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# Remove most objects from the working environment
rm(list = ls())
options(stringsAsFactors = F)
# 14.2.1. Selecting the number of components to extract
library(psych)
fa.parallel(USJudgeRatings[,-1], fa="pc", n.iter=100,
show.legend=FALSE, main="Scree plot with parallel analysis") # figure 14.2
# 14.2.2. Extracting principal components
# code listing 14.1. Principal components analysis of US Judge Ratings
library(psych)
pc <- principal(USJudgeRatings[, -1], nfactors = 1)
pc
# Principal Components Analysis
# Call: principal(r = USJudgeRatings[, -1], nfactors = 1)
# Standardized loadings (pattern matrix) based upon correlation matrix
# PC1 h2 u2 com
# INTG 0.92 0.84 0.1565 1
# DMNR 0.91 0.83 0.1663 1
# DILG 0.97 0.94 0.0613 1
# CFMG 0.96 0.93 0.0720 1
# DECI 0.96 0.92 0.0763 1
# PREP 0.98 0.97 0.0299 1
# FAMI 0.98 0.95 0.0469 1
# ORAL 1.00 0.99 0.0091 1
# WRIT 0.99 0.98 0.0196 1
# PHYS 0.89 0.80 0.2013 1
# RTEN 0.99 0.97 0.0275 1
#
# PC1
# SS loadings 10.13
# Proportion Var 0.92
#
# Mean item complexity = 1
# Test of the hypothesis that 1 component is sufficient.
#
# The root mean square of the residuals (RMSR) is 0.04
# with the empirical chi square 6.21 with prob < 1
#
# Fit based upon off diagonal values = 1
library(psych)
fa.parallel(Harman23.cor$cov, n.obs=302, fa="pc", n.iter =100,
show.legend=FALSE, main="Scree plot with parallel analysis") # figure 14.3
# code listing 14.2. Principal components analysis of body measurements
library(psych)
PC <- principal(Harman23.cor$cov, nfactors = 2, rotate = "none")
PC
# Principal Components Analysis
# Call: principal(r = Harman23.cor$cov, nfactors = 2, rotate = "none")
# Standardized loadings (pattern matrix) based upon correlation matrix
# PC1 PC2 h2 u2 com
# height 0.86 -0.37 0.88 0.123 1.4
# arm.span 0.84 -0.44 0.90 0.097 1.5
# forearm 0.81 -0.46 0.87 0.128 1.6
# lower.leg 0.84 -0.40 0.86 0.139 1.4
# weight 0.76 0.52 0.85 0.150 1.8
# bitro.diameter 0.67 0.53 0.74 0.261 1.9
# chest.girth 0.62 0.58 0.72 0.283 2.0
# chest.width 0.67 0.42 0.62 0.375 1.7
#
# PC1 PC2
# SS loadings 4.67 1.77
# Proportion Var 0.58 0.22
# Cumulative Var 0.58 0.81
# Proportion Explained 0.73 0.27
# Cumulative Proportion 0.73 1.00
#
# Mean item complexity = 1.7
# Test of the hypothesis that 2 components are sufficient.
#
# The root mean square of the residuals (RMSR) is 0.05
#
# Fit based upon off diagonal values = 0.99
#======================================================
# 14.2.3. Rotating principal components
# code lisitng 14.3. Principal components analysis with varimax rotation
rc <- principal(Harman23.cor$cov, nfactors = 2, rotate = "varimax")
