Principal Components Analysis - No Rotation
# Pricipal Components Analysis entering raw data and extracting PCs from the correlation matrix
fit <- princomp(Jogglew01Neupsy1, cor=TRUE)
summary(fit) # print variance accounted for
loadings(fit) # pc loadings
biplot(fit,col = c("light blue","blue"))
PCA Variable Factor Map
factormap = PCA(Jogglew01Neupsy1)
Scree Plot
ev <- eigen(cor(Jogglew01Neupsy1)) # get eigenvalues
ap <- parallel(subject=nrow(Jogglew01Neupsy1),var=ncol(Jogglew01Neupsy1), rep=100,cent=.05)
nS <- nScree(x=ev$values, aparallel=ap$eigen$qevpea)
plotnScree(nS)
Eigen Values
ev <- eigen(cor(Jogglew01Neupsy1))
ev$values
## [1] 5.2504 1.6261 1.4314 1.1357 1.0998 1.0179 0.9949 0.8849 0.8265 0.7828
## [11] 0.7115 0.6174 0.6023 0.5345 0.4991 0.4677 0.4485 0.4162 0.3641 0.2884
Varimax Rotated Principal Components Analysis retaining 5 components
fit <- principal(Jogglew01Neupsy1, nfactors=5, rotate="varimax")
