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

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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=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