rc
# Principal Components Analysis
# Call: principal(r = Harman23.cor$cov, nfactors = 2, rotate = "varimax")
# Standardized loadings (pattern matrix) based upon correlation matrix
# RC1 RC2 h2 u2 com
# height 0.90 0.25 0.88 0.123 1.2
# arm.span 0.93 0.19 0.90 0.097 1.1
# forearm 0.92 0.16 0.87 0.128 1.1
# lower.leg 0.90 0.22 0.86 0.139 1.1
# weight 0.26 0.88 0.85 0.150 1.2
# bitro.diameter 0.19 0.84 0.74 0.261 1.1
# chest.girth 0.11 0.84 0.72 0.283 1.0
# chest.width 0.26 0.75 0.62 0.375 1.2
#
# RC1 RC2
# SS loadings 3.52 2.92
# Proportion Var 0.44 0.37
# Cumulative Var 0.44 0.81
# Proportion Explained 0.55 0.45
# Cumulative Proportion 0.55 1.00
#
# Mean item complexity = 1.1
# Test of the hypothesis that 2 components are sufficient.
#
# The root mean square of the residuals (RMSR) is 0.05
#
# Fit based upon off diagonal values = 0.99
#=======================================================
# 14.2.4. Obtaining principal components scores
# code listing 14.4. Obtaining component scores from raw data
library(psych)
pc <- principal(USJudgeRatings[,-1], nfactors=1, score=TRUE)
head(pc$scores)
# PC1
# AARONSON,L.H. -0.1857981
# ALEXANDER,J.M. 0.7469865
# ARMENTANO,A.J. 0.0704772
# BERDON,R.I. 1.1358765
# BRACKEN,J.J. -2.1586211
# BURNS,E.B. 0.7669406
cor(USJudgeRatings$CONT, pc$score)
# PC1
# [1,] -0.008815895
# code listing 14.5. Obtaining principal component scoring coefficients
library(psych)
rc <- principal(Harman23.cor$cov, nfactors = 2, rotate = "varimax")
round(unclass(rc$weights), 2)
# RC1 RC2
# height 0.28 -0.05
# arm.span 0.30 -0.08
# forearm 0.30 -0.09
# lower.leg 0.28 -0.06
# weight -0.06 0.33
# bitro.diameter -0.08 0.32
# chest.girth -0.10 0.34
# chest.width -0.04 0.27
PC1 = 0.28*height + 0.30*arm.span + 0.30*forearm + 0.28*lower.leg -
0.06*weight - 0.08*bitro.diameter - 0.10*chest.girth -
0.04*chest.width
PC2 = -0.05*height - 0.08*arm.span - 0.09*forearm - 0.06*lower.leg +
0.33*weight + 0.32*bitro.diameter + 0.34*chest.girth +
0.27*chest.width
# 14.3. Exploratory factor analysis
options(digits = 2)
covariances <- ability.cov$cov
correlations <- cov2cor(covariances)
correlations
# general picture blocks maze reading vocab
# general 1.00 0.47 0.55 0.34 0.58 0.51
# picture 0.47 1.00 0.57 0.19 0.26 0.24
# blocks 0.55 0.57 1.00 0.45 0.35 0.36
# maze 0.34 0.19 0.45 1.00 0.18 0.22
# reading 0.58 0.26 0.35 0.18 1.00 0.79
# vocab 0.51 0.24 0.36 0.22 0.79 1.00
#===================================================
# 14.3.1. Deciding how many common factors to extract
# figure 14.4
library(psych)
covatiances <- ability.cov$cov
correlations <- cov2cor(covariances)
fa.parallel(correlations, n.obs = 112, fa="both",
n.iter = 100, main = "Scree plots with parallel analysis")
# 14.3.2. Extracting common factors
# code listing 14.6. Principal axis factoring without rotation
fa <- fa(correlations, nfactors = 2, rotate = "none", fm="pa")
