Homework Assignement 7

Principal Components

You are to use the technique of Principal Components Analysis (PCA) to perform a variable reduction of at least 5 variables. If you have an idea for latent construct, state what you believe this is. Report the summary statistics and correlation matrix for your data. Report the results of the PCA, being sure to include the eigenvalues and corresponding vectors. Interpret your component(s) if possible. If deemed appropriate, conduct some testing of your index/components/latent variables.

## Use of data from IPUMS CPS is subject to conditions including that users should
## cite the data appropriately. Use command `ipums_conditions()` for more details.

## 
## 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

## Installing package into 'C:/Users/gomez/OneDrive/Documents/R/win-library/4.1'
## (as 'lib' is unspecified)

## package 'factoextra' successfully unpacked and MD5 sums checked
## 
## The downloaded binary packages are in
##  C:\Users\gomez\AppData\Local\Temp\RtmpoNJeET\downloaded_packages

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

## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

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

cps20.pca <- PCA(cps20[, c(1:8)], scale.unit = T, graph = F)

eigenvalues <- cps20.pca$eig

head(eigenvalues)

##        eigenvalue percentage of variance cumulative percentage of variance
## comp 1  1.9619707               24.52463                          24.52463
## comp 2  1.3969473               17.46184                          41.98648
## comp 3  1.0295559               12.86945                          54.85592
## comp 4  0.9997535               12.49692                          67.35284
## comp 5  0.9851715               12.31464                          79.66749
## comp 6  0.9281263               11.60158                          91.26906

When looking at CPS data for the voter turnout in 2020, the first two components are well over 1, while the third is just over 1 and the remaining three components are not quite at 1, but very close. This suggest there are three real components among these variables that account for 54.8% of the variance.

cps20.pca$var

## $coord
##                Dim.1       Dim.2       Dim.3        Dim.4       Dim.5
## sex      -0.02339637 -0.04654566  0.69046256  0.590575799 -0.07992099
## metro     0.03394063  0.19283709 -0.41478148  0.770532748 -0.11511364
## statefip -0.15678632  0.08976974 -0.03943124  0.143677603  0.97092053
## age       0.14712304  0.80086497  0.02511979 -0.139569393 -0.02300409
## voteres   0.09256366  0.80747174 -0.02702460  0.007143369 -0.05926513
## citizen   0.96340875 -0.11093364 -0.06107593  0.032336300  0.07379757
## nativity -0.96249416  0.12220044  0.05788486 -0.027830818 -0.07500640
## educ      0.22566513  0.17000445  0.60891425 -0.123527131  0.08793567
## 
## $cor
##                Dim.1       Dim.2       Dim.3        Dim.4       Dim.5
## sex      -0.02339637 -0.04654566  0.69046256  0.590575799 -0.07992099
## metro     0.03394063  0.19283709 -0.41478148  0.770532748 -0.11511364
## statefip -0.15678632  0.08976974 -0.03943124  0.143677603  0.97092053
## age       0.14712304  0.80086497  0.02511979 -0.139569393 -0.02300409
## voteres   0.09256366  0.80747174 -0.02702460  0.007143369 -0.05926513
## citizen   0.96340875 -0.11093364 -0.06107593  0.032336300  0.07379757
## nativity -0.96249416  0.12220044  0.05788486 -0.027830818 -0.07500640
## educ      0.22566513  0.17000445  0.60891425 -0.123527131  0.08793567
## 
## $cos2
##                 Dim.1       Dim.2        Dim.3        Dim.4        Dim.5
## sex      0.0005473903 0.002166498 0.4767385485 3.487798e-01 0.0063873654
## metro    0.0011519661 0.037186143 0.1720436721 5.937207e-01 0.0132511509
## statefip 0.0245819508 0.008058606 0.0015548228 2.064325e-02 0.9426866686
## age      0.0216451880 0.641384705 0.0006310037 1.947962e-02 0.0005291882
## voteres  0.0085680319 0.652010617 0.0007303289 5.102772e-05 0.0035123553
## citizen  0.9281564205 0.012306272 0.0037302698 1.045636e-03 0.0054460811
## nativity 0.9263950032 0.014932948 0.0033506573 7.745544e-04 0.0056259594
## educ     0.0509247507 0.028901513 0.3707765625 1.525895e-02 0.0077326818
## 
## $contrib
##                Dim.1      Dim.2       Dim.3       Dim.4       Dim.5
## sex       0.02790003  0.1550881 46.30526272 34.88657596  0.64835064
## metro     0.05871475  2.6619575 16.71047467 59.38670862  1.34506039
## statefip  1.25292140  0.5768726  0.15101879  2.06483428 95.68757478
## age       1.10323706 45.9133071  0.06128892  1.94844178  0.05371534
## voteres   0.43670540 46.6739594  0.07093631  0.00510403  0.35652224
## citizen  47.30735376  0.8809403  0.36231835  0.10458941  0.55280541
## nativity 47.21757580  1.0689701  0.32544687  0.07747454  0.57106400
## educ      2.59559180  2.0689050 36.01325338  1.52627138  0.78490721

Based on these laogin (eigen) vectors, the voter index would consist of state the voter lives in, age, citizenship (native vs. naturalized), nativity of the voter, education, and maybe voter residence. I would definitely off sex and metro from the index.