Let’s review the syntax to produce an EFA on the engineering data using oblimin rotation. We will use iterated PAF extraction, although any method would provide similar results. (See Chapter 2.) We selected oblimin rotation for this data because the factors are known to be correlated. The syntax to run this analysis is presented below.[8]

proc factor data = engdata  nfactors = 2  method = PRINIT  priors = SMC
      rotate = OBLIMIN;
   var EngProb: INTERESTeng: ;
run;

This syntax produces a series of results—within which we can see the four factor loading matrices (discussed above) of particular interest to us. These matrices are presented in Figure 4.2 Factor loading matrices output. Notice how the coefficients in the unrotated factor matrix (in the upper left of the figure) are generally larger in absolute magnitude and more difficult to interpret than the other matrices. Although we expected (and see evidence of) two very clear factors, the unrotated factor loadings do not immediately identify these two separate factors. All 14 items seem to have similar loadings on factor 1 and it is only in combination with the loadings on factor 2 where the two factors separate. If we plot each item in two-dimensional space (as done in Figure 4.1 Example of unrotated and rotated factor loading plot from engineering data), we clearly see the separation; however, rotation can help us interpret these results without plotting.

The rotated pattern matrix and the rotated structure matrix (in the upper right and lower right of the figure) support the same two-factor solution proposed in the plot of the unrotated results: the eight engineering problem-solving items load on factor 1, and the six interest items load on factor 2. However, notice that the matrices have slight differences in their factor loadings. The off-factor loadings (e.g., loading of EngProbSolv1 on factor 2) in the rotated structure matrix are higher than those in the rotated pattern matrix. As we stated previously, this is because the structure matrix represents the simple correlation between the item and the factor, but the pattern matrix reflects the unique relationship between the two, holding the other factor constant. The factors in this example analysis are correlated, r = 0.37. Thus, holding one constant removes the shared variance from the analysis and clarifies the unique factor structure. It is important to interpret both in order to diagnose how the correlation between factors might be affecting the results.

The reference structure matrix (in the lower left of the figure) presents a nearly identical picture as the rotated pattern matrix. There are slight differences in the loadings that can be traced to the difference in metric, but it does not tell us anything new. This matrix is not traditionally interpreted, and it is not output by other programs. We present the results below only in order to provide an explanation of the results that you might see. Since it does not provide any information beyond what is provided in the pattern matrix, we will not include it in subsequent examples.