fa
# Factor Analysis using method = pa
# Call: fa(r = correlations, nfactors = 2, rotate = "none", fm = "pa")
# Standardized loadings (pattern matrix) based upon correlation matrix
# PA1 PA2 h2 u2 com
# general 0.75 0.07 0.57 0.432 1.0
# picture 0.52 0.32 0.38 0.623 1.7
# blocks 0.75 0.52 0.83 0.166 1.8
# maze 0.39 0.22 0.20 0.798 1.6
# reading 0.81 -0.51 0.91 0.089 1.7
# vocab 0.73 -0.39 0.69 0.313 1.5
#
# PA1 PA2
# SS loadings 2.75 0.83
# Proportion Var 0.46 0.14
# Cumulative Var 0.46 0.60
# Proportion Explained 0.77 0.23
# Cumulative Proportion 0.77 1.00
#
# Mean item complexity = 1.5
# Test of the hypothesis that 2 factors are sufficient.
#
# The degrees of freedom for the null model are 15 and the objective function was 2.5
# The degrees of freedom for the model are 4 and the objective function was 0.07
#
# The root mean square of the residuals (RMSR) is 0.03
# The df corrected root mean square of the residuals is 0.06
#
# Fit based upon off diagonal values = 0.99
# Measures of factor score adequacy
# PA1 PA2
# Correlation of (regression) scores with factors 0.96 0.92
# Multiple R square of scores with factors 0.93 0.84
# Minimum correlation of possible factor scores 0.86 0.68
#===========================================================
# 14.3.3. Rotating factors
# code listing 14.7. Factor extraction with orthogonal rotation
fa.varimax <- fa(correlations, nfactors = 2, rotate="varimax", fm="pa")
fa.varimax
# Factor Analysis using method = pa
# Call: fa(r = correlations, nfactors = 2, rotate = "varimax", fm = "pa")
# Standardized loadings (pattern matrix) based upon correlation matrix
# PA1 PA2 h2 u2 com
# general 0.49 0.57 0.57 0.432 2.0
# picture 0.16 0.59 0.38 0.623 1.1
# blocks 0.18 0.89 0.83 0.166 1.1
# maze 0.13 0.43 0.20 0.798 1.2
# reading 0.93 0.20 0.91 0.089 1.1
# vocab 0.80 0.23 0.69 0.313 1.2
#
# PA1 PA2
# SS loadings 1.83 1.75
# Proportion Var 0.30 0.29
# Cumulative Var 0.30 0.60
# Proportion Explained 0.51 0.49
# Cumulative Proportion 0.51 1.00
#
# Mean item complexity = 1.3
# Test of the hypothesis that 2 factors are sufficient.
#
# The degrees of freedom for the null model are 15 and the objective function was 2.5
# The degrees of freedom for the model are 4 and the objective function was 0.07
#
# The root mean square of the residuals (RMSR) is 0.03
# The df corrected root mean square of the residuals is 0.06
#
# Fit based upon off diagonal values = 0.99
# Measures of factor score adequacy
# PA1 PA2
# Correlation of (regression) scores with factors 0.96 0.92
# Multiple R square of scores with factors 0.91 0.85
# Minimum correlation of possible factor scores 0.82 0.71
#============================================================
# code listing 14.8. Factor extraction with oblique rotation
# install.packages("GPArotation")
fa.promax <- fa(correlations, nfactors = 2, rotate="promax", fm="pa")
fa.promax
# Factor Analysis using method = pa
# Call: fa(r = correlations, nfactors = 2, rotate = "promax", fm = "pa")
# Standardized loadings (pattern matrix) based upon correlation matrix
# PA1 PA2 h2 u2 com
# general 0.37 0.48 0.57 0.432 1.9
# picture -0.03 0.63 0.38 0.623 1.0
# blocks -0.10 0.97 0.83 0.166 1.0
# maze 0.00 0.45 0.20 0.798 1.0
# reading 1.00 -0.09 0.91 0.089 1.0
# vocab 0.84 -0.01 0.69 0.313 1.0
#
# PA1 PA2
# SS loadings 1.83 1.75
# Proportion Var 0.30 0.29
# Cumulative Var 0.30 0.60
# Proportion Explained 0.51 0.49
# Cumulative Proportion 0.51 1.00
#
# With factor correlations of
# PA1 PA2
# PA1 1.00 0.55
# PA2 0.55 1.00
#
# Mean item complexity = 1.2
# Test of the hypothesis that 2 factors are sufficient.
#
# The degrees of freedom for the null model are 15 and the objective function was 2.5
# The degrees of freedom for the model are 4 and the objective function was 0.07
#
# The root mean square of the residuals (RMSR) is 0.03
# The df corrected root mean square of the residuals is 0.06
#
# Fit based upon off diagonal values = 0.99
# Measures of factor score adequacy
# PA1 PA2
# Correlation of (regression) scores with factors 0.97 0.94
# Multiple R square of scores with factors 0.93 0.88
# Minimum correlation of possible factor scores 0.86 0.77
fsm <- function(oblique) {
if (class(oblique)[2] == "fa" & is.null(oblique$Phi)) {
warning("Object doesn't look like oblique EFA")
} else {
P <- unclass(oblique$loading)
F <- P %*% oblique$Phi
colnames(F) <- c("PA1", "PA2")
return(F)
}
}
fsm(fa.promax)
# PA1 PA2
# general 0.64 0.69
# picture 0.32 0.61
# blocks 0.43 0.91
# maze 0.25 0.45
# reading 0.95 0.46
# vocab 0.83 0.45
factor.plot(fa.promax, labels=rownames(fa.promax$loadings)) # figure 14.5
fa.diagram(fa.promax, simple = F)
# 14.3.4. Factor scores
fa.promax$weights
# PA1 PA2
# general 0.078 0.211
# picture 0.020 0.090
# blocks 0.037 0.702
# maze 0.027 0.035
# reading 0.743 0.030
# vocab 0.177 0.036`
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