STAT 360: Computational Statistics and Data Analysis
Load R Libraries, Import and Attach Relevant Data, and Specify Seed
library(rmarkdown); library(knitr); library(readxl)
set.seed(37)EXERCISE 01
Part (a)
DCF <- matrix(c(1.00, -.72, -.09, -.38, -.72, 1.00, 0.23, 0.49, -.09, 0.23, 1.0, -.46, -.38, 0.49, -.46, 1.00), nrow = 4, ncol = 4)
rownames(DCF)<- c("DBH", "TYPE", "SLOPE", "ASPECT")
colnames(DCF)<- c("DBH", "TYPE", "SLOPE", "ASPECT")
DCF## DBH TYPE SLOPE ASPECT
## DBH 1.00 -0.72 -0.09 -0.38
## TYPE -0.72 1.00 0.23 0.49
## SLOPE -0.09 0.23 1.00 -0.46
## ASPECT -0.38 0.49 -0.46 1.00Part (b)
eigen(DCF)$values## [1] 2.0746888 1.3654830 0.3926504 0.1671779library(psych)## Warning: package 'psych' was built under R version 4.0.5Arot <- pca(DCF, 2, rotate = "varimax")$loadings[]
Arot## RC1 RC2
## DBH -0.8854395 0.02236984
## TYPE 0.9433740 0.02898144
## SLOPE 0.2220135 0.92760753
## ASPECT 0.5426109 -0.74879998Part (c)
Yes, this rotated matrix has complexity. The Aspect dimension loads significantly onto both factors. Since there is only one that is complex, we want to add it as a measured variable.Part (d)
library(knitr)
include_graphics("C:/Users/Sarah Chock/OneDrive - University of St. Thomas/Senior Year/STAT 360 Comp Stat and Data Analysis/Structural Equation Models/PS12 Q1.png")
EXERCISE 02
Part (a)
library(readxl)
DistressData <- read_excel("C:/Users/Sarah Chock/OneDrive - University of St. Thomas/Senior Year/STAT 360 Comp Stat and Data Analysis/Exploratory Data Analysis/DistressData.xlsx")
dd <- as.matrix(DistressData)
dd[which(dd==3)] = 6
dd[which(dd==5)] = 3
dd[which(dd==6)] = 5
head(DistressData)## # A tibble: 6 x 10
## Hopelessness Overwhelmed Exhausted VeryLonely VerySad Depressed Anxiety
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 5 3 3 5 5 5 4
## 2 4 4 4 5 5 2 3
## 3 1 4 3 4 2 2 2
## 4 5 3 4 3 4 2 3
## 5 5 5 5 5 5 1 1
## 6 3 4 4 3 3 4 3
## # ... with 3 more variables: SelfHarm <dbl>, SuicidalThoughts <dbl>,
## # SuicidalAttempts <dbl>Part (b)
R <- cor(dd)
eigen(R)$values## [1] 4.8390360 1.6895478 0.8790759 0.4963882 0.4727502 0.4058604 0.3471259
## [8] 0.3220688 0.2873694 0.2607774#The intrinsic dimensionality using Kaiser's criterion is 2
library(psych)
A <- pca(R, 2, rotate = "varimax")$loadings[]
A## RC1 RC2
## Hopelessness 0.736782679 0.35542955
## Overwhelmed 0.750335749 -0.10891644
## Exhausted 0.766728049 -0.04801078
## VeryLonely 0.763674783 0.25509207
## VerySad 0.801576138 0.26866823
## Depressed 0.691210438 0.47498740
## Anxiety 0.742612047 0.22977641
## SelfHarm 0.162446158 0.78686905
## SuicidalThoughts 0.232894533 0.82454863
## SuicidalAttempts 0.005108308 0.80251514Part (c)
library(GPArotation)
Aoblique <- pca(R, 2, rotate = "oblimin")$loadings[]
Aoblique## TC1 TC2
## Hopelessness 0.72687711 0.20895915
## Overwhelmed 0.80001663 -0.27378107
## Exhausted 0.80946160 -0.21436714
## VeryLonely 0.76779000 0.09949767
## VerySad 0.80577935 0.10538218
## Depressed 0.66395180 0.34229585
## Anxiety 0.74893402 0.07786394
## SelfHarm 0.07033278 0.77819767
## SuicidalThoughts 0.13936505 0.80199550
## SuicidalAttempts -0.09650970 0.82817441Part (d)
Z <- scale(dd)
B <- solve(R) %*% Aoblique
factorscores <- Z %*% B
phi <- cor(factorscores)
phi## TC1 TC2
## TC1 1.0000000 -0.3198569
## TC2 -0.3198569 1.0000000Part (e)
Yes, oblique rotation was necessary. Our factors are correlated with each other, with correlations of -.32, meaning that they are not independent since they are below the -.3 threshold. If they were independent, orthogonal rotation would be better. In our case, oblique rotation was the correct choice.Part (f)
include_graphics("C:/Users/Sarah Chock/OneDrive - University of St. Thomas/Senior Year/STAT 360 Comp Stat and Data Analysis/Structural Equation Models/PS12 Q2.png")