Maximum Likelihood Factor Analysis Example -- Harman -- pg
Variables:
y1 total population
y2 median school years
y3 total employment
y4 misc. profess services
y5 median value house
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data socfac (type=corr);
_type_='corr';
input _name_ $ y1 y2 y3 y4 y5;
cards;
y1 1.00000 0.00975 0.97245 0.43887 0.02241
y2 0.00975 1.00000 0.15428 0.69141 0.86307
y3 0.97245 0.15428 1.00000 0.51472 0.12193
y4 0.43887 0.69141 0.51472 1.00000 0.77765
y5 0.02241 0.86307 0.12193 0.77765 1.00000
;
proc factor method=ml priors=smc heywood rotate=varimax rotate=promax;
title 'ML Factor Analysis Example';
run;
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ML Factor Analysis Example
Initial Factor Method: Maximum Likelihood
Prior Communality Estimates: SMC
Y1 Y2 Y3 Y4 Y5
0.968593 0.822273 0.969182 0.785718 0.847008
Preliminary Eigenvalues: Total = 76.1173815 Average = 15.2234763
1 2 3 4 5
Eigenvalue 63.7028 13.0539 0.3275 -0.3473 -0.6195
Difference 50.6489 12.7263 0.6748 0.2722
Proportion 0.8369 0.1715 0.0043 -0.0046 -0.0081
Cumulative 0.8369 1.0084 1.0127 1.0081 1.0000
2 factors will be retained by the PROPORTION criterion.
Iter Criterion Ridge Change Communalities
1 0.34305 0.000 0.04710 1.00000 0.80672 0.95058 0.79348 0.89411
2 0.30713 0.000 0.03069 1.00000 0.80821 0.96024 0.81047 0.92480
3 0.30670 0.000 0.00629 1.00000 0.81149 0.95949 0.81676 0.92023
4 0.30665 0.000 0.00218 1.00000 0.80985 0.95963 0.81498 0.92241
5 0.30665 0.000 0.00071 1.00000 0.81019 0.95956 0.81569 0.92187
Convergence criterion satisfied.
Initial Factor Method: Maximum Likelihood
Significance tests based on 10000 observations:
Test of H0: No common factors.
vs HA: At least one common factor.
Chi-square = 63803.130 df = 10 Prob>chi**2 = 0.0001
Test of H0: 2 Factors are sufficient.
vs HA: More factors are needed.
Chi-square = 3065.006 df = 1 Prob>chi**2 = 0.0001
Chi-square without Bartlett's correction = 3066.1812237
Akaike's Information Criterion = 3064.1812237
Schwarz's Bayesian Criterion = 3056.9708833
Tucker and Lewis's Reliability Coefficient = 0.5196965955
Initial Factor Method: Maximum Likelihood
Squared Canonical Correlations
FACTOR1 FACTOR2
1.000000 0.951887
Eigenvalues of the Weighted Reduced Correlation Matrix:
Total = 19.7845481 Average = 4.94613703
1 2 3 4 5
Eigenvalue . 19.7845 0.5431 -0.0398 -0.5033
Difference . 19.2414 0.5829 0.4636
Proportion . 1.0000 0.0275 -0.0020 -0.0254
Cumulative . 1.0000 1.0275 1.0254 1.0000
Initial Factor Method: Maximum Likelihood
Factor Pattern
FACTOR1 FACTOR2
Y1 1.00000* 0.00000 total population
Y2 0.00975 0.90003* median school years
Y3 0.97245* 0.11797 total employment
Y4 0.43887 0.78930* misc. profess services
Y5 0.02241 0.95989* median value house
Variance explained by each factor
FACTOR1 FACTOR2
Weighted 24.435550 19.784547
Unweighted 2.138863 2.368350
Initial Factor Method: Maximum Likelihood
Final Communality Estimates and Variable Weights
Total Communality: Weighted = 44.220097 Unweighted = 4.507214
Y1 Y2 Y3 Y4 Y5
Communality 1.000000 0.810150 0.959576 0.815599 0.921888
Weight . 5.268444 24.727338 5.425519 12.798798
Prerotation Method: Varimax
Orthogonal Transformation Matrix
1 2
1 0.02133 0.99977
2 0.99977 -0.02133
Rotated Factor Pattern
FACTOR1 FACTOR2
Y1 0.02133 0.99977* total population
Y2 0.90003* -0.00945 median school years
Y3 0.13869 0.96971* total employment
Y4 0.79848* 0.42193 misc. profess services
Y5 0.96015* 0.00193 median value house
Prerotation Method: Varimax
Variance explained by each factor
FACTOR1 FACTOR2
Weighted 20.001570 24.218527
Unweighted 2.389209 2.118004
Final Communality Estimates and Variable Weights
Total Communality: Weighted = 44.220097 Unweighted = 4.507214
Y1 Y2 Y3 Y4 Y5
Communality 1.000000 0.810150 0.959576 0.815599 0.921888
Weight . 5.268444 24.727338 5.425519 12.798798
Rotation Method: Promax
Target Matrix for Procrustean Transformation
FACTOR1 FACTOR2
Y1 0.00001 1.00000
Y2 0.99984 -0.00000
Y3 0.00284 0.97075
Y4 0.69117 0.10205
Y5 1.00000 0.00000
Procrustean Transformation Matrix
1 2
1 1.03022 -0.09567
2 -0.10099 0.95877
Rotation Method: Promax
Normalized Oblique Transformation Matrix
1 2
1 -0.07781 1.01233
2 1.01678 -0.12288
Inter-factor Correlations
FACTOR1 FACTOR2
FACTOR1 1.00000 0.19589
FACTOR2 0.19589 1.00000
Rotation Method: Promax
Rotated Factor Pattern (Std Reg Coefs)
FACTOR1 FACTOR2
Y1 -0.07781 1.01233* total population
Y2 0.91438* -0.10072 median school years
Y3 0.04428 0.96994* total employment
Y4 0.76840* 0.34729 misc. profess services
Y5 0.97426* -0.09526 median value house
Reference Axis Correlations
FACTOR1 FACTOR2
FACTOR1 1.00000 -0.19589
FACTOR2 -0.19589 1.00000
Rotation Method: Promax
Reference Structure (Semipartial Correlations)
FACTOR1 FACTOR2
Y1 -0.07630 0.99271
Y2 0.89666 -0.09877
Y3 0.04343 0.95115
Y4 0.75351 0.34056
Y5 0.95538 -0.09342
Variance explained by each factor eliminating other factors
FACTOR1 FACTOR2
Weighted 19.045100 23.162829
Unweighted 2.292243 2.024633
Rotation Method: Promax
Factor Structure (Correlations)
FACTOR1 FACTOR2
Y1 0.12050 0.99708* total population
Y2 0.89465* 0.07840 median school years
Y3 0.23429 0.97862* total employment
Y4 0.83643* 0.49782 misc. profess services
Y5 0.95559* 0.09559 median value house
Variance explained by each factor ignoring other factors
FACTOR1 FACTOR2
Weighted 21.057268 25.174997
Unweighted 2.482581 2.214971
Rotation Method: Promax
Final Communality Estimates and Variable Weights
Total Communality: Weighted = 44.220097 Unweighted = 4.507214
Y1 Y2 Y3 Y4 Y5
Communality 1.000000 0.810150 0.959576 0.815599 0.921888
Weight . 5.268444 24.727338 5.425519 12.798798