Principal Components Analysis - No Rotation
# Pricipal Components Analysis entering raw data and extracting PCs from the correlation matrix
fit <- princomp(Jogglew03Neupsy1, 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(Jogglew03Neupsy1)
Scree Plot
ev <- eigen(cor(Jogglew03Neupsy1)) # get eigenvalues
ap <- parallel(subject=nrow(Jogglew03Neupsy1),var=ncol(Jogglew03Neupsy1), rep=100,cent=.05)
nS <- nScree(x=ev$values, aparallel=ap$eigen$qevpea)
plotnScree(nS)
Eigen Values
ev <- eigen(cor(Jogglew03Neupsy1))
ev$values
## [1] 5.6572 1.6108 1.2676 1.2320 1.0525 1.0162 0.9343 0.8680 0.7982 0.7436
## [11] 0.7247 0.6223 0.5882 0.5274 0.5030 0.4778 0.4135 0.3450 0.3296 0.2879
Varimax Rotated Principal Components Analysis retaining 5 components
fit <- principal(Jogglew03Neupsy1, nfactors=5, rotate="varimax")
fit
## Principal Components Analysis
## Call: principal(r = Jogglew03Neupsy1, nfactors = 5, rotate = "varimax")
## Standardized loadings (pattern matrix) based upon correlation matrix
## RC1 RC2 RC3 RC4 RC5 h2 u2
## BARTaccuracy3 0.26 0.15 0.35 0.43 -0.39 0.55 0.45
## DSSTefficiency3 0.62 0.14 0.46 0.06 0.14 0.64 0.36
## LOTefficiency3 0.72 0.05 0.17 0.17 -0.02 0.58 0.42
## PVTefficiency3 0.59 0.13 -0.08 0.11 0.30 0.47 0.53
## AMefficiency3 0.05 0.17 0.73 -0.11 -0.02 0.58 0.42
## NBACKaccuracy3 -0.12 0.05 0.12 0.71 0.12 0.55 0.45
## VOLTefficiency3 0.24 -0.11 0.70 0.08 0.14 0.58 0.42
## MPTspeed3 0.60 -0.14 0.30 0.00 0.06 0.47 0.53
## BVRtot3 -0.72 -0.20 0.15 -0.08 -0.19 0.62 0.38
## StroopColors3 0.57 0.30 0.24 -0.26 0.19 0.58 0.42
## StroopWords3 0.45 0.48 0.06 -0.04 0.03 0.43 0.57
## StroopMixed3 0.42 0.44 0.24 -0.10 0.15 0.46 0.54
## CVLtca3 0.20 0.35 0.37 0.08 0.47 0.53 0.47
## DigitSpanFwd3 0.07 0.76 0.08 0.12 -0.05 0.60 0.40
## DigitSpanBck3 0.08 0.82 -0.09 0.07 0.22 0.75 0.25
## FluencyWord3 0.21 0.06 0.09 0.10 0.76 0.64 0.36
## TrailsAtestSec3 -0.57 -0.18 -0.28 -0.08 -0.05 0.44 0.56
## TrailsBminusA3 -0.39 -0.34 -0.25 -0.14 0.09 0.36 0.64
## Attention3 0.57 0.44 0.04 -0.13 -0.07 0.54 0.46
## ClockTotal3 0.23 0.03 -0.20 0.60 0.04 0.45 0.55
##
## RC1 RC2 RC3 RC4 RC5
## SS loadings 3.92 2.39 1.97 1.28 1.26
## Proportion Var 0.20 0.12 0.10 0.06 0.06
## Cumulative Var 0.20 0.32 0.41 0.48 0.54
## Proportion Explained 0.36 0.22 0.18 0.12 0.12
## Cumulative Proportion 0.36 0.58 0.77 0.88 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 5.7
## The degrees of freedom for the model are 100 and the objective function was 1.75
## The total number of observations was 229 with MLE Chi Square = 379 with prob < 4.5e-34
##
## Fit based upon off diagonal values = 0.92
Promax Rotated Principal Components Analysis retaining 5 components