fit
## Principal Components Analysis
## Call: principal(r = Jogglew01Neupsy1, nfactors = 5, rotate = "varimax")
## Standardized loadings (pattern matrix) based upon correlation matrix
## RC1 RC3 RC2 RC4 RC5 h2 u2
## BARTaccuracy1 0.09 0.17 0.41 0.31 0.22 0.35 0.65
## DSSTefficiency1 0.57 0.16 0.55 0.05 0.05 0.66 0.34
## LOTefficiency1 0.61 0.18 0.32 0.18 0.02 0.54 0.46
## PVTefficiency1 0.64 0.10 -0.02 0.08 0.05 0.43 0.57
## AMefficiency1 -0.05 0.28 0.73 -0.17 -0.09 0.65 0.35
## NBACKaccuracy1 -0.02 -0.02 0.01 -0.09 0.87 0.77 0.23
## VOLTefficiency1 0.24 -0.14 0.71 0.02 0.07 0.58 0.42
## MPTspeed1 0.56 -0.10 0.40 -0.02 -0.16 0.51 0.49
## BVRtot1 -0.72 -0.19 -0.12 -0.14 0.10 0.60 0.40
## CrdRot1 0.68 0.28 0.10 0.04 0.12 0.56 0.44
## IdentPicCor1 0.55 0.19 0.30 -0.07 0.29 0.52 0.48
## IdentPicErr1 -0.07 -0.04 0.03 -0.79 0.08 0.63 0.37
## CVLtca1 0.35 0.27 0.22 0.27 0.22 0.37 0.63
## DigitSpanFwd1 0.05 0.63 0.12 0.22 0.02 0.46 0.54
## DigitSpanBck1 0.16 0.79 0.00 0.13 0.04 0.68 0.32
## FluencyWord1 0.39 -0.03 0.02 0.35 0.38 0.42 0.58
## TrailsAtestSec1 -0.56 -0.22 -0.07 0.35 -0.09 0.50 0.50
## TrailsBminusA1 -0.29 -0.43 -0.07 0.12 -0.06 0.29 0.71
## Attention1 0.24 0.68 0.01 -0.16 -0.13 0.57 0.43
## ClockTotal1 0.49 0.15 -0.27 0.33 -0.07 0.45 0.55
##
## RC1 RC3 RC2 RC4 RC5
## SS loadings 3.78 2.19 2.02 1.36 1.20
## Proportion Var 0.19 0.11 0.10 0.07 0.06
## Cumulative Var 0.19 0.30 0.40 0.47 0.53
## Proportion Explained 0.36 0.21 0.19 0.13 0.11
## Cumulative Proportion 0.36 0.57 0.76 0.89 1.00
##
## Test of the hypothesis that 5 components are sufficient.
##
## The degrees of freedom for the null model are 190 and the objective function was 5.16
## The degrees of freedom for the model are 100 and the objective function was 1.58
## The total number of observations was 229 with MLE Chi Square = 343.1 with prob < 2.2e-28
##
## Fit based upon off diagonal values = 0.91
Promax Rotated Principal Components Analysis retaining 5 components
fit <- principal(Jogglew01Neupsy1, nfactors=5, rotate="promax")
fit
## Principal Components Analysis
## Call: principal(r = Jogglew01Neupsy1, nfactors = 5, rotate = "promax")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC3 PC2 PC4 PC5 h2 u2
