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The following example demonstrates how you can use the FACTOR procedure to perform common factor analysis and use a transformation to rotate the extracted factors.

Suppose that you want to use factor analysis to explore the relationship among assessment scores of a group of students. For each student in the group, you record six homework scores, two midterm examination scores, and the final exam score.

The following DATA step creates the SAS data set Grades:

   data Grades;                                                 
      input HomeWork1 - HomeWork6 MidTerm1 MidTerm2 FinalExam;     
      datalines;                                                   
   15  18  36  29  44  30  78  87  70                           
   15  16  24  30  41  30  71  73  89                           
   15  14  23  34  28  24  84  72  76                           
   15  20  39  35  50  30  74  79  96                           
   15  20  39  35  46  30  76  77  94                           
   15  20  28  30  49  28  40  44  66                           
   15  15  29  25  36  30  88  69  93                           
   15  20  37  35  50  30  97  95  98                           
   14  16  24  30  44  28  57  78  85                           
   15  17  29  26  38  28  56  78  76                           
   15  17  31  34  40  27  72  67  84                           
   11  16  29  34  31  27  83  68  75                           
   15  18  31  18  40  30  75  43  67                           
   14  14  29  25  49  30  71  93  93                           
   15  18  36  29  44  30  85  64  75                           
   ;

The data set Grades contains the variables representing homework scores (HomeWork1 -HomeWork6), the two midterm exam scores (MidTerm1 and MidTerm2), and the final exam score (FinalExam).

The following statements invoke the FACTOR procedure:

   proc factor data=Grades priors=smc rotate=varimax nfactors=2;
   run;

The DATA= option in PROC FACTOR specifies the SAS data set Grades as the input data set. The PRIORS= option specifies that the squared multiple correlations (SMC) of each variable with all the other variables are used as the prior communality estimates and also that PROC FACTOR gives a principal factor solution to the common factor model. The ROTATE= option specifies the VARIMAX orthogonal factor rotation method. To see if two latent factors can explain the observed variation in the data, the NFACTOR= option specifies that two factors be retained. All variables in the data set are analyzed.

The output from this analysis is displayed in the following figures.

As displayed in Figure 26.1, the prior communality estimates are set to the squared multiple correlations. Figure 26.1 also displays the table of eigenvalues, which are the variances of the principal factors, of the reduced correlation matrix. Each row of the table pertains to a single eigenvalue. Following the column of eigenvalues are three measures of each eigenvalue's relative size and importance. The first of these displays the difference between the eigenvalue and its successor. The last two columns display the individual and cumulative proportions that the corresponding factor contributes to the total variation. The last line displayed in Figure 26.1 states that two factors are retained, as specified by the NFACTORS= option in the PROC FACTOR statement.

Figure 26.2 displays the factor pattern matrix. The factor pattern matrix is the matrix of correlations between variables and the common factors. When the factors are orthogonal, the pattern matrix is also equal to the matrix of standardized regression coefficients for predicting the variables using the extracted factors.

The pattern matrix suggests that the first factor represents general ability, with positive loadings from all variables. The second factor is more difficult to interpret, but it may represent a contrast between exam and homework scores, with the exception of the score for HomeWork4.

Figure 26.3 displays the variance explained by each factor and the final communality estimates, including the total communality. The final communality estimates are the proportion of variance of the variables accounted for by the common factors. When the factors are orthogonal, the final communalities are calculated by taking the sum of squares of each row of the factor pattern matrix. For example, the final communality estimate for the variable FinalExam is computed as follows:

0.60256254 = (0.52745)² + (0.56953)²

Figure 26.4 displays the results of the VARIMAX rotation of the two extracted factors and the final communality estimates of the rotated factors.

The rotated factor pattern matrix is calculated by postmultiplying the original factor pattern matrix (Figure 26.2) by the orthogonal transformation matrix (Figure 26.4).

The rotated factor pattern matrix is somewhat simpler to interpret: the rotated Factor1 can now be interpreted as general ability in homework performance. The homework variables load higher on Factor1 (with the single exception of the variable HomeWork4), with small loadings for the exam score variables. The rotated Factor2 seems to measure exam performance or test-taking ability. The exam score variables load heavily on Factor2, as does HomeWork4.

Figure 26.5 displays the variance explained by each factor and the final communality estimates. Even though the variance explained by the rotated Factor1 is less than that explained by the unrotated factor (compare with Figure 26.3), the cumulative variance explained by both common factors remains the same after the orthogonal rotation. Also note that the VARIMAX rotation, as with any orthogonal rotation, has not changed the final communalities.

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