fit <- principal(Jogglew03Neupsy1, nfactors=5, rotate="promax")
fit
## Principal Components Analysis
## Call: principal(r = Jogglew03Neupsy1, nfactors = 5, rotate = "promax")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 PC3 PC4 PC5 h2 u2
## BARTaccuracy3 0.20 0.07 0.27 0.39 -0.39 0.55 0.45
## DSSTefficiency3 0.58 -0.08 0.37 0.01 0.09 0.64 0.36
## LOTefficiency3 0.81 -0.17 0.02 0.10 -0.08 0.58 0.42
## PVTefficiency3 0.67 -0.01 -0.22 0.07 0.25 0.47 0.53
## AMefficiency3 -0.18 0.08 0.80 -0.10 -0.03 0.58 0.42
## NBACKaccuracy3 -0.23 0.14 0.11 0.75 0.18 0.55 0.45
## VOLTefficiency3 0.11 -0.26 0.74 0.09 0.15 0.58 0.42
## MPTspeed3 0.67 -0.37 0.22 -0.05 0.03 0.47 0.53
## BVRtot3 -0.84 -0.03 0.33 -0.02 -0.11 0.62 0.38
## StroopColors3 0.54 0.12 0.15 -0.30 0.10 0.58 0.42
## StroopWords3 0.40 0.39 -0.06 -0.08 -0.05 0.43 0.57
## StroopMixed3 0.32 0.34 0.16 -0.12 0.07 0.46 0.54
## CVLtca3 0.00 0.31 0.37 0.11 0.45 0.53 0.47
## DigitSpanFwd3 -0.14 0.83 0.02 0.12 -0.11 0.60 0.40
## DigitSpanBck3 -0.11 0.92 -0.16 0.08 0.16 0.75 0.25
## FluencyWord3 0.15 0.03 0.08 0.15 0.76 0.64 0.36
## TrailsAtestSec3 -0.56 0.00 -0.18 -0.04 0.01 0.44 0.56
## TrailsBminusA3 -0.33 -0.24 -0.16 -0.10 0.13 0.36 0.64
## Attention3 0.58 0.29 -0.10 -0.19 -0.17 0.54 0.46
## ClockTotal3 0.30 0.03 -0.31 0.58 0.06 0.45 0.55
##
## PC1 PC2 PC3 PC4 PC5
## SS loadings 4.18 2.22 1.98 1.25 1.19
## Proportion Var 0.21 0.11 0.10 0.06 0.06
## Cumulative Var 0.21 0.32 0.42 0.48 0.54
## Proportion Explained 0.39 0.21 0.18 0.12 0.11
## Cumulative Proportion 0.39 0.59 0.77 0.89 1.00
##
## With component correlations of
## PC1 PC2 PC3 PC4 PC5
## PC1 1.00 0.50 0.47 0.09 0.13
## PC2 0.50 1.00 0.31 -0.05 0.09
## PC3 0.47 0.31 1.00 0.05 -0.02
## PC4 0.09 -0.05 0.05 1.00 -0.15
## PC5 0.13 0.09 -0.02 -0.15 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 5.7
## The degrees of freedom for the model are 100 and the objective function was 1.75
## The total number of observations was 229 with MLE Chi Square = 379 with prob < 4.5e-34
##
## Fit based upon off diagonal values = 0.92
Maximum Likelihood Factor Analysis:
Enter raw data and extract 5 factors with varimax rotation
fit <- factanal(Jogglew03Neupsy1, 5, rotation="varimax")
print(fit, digits=2, cutoff=.3, sort=TRUE)