## BARTaccuracy1 -0.05 0.13 0.37 0.30 0.17 0.35 0.65
## DSSTefficiency1 0.54 0.00 0.46 0.01 -0.04 0.66 0.34
## LOTefficiency1 0.60 0.04 0.21 0.14 -0.07 0.54 0.46
## PVTefficiency1 0.70 -0.03 -0.15 0.01 -0.01 0.43 0.57
## AMefficiency1 -0.15 0.24 0.79 -0.14 -0.12 0.65 0.35
## NBACKaccuracy1 -0.15 -0.05 -0.09 -0.20 0.96 0.77 0.23
## VOLTefficiency1 0.20 -0.27 0.69 0.02 0.01 0.58 0.42
## MPTspeed1 0.63 -0.26 0.34 -0.04 -0.24 0.51 0.49
## BVRtot1 -0.77 -0.04 0.01 -0.09 0.19 0.60 0.40
## CrdRot1 0.69 0.13 -0.04 -0.04 0.06 0.56 0.44
## IdentPicCor1 0.53 0.04 0.17 -0.16 0.26 0.52 0.48
## IdentPicErr1 0.01 -0.06 0.07 -0.82 0.18 0.63 0.37
## CVLtca1 0.26 0.19 0.12 0.22 0.17 0.37 0.63
## DigitSpanFwd1 -0.10 0.65 0.10 0.21 -0.01 0.46 0.54
## DigitSpanBck1 0.02 0.80 -0.05 0.10 0.02 0.68 0.32
## FluencyWord1 0.35 -0.12 -0.11 0.28 0.33 0.42 0.58
## TrailsAtestSec1 -0.63 -0.08 0.03 0.44 -0.09 0.50 0.50
## TrailsBminusA1 -0.25 -0.38 -0.02 0.17 -0.05 0.29 0.71
## Attention1 0.19 0.67 -0.02 -0.19 -0.14 0.57 0.43
## ClockTotal1 0.54 0.09 -0.38 0.29 -0.14 0.45 0.55
##
## PC1 PC3 PC2 PC4 PC5
## SS loadings 4.09 2.02 1.92 1.33 1.18
## Proportion Var 0.20 0.10 0.10 0.07 0.06
## Cumulative Var 0.20 0.31 0.40 0.47 0.53
## Proportion Explained 0.39 0.19 0.18 0.13 0.11
## Cumulative Proportion 0.39 0.58 0.76 0.89 1.00
##
## With component correlations of
## PC1 PC3 PC2 PC4 PC5
## PC1 1.00 0.38 0.33 0.25 0.31
## PC3 0.38 1.00 0.17 0.07 0.13
## PC2 0.33 0.17 1.00 0.09 0.24
## PC4 0.25 0.07 0.09 1.00 0.28
## PC5 0.31 0.13 0.24 0.28 1.00
##
## Test of the hypothesis that 5 components are sufficient.
##
## The degrees of freedom for the null model are 190 and the objective function was 5.16
## The degrees of freedom for the model are 100 and the objective function was 1.58
## The total number of observations was 229 with MLE Chi Square = 343.1 with prob < 2.2e-28
##
## Fit based upon off diagonal values = 0.91
Maximum Likelihood Factor Analysis:
Enter raw data and extract 5 factors with varimax rotation
fit <- factanal(Jogglew01Neupsy1, 5, rotation="varimax")
print(fit, digits=2, cutoff=.3, sort=TRUE)
##
## Call:
## factanal(x = Jogglew01Neupsy1, factors = 5, rotation = "varimax")
##
## Uniquenesses:
## BARTaccuracy1 DSSTefficiency1 LOTefficiency1 PVTefficiency1
## 0.88 0.38 0.48 0.68
## AMefficiency1 NBACKaccuracy1 VOLTefficiency1 MPTspeed1
## 0.62 0.97 0.58 0.65
## BVRtot1 CrdRot1 IdentPicCor1 IdentPicErr1
## 0.46 0.46 0.00 0.85
## CVLtca1 DigitSpanFwd1 DigitSpanBck1 FluencyWord1
## 0.74 0.73 0.19 0.71
## TrailsAtestSec1 TrailsBminusA1 Attention1 ClockTotal1
## 0.71 0.82 0.61 0.71
##
## Loadings:
## Factor1 Factor2 Factor3 Factor4 Factor5
## LOTefficiency1 0.57 0.37
## BVRtot1 -0.66
## CrdRot1 0.63
## ClockTotal1 0.50
## DSSTefficiency1 0.37 0.60
## AMefficiency1 0.52
## VOLTefficiency1 0.63
## DigitSpanBck1 0.88
## IdentPicCor1 0.90
## BARTaccuracy1
## PVTefficiency1 0.47
## NBACKaccuracy1
## MPTspeed1 0.31 0.48
## IdentPicErr1 -0.34
## CVLtca1 0.30
## DigitSpanFwd1 0.48
## FluencyWord1 0.44
## TrailsAtestSec1 -0.34 -0.35
## TrailsBminusA1
## Attention1 0.32 0.43
##
## Factor1 Factor2 Factor3 Factor4 Factor5
## SS loadings 2.49 1.79 1.56 1.22 0.70
## Proportion Var 0.12 0.09 0.08 0.06 0.04
## Cumulative Var 0.12 0.21 0.29 0.35 0.39
##
## Test of the hypothesis that 5 factors are sufficient.
## The chi square statistic is 116.9 on 100 degrees of freedom.
## The p-value is 0.119
Maximum Likelihood Factor Analysis:
Enter raw data and extract 5 factors with promax rotation
fit <- factanal(Jogglew01Neupsy1, 5, rotation="promax")
print(fit, digits=2, cutoff=.3, sort=TRUE)
