Principal Components Example -- Harman -- pg 164
Same data as Assignment 10
Based on 172 9 to 12 year old children
Variables:
y1 sociability
y2 sorrow
y3 tenderness
y4 joy
y5 wonder
y6 disgust
y6 anger
y8 fear
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data emofac (type=corr);
_type_='corr';
input _name_ $ y1 y2 y3 y4 y5 y6 y7 y8;
cards;
y1 1.00 0.83 0.81 0.80 0.71 0.54 0.53 0.24
y2 0.83 1.00 0.87 0.62 0.59 0.58 0.44 0.45
y3 0.81 0.87 1.00 0.63 0.37 0.30 0.12 0.33
y4 0.80 0.62 0.63 1.00 0.49 0.30 0.28 0.29
y5 0.71 0.59 0.37 0.49 1.00 0.34 0.55 0.19
y6 0.54 0.58 0.30 0.30 0.34 1.00 0.32 0.21
y7 0.53 0.44 0.12 0.28 0.55 0.32 1.00 0.10
y8 0.24 0.45 0.33 0.29 0.19 0.21 0.10 1.00
;
proc factor method=prin rotate=varimax rotate=promax;
var y1 y2 y3 y4 y5 y6 y7 y8;
run;
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Initial Factor Method: Principal Components
Prior Communality Estimates: ONE
Eigenvalues of the Correlation Matrix: Total = 8 Average = 1
1 2 3 4
Eigenvalue 4.4178 1.1362 0.8436 0.7236
Difference 3.2816 0.2926 0.1200 0.2976
Proportion 0.5522 0.1420 0.1054 0.0905
Cumulative 0.5522 0.6942 0.7997 0.8901
5 6 7 8
Eigenvalue 0.4261 0.3887 0.0842 -0.0201
Difference 0.0374 0.3044 0.1044
Proportion 0.0533 0.0486 0.0105 -0.0025
Cumulative 0.9434 0.9920 1.0025 1.0000
2 factors will be retained by the MINEIGEN criterion.
Initial Factor Method: Principal Components
Factor Pattern
FACTOR1 FACTOR2
Y1 0.96033* 0.06262 sociability
Y2 0.93312* -0.14613 sorrow
Y3 0.80106* -0.44173 tenderness
Y4 0.78012* -0.16374 joy
Y5 0.73055* 0.40061 wonder
Y6 0.59903* 0.16011 disgust
Y7 0.54723* 0.68577* anger
Y8 0.42726 -0.48224 fear
Variance explained by each factor
FACTOR1 FACTOR2
4.417774 1.136176
Initial Factor Method: Principal Components
Final Communality Estimates: Total = 5.553950
Y1 Y2 Y3 Y4
0.926151 0.892059 0.836820 0.635396
Y5 Y6 Y7 Y8
0.694193 0.384476 0.769743 0.415112
Prerotation Method: Varimax
Orthogonal Transformation Matrix
1 2
1 0.72833 0.68523
2 -0.68523 0.72833
Prerotation Method: Varimax
Rotated Factor Pattern
FACTOR1 FACTOR2
Y1 0.65652* 0.70365* sociability
Y2 0.77975* 0.53297* sorrow
Y3 0.88612* 0.22718 tenderness
Y4 0.68038* 0.41530 joy
Y5 0.25757 0.79237* wonder
Y6 0.32659 0.52708* disgust
Y7 -0.07134 0.87444* anger
Y8 0.64163* -0.05846 fear
Variance explained by each factor
FACTOR1 FACTOR2
2.876948 2.677002
Prerotation Method: Varimax
Final Communality Estimates: Total = 5.553950
Y1 Y2 Y3 Y4
0.926151 0.892059 0.836820 0.635396
Y5 Y6 Y7 Y8
0.694193 0.384476 0.769743 0.415112
Rotation Method: Promax
Target Matrix for Procrustean Transformation
FACTOR1 FACTOR2
Y1 0.32145 0.39480
Y2 0.56971 0.18148
Y3 0.92027 0.01547
Y4 0.62962 0.14284
Y5 0.02991 0.86873
Y6 0.14794 0.62038
Y7 -0.00054 1.00000
Y8 1.00000 -0.00075
Rotation Method: Promax
Procrustean Transformation Matrix
1 2
1 1.04462 -0.29739
2 -0.24943 1.05127
Normalized Oblique Transformation Matrix
1 2
1 0.62939 0.52835
2 -0.95752 1.01677
Rotation Method: Promax
Inter-factor Correlations
FACTOR1 FACTOR2
FACTOR1 1.00000 0.48824
FACTOR2 0.48824 1.00000
Rotation Method: Promax
Rotated Factor Pattern (Std Reg Coefs)
FACTOR1 FACTOR2
Y1 0.54446* 0.57106* sociability
Y2 0.72722* 0.34443 sorrow
Y3 0.92715* -0.02591 tenderness
Y4 0.64778* 0.24569 joy
Y5 0.07620 0.79332* wonder
Y6 0.22372 0.47929 disgust
Y7 -0.31221 0.98640* anger
Y8 0.73067* -0.26459 fear
Rotation Method: Promax
Reference Axis Correlations
FACTOR1 FACTOR2
FACTOR1 1.00000 -0.48824
FACTOR2 -0.48824 1.00000
Rotation Method: Promax
Reference Structure (Semipartial Correlations)
FACTOR1 FACTOR2
Y1 0.47516 0.49837* sociability
Y2 0.63465* 0.30059 sorrow
Y3 0.80913* -0.02261 tenderness
Y4 0.56533* 0.21442 joy
Y5 0.06650 0.69234* wonder
Y6 0.19525 0.41828 disgust
Y7 -0.27247 0.86084* anger
Y8 0.63767* -0.23091 fear
Variance explained by each factor eliminating other factors
FACTOR1 FACTOR2
2.126251 1.833875
Rotation Method: Promax
Factor Structure (Correlations)
FACTOR1 FACTOR2
Y1 0.82327* 0.83689* sociability
Y2 0.89538* 0.69948* sorrow
Y3 0.91450* 0.42676 tenderness
Y4 0.76774* 0.56196* joy
Y5 0.46353 0.83052* wonder
Y6 0.45773 0.58852* disgust
Y7 0.16938 0.83397* anger
Y8 0.60149* 0.09215 fear
Variance explained by each factor ignoring other factors
FACTOR1 FACTOR2
3.720075 3.427699
Rotation Method: Promax
Final Communality Estimates: Total = 5.553950
Y1 Y2 Y3 Y4
0.926151 0.892059 0.836820 0.635396
Y5 Y6 Y7 Y8
0.694193 0.384476 0.769743 0.415112