##
## Call:
## factanal(x = Jogglew03Neupsy1, factors = 5, rotation = "varimax")
##
## Uniquenesses:
## BARTaccuracy3 DSSTefficiency3 LOTefficiency3 PVTefficiency3
## 0.84 0.39 0.00 0.64
## AMefficiency3 NBACKaccuracy3 VOLTefficiency3 MPTspeed3
## 0.63 0.98 0.63 0.60
## BVRtot3 StroopColors3 StroopWords3 StroopMixed3
## 0.50 0.22 0.64 0.50
## CVLtca3 DigitSpanFwd3 DigitSpanBck3 FluencyWord3
## 0.67 0.65 0.04 0.87
## TrailsAtestSec3 TrailsBminusA3 Attention3 ClockTotal3
## 0.61 0.74 0.62 0.95
##
## Loadings:
## Factor1 Factor2 Factor3 Factor4 Factor5
## PVTefficiency3 0.55
## BVRtot3 -0.54 -0.42
## Attention3 0.52
## DSSTefficiency3 0.50 0.54
## AMefficiency3 0.59
## VOLTefficiency3 0.57
## DigitSpanFwd3 0.55
## DigitSpanBck3 0.95
## LOTefficiency3 0.34 0.89
## StroopColors3 0.52 0.31 0.63
## BARTaccuracy3 0.31
## NBACKaccuracy3
## MPTspeed3 0.50 0.33
## StroopWords3 0.48
## StroopMixed3 0.33 0.45
## CVLtca3 0.38 0.31 0.31
## FluencyWord3
## TrailsAtestSec3 -0.47 -0.35
## TrailsBminusA3 -0.35
## ClockTotal3
##
## Factor1 Factor2 Factor3 Factor4 Factor5
## SS loadings 2.76 1.81 1.66 1.29 0.77
## Proportion Var 0.14 0.09 0.08 0.06 0.04
## Cumulative Var 0.14 0.23 0.31 0.38 0.41
##
## Test of the hypothesis that 5 factors are sufficient.
## The chi square statistic is 109.7 on 100 degrees of freedom.
## The p-value is 0.238
Maximum Likelihood Factor Analysis:
Enter raw data and extract 5 factors with promax rotation
fit <- factanal(Jogglew03Neupsy1, 5, rotation="promax")
print(fit, digits=2, cutoff=.3, sort=TRUE)