##
## Call:
## factanal(x = Jogglew01Neupsy1, factors = 5, rotation = "promax")
##
## Uniquenesses:
## BARTaccuracy1 DSSTefficiency1 LOTefficiency1 PVTefficiency1
## 0.88 0.38 0.48 0.68
## AMefficiency1 NBACKaccuracy1 VOLTefficiency1 MPTspeed1
## 0.62 0.97 0.58 0.65
## BVRtot1 CrdRot1 IdentPicCor1 IdentPicErr1
## 0.46 0.46 0.00 0.85
## CVLtca1 DigitSpanFwd1 DigitSpanBck1 FluencyWord1
## 0.74 0.73 0.19 0.71
## TrailsAtestSec1 TrailsBminusA1 Attention1 ClockTotal1
## 0.71 0.82 0.61 0.71
##
## Loadings:
## Factor1 Factor2 Factor3 Factor4 Factor5
## LOTefficiency1 0.61
## BVRtot1 -0.75
## CrdRot1 0.71
## ClockTotal1 0.64
## DSSTefficiency1 0.55
## AMefficiency1 0.58
## VOLTefficiency1 0.71
## DigitSpanFwd1 0.55
## DigitSpanBck1 1.01
## IdentPicCor1 0.98
## BARTaccuracy1
## PVTefficiency1 0.48
## NBACKaccuracy1
## MPTspeed1 0.48
## IdentPicErr1 -0.35
## CVLtca1
## FluencyWord1 0.45
## TrailsAtestSec1 -0.34
## TrailsBminusA1 -0.31
## Attention1 0.35 0.30 -0.33
##
## Factor1 Factor2 Factor3 Factor4 Factor5
## SS loadings 2.70 1.63 1.58 1.16 0.68
## Proportion Var 0.13 0.08 0.08 0.06 0.03
## Cumulative Var 0.13 0.22 0.29 0.35 0.39
##
## Factor Correlations:
## Factor1 Factor2 Factor3 Factor4 Factor5
## Factor1 1.00 0.39 0.473 -0.47 0.145
## Factor2 0.39 1.00 0.347 -0.60 0.122
## Factor3 0.47 0.35 1.000 -0.53 -0.032
## Factor4 -0.47 -0.60 -0.529 1.00 0.182
## Factor5 0.15 0.12 -0.032 0.18 1.000
##
## Test of the hypothesis that 5 factors are sufficient.
## The chi square statistic is 116.9 on 100 degrees of freedom.
## The p-value is 0.119
Correlations
## BARTaccuracy1 DSSTefficiency1 LOTefficiency1
## BARTaccuracy1 1.00 0.26*** 0.31***
## DSSTefficiency1 0.26*** 1.00 0.54***
## LOTefficiency1 0.31*** 0.54*** 1.00
## PVTefficiency1 0.08 0.36*** 0.35***
## AMefficiency1 0.16* 0.33*** 0.22***
## NBACKaccuracy1 0.12 0.05 0.02
## VOLTefficiency1 0.17* 0.41*** 0.25***
## MPTspeed1 0.15* 0.47*** 0.34***
## BVRtot1 -0.22*** -0.47*** -0.52***
## CrdRot1 0.24*** 0.42*** 0.57***
## IdentPicCor1 0.20** 0.54*** 0.40***
## IdentPicErr1 -0.13 -0.11 -0.10
## CVLtca1 0.16* 0.37*** 0.27***
## DigitSpanFwd1 0.09 0.17** 0.22***
## DigitSpanBck1 0.14* 0.27*** 0.25***
## FluencyWord1 0.09 0.24*** 0.28***
## TrailsAtestSec1 -0.12 -0.37*** -0.28***
## TrailsBminusA1 -0.09 -0.22*** -0.21**
## Attention1 0.14* 0.26*** 0.22***
## ClockTotal1 0.08 0.13 0.26***
## PVTefficiency1 AMefficiency1 NBACKaccuracy1
## BARTaccuracy1 0.08 0.16* 0.12
## DSSTefficiency1 0.36*** 0.33*** 0.05
## LOTefficiency1 0.35*** 0.22*** 0.02
## PVTefficiency1 1.00 0.04 0.08
## AMefficiency1 0.04 1.00 -0.04
## NBACKaccuracy1 0.08 -0.04 1.00
## VOLTefficiency1 0.16* 0.34*** 0.11
## MPTspeed1 0.35*** 0.17* -0.02
## BVRtot1 -0.41*** -0.09 0.05
## CrdRot1 0.37*** 0.14* 0.03
## IdentPicCor1 0.26*** 0.24*** 0.10
## IdentPicErr1 -0.09 0.15* 0.07
## CVLtca1 0.23*** 0.16* 0.08
## DigitSpanFwd1 0.19** 0.14* 0.02
## DigitSpanBck1 0.24*** 0.15* -0.02
## FluencyWord1 0.24*** 0.02 0.07
## TrailsAtestSec1 -0.24*** -0.07 -0.03
## TrailsBminusA1 -0.25*** -0.15* -0.09
## Attention1 0.16* 0.19** -0.07
## ClockTotal1 0.28*** -0.07 0.00
## VOLTefficiency1 MPTspeed1 BVRtot1 CrdRot1 IdentPicCor1
## BARTaccuracy1 0.17* 0.15* -0.22*** 0.24*** 0.20**
## DSSTefficiency1 0.41*** 0.47*** -0.47*** 0.42*** 