 1.1313 1.1281 1.1043
+## Proportion of Variance 0.02346 0.02234 0.02218 0.02101 0.0191 0.0190 0.0182
+## Cumulative Proportion  0.63808 0.66042 0.68260 0.70361 0.7227 0.7417 0.7599
+##                           PC22    PC23    PC24    PC25    PC26    PC27    PC28
+## Standard deviation     1.06319 1.01168 0.99666 0.96528 0.95049 0.93257 0.90508
+## Proportion of Variance 0.01687 0.01528 0.01483 0.01391 0.01348 0.01298 0.01223
+## Cumulative Proportion  0.77678 0.79205 0.80688 0.82079 0.83427 0.84725 0.85948
+##                           PC29   PC30    PC31    PC32    PC33    PC34   PC35
+## Standard deviation     0.85161 0.8348 0.81880 0.78539 0.76079 0.73351 0.7228
+## Proportion of Variance 0.01082 0.0104 0.01001 0.00921 0.00864 0.00803 0.0078
+## Cumulative Proportion  0.87030 0.8807 0.89071 0.89992 0.90856 0.91659 0.9244
+##                           PC36    PC37    PC38    PC39    PC40    PC41    PC42
+## Standard deviation     0.67319 0.66343 0.64839 0.62449 0.60331 0.56847 0.55769
+## Proportion of Variance 0.00676 0.00657 0.00627 0.00582 0.00543 0.00482 0.00464
+## Cumulative Proportion  0.93115 0.93772 0.94399 0.94981 0.95524 0.96007 0.96471
+##                           PC43    PC44    PC45    PC46   PC47    PC48    PC49
+## Standard deviation     0.51032 0.49443 0.47128 0.44551 0.4329 0.41344 0.37260
+## Proportion of Variance 0.00389 0.00365 0.00332 0.00296 0.0028 0.00255 0.00207
+## Cumulative Proportion  0.96860 0.97224 0.97556 0.97852 0.9813 0.98387 0.98594
+##                           PC50    PC51    PC52    PC53    PC54    PC55    PC56
+## Standard deviation     0.36654 0.35016 0.33278 0.32800 0.30414 0.28040 0.27067
+## Proportion of Variance 0.00201 0.00183 0.00165 0.00161 0.00138 0.00117 0.00109
+## Cumulative Proportion  0.98795 0.98978 0.99143 0.99304 0.99442 0.99559 0.99668
+##                           PC57    PC58    PC59    PC60    PC61   PC62    PC63
+## Standard deviation     0.23730 0.21156 0.17617 0.16542 0.14778 0.1420 0.11093
+## Proportion of Variance 0.00084 0.00067 0.00046 0.00041 0.00033 0.0003 0.00018
+## Cumulative Proportion  0.99752 0.99819 0.99866 0.99906 0.99939 0.9997 0.99987
+##                           PC64    PC65    PC66    PC67
+## Standard deviation     0.07055 0.04430 0.03589 0.01241
+## Proportion of Variance 0.00007 0.00003 0.00002 0.00000
+## Cumulative Proportion  0.99995 0.99998 1.00000 1.00000
+
plot(PCA_T, type="lines")
+

+
#PCA_T$rotation
+PCAT_loadings <- abs(PCA_T$rotation)
+biplot(PCA_T)
+

+
#from the biplot you can see how students identify similar programs as "similar" programs point to the same direction. For example, you can see Kinesiology, Health Education and Nutrition on the top left, and Psychology programs on the top left.
+
+ + + + +
+ + + + + + + + + + + + + + +