##
## Call:
## factanal(x = Jogglew03Neupsy1, factors = 5, rotation = "promax")
##
## Uniquenesses:
## BARTaccuracy3 DSSTefficiency3 LOTefficiency3 PVTefficiency3
## 0.84 0.39 0.00 0.64
## AMefficiency3 NBACKaccuracy3 VOLTefficiency3 MPTspeed3
## 0.63 0.98 0.63 0.60
## BVRtot3 StroopColors3 StroopWords3 StroopMixed3
## 0.50 0.22 0.64 0.50
## CVLtca3 DigitSpanFwd3 DigitSpanBck3 FluencyWord3
## 0.67 0.65 0.04 0.87
## TrailsAtestSec3 TrailsBminusA3 Attention3 ClockTotal3
## 0.61 0.74 0.62 0.95
##
## Loadings:
## Factor1 Factor2 Factor3 Factor4 Factor5
## PVTefficiency3 0.67
## MPTspeed3 0.57
## BVRtot3 -0.62
## StroopWords3 0.56
## Attention3 0.57
## DigitSpanFwd3 0.57
## DigitSpanBck3 1.03
## AMefficiency3 0.67
## VOLTefficiency3 0.58
## LOTefficiency3 0.89
## StroopColors3 0.61 0.67
## BARTaccuracy3
## DSSTefficiency3 0.44 0.42
## NBACKaccuracy3
## StroopMixed3 0.44
## CVLtca3 0.32
## FluencyWord3 0.30
## TrailsAtestSec3 -0.46
## TrailsBminusA3
## ClockTotal3
##
## Factor1 Factor2 Factor3 Factor4 Factor5
## SS loadings 3.05 1.60 1.42 0.96 0.81
## Proportion Var 0.15 0.08 0.07 0.05 0.04
## Cumulative Var 0.15 0.23 0.30 0.35 0.39
##
## Factor Correlations:
## Factor1 Factor2 Factor3 Factor4 Factor5
## Factor1 1.000 -0.19 0.497 0.012 -0.262
## Factor2 -0.189 1.00 -0.477 -0.217 0.267
## Factor3 0.497 -0.48 1.000 -0.013 -0.551
## Factor4 0.012 -0.22 -0.013 1.000 -0.039
## Factor5 -0.262 0.27 -0.551 -0.039 1.000
##
## Test of the hypothesis that 5 factors are sufficient.
## The chi square statistic is 109.7 on 100 degrees of freedom.
## The p-value is 0.238
Correlations
zCor(Jogglew03Neupsy1)
## BARTaccuracy3 DSSTefficiency3 LOTefficiency3
## BARTaccuracy3 1.00 0.26*** 0.31***
## DSSTefficiency3 0.26*** 1.00 0.54***
## LOTefficiency3 0.31*** 0.54*** 1.00
## PVTefficiency3 0.08 0.36*** 0.35***
## AMefficiency3 0.16* 0.33*** 0.22***
## NBACKaccuracy3 0.12 0.05 0.02
## VOLTefficiency3 0.17* 0.41*** 0.25***
## MPTspeed3 0.16* 0.44*** 0.33***
## BVRtot3 -0.12 -0.43*** -0.53***
## StroopColors3 0.11 0.48*** 0.37***
## StroopWords3 0.18** 0.32*** 0.25***
## StroopMixed3 0.13* 0.40*** 0.38***
## CVLtca3 0.19** 0.40*** 0.20**
## DigitSpanFwd3 0.17* 0.24*** 0.18**
## DigitSpanBck3 0.04 0.18** 0.14*
## FluencyWord3 0.06 0.24*** 0.23***
## TrailsAtestSec3 -0.23*** -0.51*** -0.40***
## TrailsBminusA3 -0.26*** -0.32*** -0.30***
## Attention3 0.21** 0.39*** 0.33***
## ClockTotal3 0.11 0.08 0.17*
## PVTefficiency3 AMefficiency3 NBACKaccuracy3
## BARTaccuracy3 0.08 0.16* 0.12
## DSSTefficiency3 0.36*** 0.33*** 0.05
## LOTefficiency3 0.35*** 0.22*** 0.02
## PVTefficiency3 1.00 0.04 0.08
## AMefficiency3 0.04 1.00 -0.04
## NBACKaccuracy3 0.08 -0.04 1.00
## VOLTefficiency3 0.16* 0.34*** 0.11
## MPTspeed3 0.33*** 0.19** -0.02
## BVRtot3 -0.41*** -0.02 0.01
## StroopColors3 0.33*** 0.22*** -0.08
## StroopWords3 0.31*** 0.15* 0.07
## StroopMixed3 0.23*** 0.24*** 0.06
## CVLtca3 0.23*** 0.19** 0.05
## DigitSpanFwd3 0.20** 0.13* 0.04
## DigitSpanBck3 0.23*** 0.10 0.02
## FluencyWord3 0.26*** 0.10 0.02
## TrailsAtestSec3 -0.29*** -0.18** -0.01
## TrailsBminusA3 -0.28*** -0.23*** -0.03
## Attention3 0.33*** 0.13* -0.03
## ClockTotal3 0.10 -0.02 0.13*
## VOLTefficiency3 MPTspeed3 BVRtot3 StroopColors3
## BARTaccuracy3 0.17* 0.16* -0.12 0.11
## DSSTefficiency3 0.41*** 0.44*** -0.43*** 0.48***
## LOTefficiency3 0.25*** 