0.54***
## LOTefficiency1 0.25*** 0.34*** -0.52*** 0.57*** 0.40***
## PVTefficiency1 0.16* 0.35*** -0.41*** 0.37*** 0.26***
## AMefficiency1 0.34*** 0.17* -0.09 0.14* 0.24***
## NBACKaccuracy1 0.11 -0.02 0.05 0.03 0.10
## VOLTefficiency1 1.00 0.36*** -0.22*** 0.21** 0.25***
## MPTspeed1 0.36*** 1.00 -0.35*** 0.23*** 0.26***
## BVRtot1 -0.22*** -0.35*** 1.00 -0.50*** -0.39***
## CrdRot1 0.21** 0.23*** -0.50*** 1.00 0.48***
## IdentPicCor1 0.25*** 0.26*** -0.39*** 0.48*** 1.00
## IdentPicErr1 -0.01 -0.02 0.14* -0.06 -0.06
## CVLtca1 0.22*** 0.22*** -0.34*** 0.33*** 0.29***
## DigitSpanFwd1 0.09 0.14* -0.18** 0.19** 0.17*
## DigitSpanBck1 -0.04 0.09 -0.27*** 0.31*** 0.24***
## FluencyWord1 0.09 0.17* -0.22*** 0.26*** 0.35***
## TrailsAtestSec1 -0.14* -0.30*** 0.33*** -0.37*** -0.45***
## TrailsBminusA1 -0.13* -0.18** 0.31*** -0.29*** -0.21**
## Attention1 -0.00 0.09 -0.27*** 0.28*** 0.22***
## ClockTotal1 0.02 0.15* -0.34*** 0.35*** 0.13
## IdentPicErr1 CVLtca1 DigitSpanFwd1 DigitSpanBck1
## BARTaccuracy1 -0.13 0.16* 0.09 0.14*
## DSSTefficiency1 -0.11 0.37*** 0.17** 0.27***
## LOTefficiency1 -0.10 0.27*** 0.22*** 0.25***
## PVTefficiency1 -0.09 0.23*** 0.19** 0.24***
## AMefficiency1 0.15* 0.16* 0.14* 0.15*
## NBACKaccuracy1 0.07 0.08 0.02 -0.02
## VOLTefficiency1 -0.01 0.22*** 0.09 -0.04
## MPTspeed1 -0.02 0.22*** 0.14* 0.09
## BVRtot1 0.14* -0.34*** -0.18** -0.27***
## CrdRot1 -0.06 0.33*** 0.19** 0.31***
## IdentPicCor1 -0.06 0.29*** 0.17* 0.24***
## IdentPicErr1 1.00 -0.14* -0.12 -0.10
## CVLtca1 -0.14* 1.00 0.17** 0.28***
## DigitSpanFwd1 -0.12 0.17** 1.00 0.46***
## DigitSpanBck1 -0.10 0.28*** 0.46*** 1.00
## FluencyWord1 -0.14* 0.30*** 0.09 0.16*
## TrailsAtestSec1 -0.04 -0.21** -0.17* -0.22***
## TrailsBminusA1 0.05 -0.17* -0.19** -0.26***
## Attention1 0.01 0.24*** 0.20** 0.42***
## ClockTotal1 -0.17* 0.22** 0.15* 0.17*
## FluencyWord1 TrailsAtestSec1 TrailsBminusA1 Attention1
## BARTaccuracy1 0.09 -0.12 -0.09 0.14*
## DSSTefficiency1 0.24*** -0.37*** -0.22*** 0.26***
## LOTefficiency1 0.28*** -0.28*** -0.21** 0.22***
## PVTefficiency1 0.24*** -0.24*** -0.25*** 0.16*
## AMefficiency1 0.02 -0.07 -0.15* 0.19**
## NBACKaccuracy1 0.07 -0.03 -0.09 -0.07
## VOLTefficiency1 0.09 -0.14* -0.13* -0.00
## MPTspeed1 0.17* -0.30*** -0.18** 0.09
## BVRtot1 -0.22*** 0.33*** 0.31*** -0.27***
## CrdRot1 0.26*** -0.37*** -0.29*** 0.28***
## IdentPicCor1 0.35*** -0.45*** -0.21** 0.22***
## IdentPicErr1 -0.14* -0.04 0.05 0.01
## CVLtca1 0.30*** -0.21** -0.17* 0.24***
## DigitSpanFwd1 0.09 -0.17* -0.19** 0.20**
## DigitSpanBck1 0.16* -0.22*** -0.26*** 0.42***
## FluencyWord1 1.00 -0.11 -0.05 0.01
## TrailsAtestSec1 -0.11 1.00 0.17* -0.28***
## TrailsBminusA1 -0.05 0.17* 1.00 -0.28***
## Attention1 0.01 -0.28*** -0.28*** 1.00
## ClockTotal1 0.16* -0.09 -0.12 0.21**
## ClockTotal1
## BARTaccuracy1 0.08
## DSSTefficiency1 0.13
## LOTefficiency1 0.26***
## PVTefficiency1 0.28***
## AMefficiency1 -0.07
## NBACKaccuracy1 0.00
## VOLTefficiency1 0.02
## MPTspeed1 0.15*
## BVRtot1 -0.34***
## CrdRot1 0.35***
## IdentPicCor1 0.13
## IdentPicErr1 -0.17*
## CVLtca1 0.22**
## DigitSpanFwd1 0.15*
## DigitSpanBck1 0.17*
## FluencyWord1 0.16*
## TrailsAtestSec1 -0.09
## TrailsBminusA1 -0.12
## Attention1 0.21**
## ClockTotal1 1.00
##
## n = 229