Iterated Factored Analysis 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
-----------------------------------------------------------------------------
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=prinit rotate=varimax rotate=promax;
title 'Iterated Factor Analysis Example';
run;
-----------------------------------------------------------------------------
Iterated Factor Analysis Example
Initial Factor Method: Iterated Principal Factor Analysis
Prior Communality Estimates: ONE
Preliminary Eigenvalues: 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: Iterated Principal Factor Analysis
Iter Change Communalities
1 0.615524 0.92615 0.89206 0.83682 0.63540
0.69419 0.38448 0.76974 0.41511
2 0.227148 0.94327 0.89470 0.84753 0.55976
0.61043 0.28851 0.72103 0.18796
3 0.035240 0.95828 0.89999 0.87106 0.54126
0.58074 0.27798 0.69641 0.15272
4 0.019137 0.96591 0.90138 0.89020 0.53517
0.57094 0.27761 0.67750 0.14803
5 0.016398 0.96953 0.90090 0.90531 0.53254
0.56894 0.27819 0.66110 0.14699
6 0.014608 0.97132 0.89983 0.91765 0.53109
0.56997 0.27874 0.64649 0.14647
7 0.012995 0.97227 0.89869 0.92803 0.53011
0.57209 0.27920 0.63350 0.14608
8 0.011448 0.97285 0.89763 0.93694 0.52936
0.57449 0.27959 0.62205 0.14575
9 0.009970 0.97324 0.89668 0.94470 0.52875
0.57684 0.27991 0.61208 0.14547
10 0.008592 0.97353 0.89586 0.95151 0.52823
0.57898 0.28020 0.60349 0.14521
11 0.007337 0.97375 0.89512 0.95751 0.52778
0.58088 0.28044 0.59615 0.14498
12 0.006221 0.97393 0.89448 0.96281 0.52740
0.58252 0.28065 0.58993 0.14477
13 0.005246 0.97408 0.89390 0.96749 0.52706
0.58391 0.28082 0.58469 0.14458
14 0.004407 0.97419 0.89339 0.97162 0.52677
0.58509 0.28098 0.58028 0.14440
15 0.003694 0.97428 0.89293 0.97526 0.52652
0.58607 0.28111 0.57658 0.14425
16 0.003201 0.97435 0.89253 0.97846 0.52630
0.58689 0.28122 0.57349 0.14411
17 0.002812 0.97441 0.89217 0.98127 0.52610
0.58757 0.28132 0.57090 0.14399
18 0.002466 0.97445 0.89186 0.98374 0.52594
0.58814 0.28141 0.56874 0.14388
19 0.002159 0.97449 0.89158 0.98590 0.52580
0.58860 0.28148 0.56692 0.14378
20 0.001888 0.97452 0.89133 0.98779 0.52567
0.58899 0.28154 0.56540 0.14370
21 0.001649 0.97454 0.89112 0.98944 0.52556
0.58931 0.28160 0.56411 0.14362
22 0.001438 0.97455 0.89093 0.99087 0.52547
0.58957 0.28164 0.56303 0.14356
23 0.001253 0.97457 0.89077 0.99213 0.52539
0.58979 0.28168 0.56212 0.14350
24 0.001091 0.97458 0.89062 0.99322 0.52532
0.58998 0.28172 0.56134 0.14345
25 0.000949 0.97458 0.89050 0.99417 0.52526
0.59013 0.28175 0.56069 0.14341
Convergence criterion satisfied.