0.33*** -0.53*** 0.37***
## PVTefficiency3 0.16* 0.33*** -0.41*** 0.33***
## AMefficiency3 0.34*** 0.19** -0.02 0.22***
## NBACKaccuracy3 0.11 -0.02 0.01 -0.08
## VOLTefficiency3 1.00 0.32*** -0.16* 0.28***
## MPTspeed3 0.32*** 1.00 -0.30*** 0.26***
## BVRtot3 -0.16* -0.30*** 1.00 -0.36***
## StroopColors3 0.28*** 0.26*** -0.36*** 1.00
## StroopWords3 0.11 0.22*** -0.34*** 0.49***
## StroopMixed3 0.24*** 0.16* -0.37*** 0.59***
## CVLtca3 0.24*** 0.25*** -0.27*** 0.32***
## DigitSpanFwd3 0.10 0.08 -0.21** 0.17*
## DigitSpanBck3 -0.06 0.03 -0.31*** 0.23***
## FluencyWord3 0.11 0.16* -0.24*** 0.23***
## TrailsAtestSec3 -0.26*** -0.39*** 0.32*** -0.39***
## TrailsBminusA3 -0.16* -0.26*** 0.27*** -0.29***
## Attention3 0.12 0.30*** -0.41*** 0.42***
## ClockTotal3 0.00 0.05 -0.18** 0.06
## StroopWords3 StroopMixed3 CVLtca3 DigitSpanFwd3
## BARTaccuracy3 0.18** 0.13* 0.19** 0.17*
## DSSTefficiency3 0.32*** 0.40*** 0.40*** 0.24***
## LOTefficiency3 0.25*** 0.38*** 0.20** 0.18**
## PVTefficiency3 0.31*** 0.23*** 0.23*** 0.20**
## AMefficiency3 0.15* 0.24*** 0.19** 0.13*
## NBACKaccuracy3 0.07 0.06 0.05 0.04
## VOLTefficiency3 0.11 0.24*** 0.24*** 0.10
## MPTspeed3 0.22*** 0.16* 0.25*** 0.08
## BVRtot3 -0.34*** -0.37*** -0.27*** -0.21**
## StroopColors3 0.49*** 0.59*** 0.32*** 0.17*
## StroopWords3 1.00 0.38*** 0.21** 0.33***
## StroopMixed3 0.38*** 1.00 0.28*** 0.27***
## CVLtca3 0.21** 0.28*** 1.00 0.20**
## DigitSpanFwd3 0.33*** 0.27*** 0.20** 1.00
## DigitSpanBck3 0.28*** 0.33*** 0.35*** 0.56***
## FluencyWord3 0.20** 0.19** 0.32*** 0.06
## TrailsAtestSec3 -0.31*** -0.32*** -0.32*** -0.21**
## TrailsBminusA3 -0.26*** -0.25*** -0.32*** -0.18**
## Attention3 0.42*** 0.39*** 0.30*** 0.25***
## ClockTotal3 0.07 0.08 0.03 0.06
## DigitSpanBck3 FluencyWord3 TrailsAtestSec3 TrailsBminusA3
## BARTaccuracy3 0.04 0.06 -0.23*** -0.26***
## DSSTefficiency3 0.18** 0.24*** -0.51*** -0.32***
## LOTefficiency3 0.14* 0.23*** -0.40*** -0.30***
## PVTefficiency3 0.23*** 0.26*** -0.29*** -0.28***
## AMefficiency3 0.10 0.10 -0.18** -0.23***
## NBACKaccuracy3 0.02 0.02 -0.01 -0.03
## VOLTefficiency3 -0.06 0.11 -0.26*** -0.16*
## MPTspeed3 0.03 0.16* -0.39*** -0.26***
## BVRtot3 -0.31*** -0.24*** 0.32*** 0.27***
## StroopColors3 0.23*** 0.23*** -0.39*** -0.29***
## StroopWords3 0.28*** 0.20** -0.31*** -0.26***
## StroopMixed3 0.33*** 0.19** -0.32*** -0.25***
## CVLtca3 0.35*** 0.32*** -0.32*** -0.32***
## DigitSpanFwd3 0.56*** 0.06 -0.21** -0.18**
## DigitSpanBck3 1.00 0.19** -0.22*** -0.29***
## FluencyWord3 0.19** 1.00 -0.20** -0.12
## TrailsAtestSec3 -0.22*** -0.20** 1.00 0.34***
## TrailsBminusA3 -0.29*** -0.12 0.34*** 1.00
## Attention3 0.34*** 0.13* -0.33*** -0.35***
## ClockTotal3 0.11 0.09 -0.13* -0.10
## Attention3 ClockTotal3
## BARTaccuracy3 0.21** 0.11
## DSSTefficiency3 0.39*** 0.08
## LOTefficiency3 0.33*** 0.17*
## PVTefficiency3 0.33*** 0.10
## AMefficiency3 0.13* -0.02
## NBACKaccuracy3 -0.03 0.13*
## VOLTefficiency3 0.12 0.00
## MPTspeed3 0.30*** 0.05
## BVRtot3 -0.41*** -0.18**
## StroopColors3 0.42*** 0.06
## StroopWords3 0.42*** 0.07
## StroopMixed3 0.39*** 0.08
## CVLtca3 0.30*** 0.03
## DigitSpanFwd3 0.25*** 0.06
## DigitSpanBck3 0.34*** 0.11
## FluencyWord3 0.13* 0.09
## TrailsAtestSec3 -0.33*** -0.13*
## TrailsBminusA3 -0.35*** -0.10
## Attention3 1.00 0.03
## ClockTotal3 0.03 1.00
##
## n = 229