Homework 7: Principle Component Analysis

Principal Components

For this short report, I use the technique of Principal Components Analysis (PCA) to form an index of student engagement in education using data from the 2019 National Household Education Survey (NHES), made available by the National Center for Education Statistics (NCES). The index will be comprised of 5 variables: - Days absent - Hours spent completing homework - Enjoyment of school - Academic performance - Academic engagement

Data was filtered to include children only enrolled in public schools, grades 6-8 (middle school).

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## Attaching package: 'car'

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## Please cite as:

##  Hlavac, Marek (2018). stargazer: Well-Formatted Regression and Summary Statistics Tables.

##  R package version 5.2.2. https://CRAN.R-project.org/package=stargazer

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Analysis

Summary Statistics and Correlation Matrix

In examining the summary table of eigenvalues, the first three components of the PCA model account for over 80% of the variation in the input variables. The first variable has an eigenvalue greater than 2, and the next two contributing variables have eigenvalues very close to 1.

According to summary statistics, academic engagement and academic performance account for the most variation in the index. Number of hours spent on homework and number of days absent are weaker explanatory variables for the overarching latent variable, student engagement. Number of hours spent on homework is, in fact, negatively correlated to the other factors indicating a poor fit of this variable for the index.

## Warning: package 'FactoMineR' was built under R version 4.1.3

## 
## Call:
## PCA(X = nhes19_pca.2[, c(1:5)], scale.unit = T, graph = F) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5
## Variance               2.115   0.972   0.935   0.717   0.261
## % of var.             42.291  19.442  18.704  14.339   5.224
## Cumulative % of var.  42.291  61.733  80.437  94.776 100.000
## 
## Individuals (the 10 first)
##               Dist    Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1         |  1.821 | -1.287  0.076  0.500 |  0.879  0.077  0.233 |  0.064
## 2         |  1.358 | -0.256  0.003  0.036 | -0.243  0.006  0.032 | -0.259
## 3         |  1.740 |  0.271  0.003  0.024 |  0.337  0.011  0.037 | -0.599
## 4         |  1.928 |  0.135  0.001  0.005 | -0.990  0.098  0.264 |  1.496
## 5         |  6.178 |  3.761  0.650  0.371 |  0.710  0.050  0.013 |  4.835
## 6         |  1.583 | -1.210  0.067  0.584 |  0.426  0.018  0.072 |  0.008
## 7         |  2.440 | -2.042  0.192  0.700 |  1.260  0.159  0.267 |  0.176
## 8         |  1.459 | -1.132  0.059  0.603 | -0.028  0.000  0.000 | -0.048
## 9         |  1.075 | -0.375  0.006  0.122 | -0.609  0.037  0.320 | -0.358
## 10        |  2.312 |  0.820  0.031  0.126 | -0.271  0.007  0.014 |  1.226
##              ctr   cos2  
## 1          0.000  0.001 |
## 2          0.007  0.036 |
## 3          0.037  0.119 |
## 4          0.232  0.602 |
## 5          2.429  0.612 |
## 6          0.000  0.000 |
## 7          0.003  0.005 |
## 8          0.000  0.001 |
## 9          0.013  0.111 |
## 10         0.156  0.281 |
## 
## Variables
##              Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3    ctr   cos2
## daysabsnt |  0.351  5.821  0.123 | -0.078  0.623  0.006 |  0.931 92.594  0.866
## hwkhrswk  | -0.243  2.783  0.059 |  0.962 95.175  0.925 |  0.117  1.455  0.014
## schlenjoy |  0.656 20.353  0.430 |  0.048  0.233  0.002 | -0.036  0.138  0.001
## grades    |  0.858 34.845  0.737 |  0.174  3.100  0.030 | -0.111  1.313  0.012
## engage    |  0.875 36.198  0.765 |  0.092  0.869  0.008 | -0.205  4.500  0.042
##            
## daysabsnt |
## hwkhrswk  |
## schlenjoy |
## grades    |
## engage    |

##        eigenvalue percentage of variance cumulative percentage of variance
## comp 1  2.1145631              42.291262                          42.29126
## comp 2  0.9721076              19.442152                          61.73341
## comp 3  0.9351943              18.703887                          80.43730
## comp 4  0.7169327              14.338654                          94.77595
## comp 5  0.2612023               5.224045                         100.00000

## $coord
##                Dim.1       Dim.2       Dim.3       Dim.4       Dim.5
## daysabsnt  0.3508348 -0.07779619  0.93055380 -0.06166391  0.03361407
## hwkhrswk  -0.2425999  0.96187562  0.11664665  0.03551630  0.03275247
## schlenjoy  0.6560264  0.04755240 -0.03592329  0.75151121 -0.03617294
## grades     0.8583840  0.17360181 -0.11081292 -0.31256206 -0.35080591
## engage     0.8748883  0.09193390 -0.20515247 -0.22227164  0.36691465
## 
## $cor
##                Dim.1       Dim.2       Dim.3       Dim.4       Dim.5
## daysabsnt  0.3508348 -0.07779619  0.93055380 -0.06166391  0.03361407
## hwkhrswk  -0.2425999  0.96187562  0.11664665  0.03551630  0.03275247
## schlenjoy  0.6560264  0.04755240 -0.03592329  0.75151121 -0.03617294
## grades     0.8583840  0.17360181 -0.11081292 -0.31256206 -0.35080591
## engage     0.8748883  0.09193390 -0.20515247 -0.22227164  0.36691465
## 
## $cos2
##                Dim.1       Dim.2       Dim.3       Dim.4       Dim.5
## daysabsnt 0.12308504 0.006052247 0.865930374 0.003802438 0.001129906
## hwkhrswk  0.05885471 0.925204715 0.013606440 0.001261407 0.001072724
## schlenjoy 0.43037070 0.002261231 0.001290483 0.564769104 0.001308482
## grades    0.73682308 0.030137588 0.012279503 0.097695044 0.123064789
## engage    0.76542958 0.008451842 0.042087535 0.049404683 0.134626363
## 
## $contrib
##               Dim.1      Dim.2      Dim.3      Dim.4      Dim.5
## daysabsnt  5.820826  0.6225902 92.5936291  0.5303759  0.4325789
## hwkhrswk   2.783304 95.1751322  1.4549318  0.1759450  0.4106872
## schlenjoy 20.352701  0.2326111  0.1379909 78.7757515  0.5009457
## grades    34.845168  3.1002317  1.3130429 13.6268087 47.1147483
## engage    36.198001  0.8694348  4.5004053  6.8911189 51.5410399

## $quanti
##           correlation       p.value
## engage      0.8748883  0.000000e+00
## grades      0.8583840 5.747323e-300
## schlenjoy   0.6560264 1.189123e-127
## daysabsnt   0.3508348  3.602935e-31
## hwkhrswk   -0.2425999  3.001607e-15
## 
## attr(,"class")
## [1] "condes" "list"

Results

Below are the screeplot and radial plots displaying the eigenvalues configured for this PCA.

## Warning: package 'factoextra' was built under R version 4.1.3

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