Initial Extraction of the Components - No Rotation
# Pricipal Components Analysis entering raw data and extracting PCs from the correlation matrix
JogglePCA <- princomp(JoggleScores, cor=TRUE)
Summary(Variance accounted for)
summary(JogglePCA)
## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
## Standard deviation 1.659 1.0599 0.9694 0.9403 0.86449 0.77378 0.76291
## Proportion of Variance 0.344 0.1404 0.1175 0.1105 0.09342 0.07484 0.07275
## Cumulative Proportion 0.344 0.4844 0.6019 0.7124 0.80580 0.88065 0.95340
## Comp.8
## Standard deviation 0.6106
## Proportion of Variance 0.0466
## Cumulative Proportion 1.0000
PC Loadings
#PC Loadings
loadings(JogglePCA)
##
## Loadings:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
## BARTaccuracy -0.197 0.901 0.356
## DSSTefficiency -0.497 -0.102 0.121 -0.329 0.777
## LOTefficiency -0.473 0.358 -0.536 -0.591
## PVTefficiency -0.180 -0.785 0.142 -0.131 -0.534 -0.119 -0.122
## AMefficiency -0.330 0.450 -0.202 0.407 0.112 -0.676 -0.107
## NBACKaccuracy -0.238 0.364 0.242 -0.816 -0.222 0.181
## VOLTefficiency -0.370 -0.219 0.132 -0.754 0.109 0.456
## MPTspeed -0.395 -0.168 0.629 0.220 0.589 -0.135
##
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
## Cumulative Var 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000
JogglePCAScores = JogglePCA$scores # the principal components
Biplot
biplot(JogglePCA,col=c("light blue","blue"))
Eigen Values
ev <- eigen(cor(JoggleScores))
ev
Scree Plot with Optimal Coordinates
ap <- parallel(subject=nrow(JoggleScores),var=ncol(JoggleScores), rep=100,cent=.05)
nS <- nScree(x=ev$values, aparallel=ap$eigen$qevpea)
plotnScree(nS)
Varimax Rotation to Final Solution
principal(JoggleScores,nfactors=2,rotate="varimax",scores=T)
## Principal Components Analysis
## Call: principal(r = JoggleScores, nfactors = 2, rotate = "varimax",
## scores = T)
## Standardized loadings (pattern matrix) based upon correlation matrix
## RC1 RC2 h2 u2
## BARTaccuracy 0.27 0.21 0.12 0.88
## DSSTefficiency 0.78 0.26 0.68 0.32
## LOTefficiency 0.71 0.34 0.62 0.38
## PVTefficiency 0.60 -0.65 0.78 0.22
## AMefficiency 0.32 0.65 0.53 0.47
## NBACKaccuracy 0.22 0.51 0.30 0.70
## VOLTefficiency 0.61 0.14 0.39 0.61
## MPTspeed 0.67 0.09 0.46 0.54
##
## RC1 RC2
## SS loadings 2.51 1.36
## Proportion Var 0.31 0.17
## Cumulative Var 0.31 0.48
## Proportion Explained 0.65 0.35
## Cumulative Proportion 0.65 1.00
##
## Test of the hypothesis that 2 components are sufficient.
##
## The degrees of freedom for the null model are 28 and the objective function was 1.39
## The degrees of freedom for the model are 13 and the objective function was 0.36
## The total number of observations was 1299 with MLE Chi Square = 463.5 with prob < 8.1e-91
##
## Fit based upon off diagonal values = 0.84
Promax Rotation to Final Solution
principal(r=JoggleScores,nfactors=2,rotate="promax",scores=T)
## Principal Components Analysis
## Call: principal(r = JoggleScores, nfactors = 2, rotate = "promax",
## scores = T)
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 h2 u2
## BARTaccuracy 0.26 0.14 0.12 0.88
## DSSTefficiency 0.81 0.06 0.68 0.32
## LOTefficiency 0.72 0.15 0.62 0.38
## PVTefficiency 0.68 -0.83 0.78 0.22
## AMefficiency 0.29 0.58 0.53 0.47
## NBACKaccuracy 0.19 0.46 0.30 0.70
## VOLTefficiency 0.63 -0.02 0.39 0.61
## MPTspeed 0.70 -0.10 0.46 0.54
##
## PC1 PC2
## SS loadings 2.65 1.23
## Proportion Var 0.33 0.15
## Cumulative Var 0.33 0.48
## Proportion Explained 0.68 0.32
## Cumulative Proportion 0.68 1.00
##
## With component correlations of
## PC1 PC2
## PC1 1.00 0.33
## PC2 0.33 1.00
##
## Test of the hypothesis that 2 components are sufficient.
##
## The degrees of freedom for the null model are 28 and the objective function was 1.39
## The degrees of freedom for the model are 13 and the objective function was 0.36
## The total number of observations was 1299 with MLE Chi Square = 463.5 with prob < 8.1e-91
##
## Fit based upon off diagonal values = 0.84
Correlations between Joggle Scores
zCor(JoggleScores)
## BARTaccuracy DSSTefficiency LOTefficiency PVTefficiency
## BARTaccuracy 1.00 0.17*** 0.18*** 0.07**
## DSSTefficiency 0.17*** 1.00 0.61*** 0.23***
## LOTefficiency 0.18*** 0.61*** 1.00 0.15***
## PVTefficiency 0.07** 0.23*** 0.15*** 1.00
## AMefficiency 0.14*** 0.36*** 0.33*** -0.09**
## NBACKaccuracy 0.11*** 0.25*** 0.26*** -0.02
## VOLTefficiency 0.11*** 0.39*** 0.39*** 0.17***
## MPTspeed 0.14*** 0.47*** 0.38*** 0.21***
## AMefficiency NBACKaccuracy VOLTefficiency MPTspeed
## BARTaccuracy 0.14*** 0.11*** 0.11*** 0.14***
## DSSTefficiency 0.36*** 0.25*** 0.39*** 0.47***
## LOTefficiency 0.33*** 0.26*** 0.39*** 0.38***
## PVTefficiency -0.09** -0.02 0.17*** 0.21***
## AMefficiency 1.00 0.14*** 0.26*** 0.26***
## NBACKaccuracy 0.14*** 1.00 0.13*** 0.15***
## VOLTefficiency 0.26*** 0.13*** 1.00 0.25***
## MPTspeed 0.26*** 0.15*** 0.25*** 1.00
##
## n = 1299