Eigenvalues of the Reduced Correlation Matrix:
Total = 4.9602207 Average = 0.62002759
1 2 3 4
Eigenvalue 4.1685 0.7920 0.2707 0.0683
Difference 3.3766 0.5212 0.2025 0.0093
Proportion 0.8404 0.1597 0.0546 0.0138
Cumulative 0.8404 1.0001 1.0546 1.0684
Initial Factor Method: Iterated Principal Factor Analysis
5 6 7 8
Eigenvalue 0.0590 -0.0409 -0.1584 -0.1990
Difference 0.0999 0.1175 0.0406
Proportion 0.0119 -0.0082 -0.0319 -0.0401
Cumulative 1.0803 1.0721 1.0401 1.0000
Initial Factor Method: Iterated Principal Factor Analysis
Factor Pattern
FACTOR1 FACTOR2
Y1 0.98399* 0.07966 sociability
Y2 0.93683* -0.11332 sorrow
Y3 0.83456* -0.54560* tenderness
Y4 0.72062* -0.07726 joy
Y5 0.67971* 0.35794 wonder
Y6 0.51651* 0.12234 disgust
Y7 0.50186* 0.55572* anger
Y8 0.35525 -0.13116 fear
Variance explained by each factor
FACTOR1 FACTOR2
4.168522 0.791953
Initial Factor Method: Iterated Principal Factor Analysis
Final Communality Estimates: Total = 4.960475
Y1 Y2 Y3 Y4
0.974583 0.890498 0.994166 0.525258
Y5 Y6 Y7 Y8
0.590129 0.281748 0.560687 0.143406
Prerotation Method: Varimax
Orthogonal Transformation Matrix
1 2
1 0.79500 0.60660
2 -0.60660 0.79500
Prerotation Method: Varimax
Rotated Factor Pattern
FACTOR1 FACTOR2
Y1 0.73395* 0.66023* sociability
Y2 0.81353* 0.47820 sorrow
Y3 0.99444* 0.07249 tenderness
Y4 0.61976* 0.37571 joy
Y5 0.32325 0.69688* wonder
Y6 0.33642 0.41058 disgust
Y7 0.06188 0.74623* anger
Y8 0.36199 0.11123 fear
Variance explained by each factor
FACTOR1 FACTOR2
2.926048 2.034427
Prerotation Method: Varimax
Final Communality Estimates: Total = 4.960475
Y1 Y2 Y3 Y4
0.974583 0.890498 0.994166 0.525258
Y5 Y6 Y7 Y8
0.590129 0.281748 0.560687 0.143406
Rotation Method: Promax
Target Matrix for Procrustean Transformation
FACTOR1 FACTOR2
Y1 0.41421 0.30221
Y2 0.64583 0.13147
Y3 1.00000 0.00039
Y4 0.63032 0.14075
Y5 0.07510 0.75425
Y6 0.25662 0.46758
Y7 0.00057 1.00000
Y8 0.88040 0.02560
Rotation Method: Promax
Procrustean Transformation Matrix
1 2
1 1.06319 -0.33134
2 -0.28686 1.14112
Normalized Oblique Transformation Matrix
1 2
1 0.71329 0.42229
2 -0.92770 1.09137
Rotation Method: Promax
Inter-factor Correlations
FACTOR1 FACTOR2
FACTOR1 1.00000 0.51938
FACTOR2 0.51938 1.00000
Rotation Method: Promax
Rotated Factor Pattern (Std Reg Coefs)
FACTOR1 FACTOR2
Y1 0.62797* 0.50247 sociability
Y2 0.77336* 0.27194 sorrow
Y3 1.10144 -0.24303 tenderness
Y4 0.58568* 0.21999 joy
Y5 0.15278 0.67767* wonder
Y6 0.25493 0.35163 disgust
Y7 -0.15757 0.81842* anger
Y8 0.37507 0.00688 fear
Rotation Method: Promax
Reference Axis Correlations
FACTOR1 FACTOR2
FACTOR1 1.00000 -0.51938
FACTOR2 -0.51938 1.00000
Rotation Method: Promax
Reference Structure (Semipartial Correlations)
FACTOR1 FACTOR2
Y1 0.53662 0.42938
Y2 0.66087 0.23238
Y3 0.94122 -0.20768
Y4 0.50049 0.18799
Y5 0.13055 0.57910
Y6 0.21785 0.30048
Y7 -0.13465 0.69938
Y8 0.32052 0.00587
Variance explained by each factor eliminating other factors
FACTOR1 FACTOR2
2.046472 1.231652
Rotation Method: Promax
Factor Structure (Correlations)
FACTOR1 FACTOR2
Y1 0.88894* 0.82862* sociability
Y2 0.91460* 0.67361* sorrow
Y3 0.97521* 0.32903 tenderness
Y4 0.69994* 0.52418* joy
Y5 0.50475 0.75702* wonder
Y6 0.43756 0.48404 disgust
Y7 0.26750 0.73658* anger
Y8 0.37864 0.20168 fear
Variance explained by each factor ignoring other factors
FACTOR1 FACTOR2
3.728823 2.914003
Rotation Method: Promax
Final Communality Estimates: Total = 4.960475
Y1 Y2 Y3 Y4
0.974583 0.890498 0.994166 0.525258
Y5 Y6 Y7 Y8
0.590129 0.281748 0.560687 0.143406