In general, second-order factor analysis consists of analysis of correlation matrices. If you are performing principal components analysis, you can save component scores and examine them as variables in a second-order analysis, as only in PCA will the component correlations and component scores match exactly (Thompson, 2004, p. 73).[4] When using common factor analysis, we must analyze the correlation matrix.

To illustrate this methodology, we will use the engineering data that we started exploring in Chapter 2, adding a third subscale that asked eight questions about feelings of belongingness in engineering. Using iterated PAF extraction and direct oblimin rotation (as before), there were 372 cases with valid data on all variables.

First-order analysis of engineering data. Since our data has changed, we start by examining the factor structure. The various plots to examine factor structure are presented in Figure 10.1 Factor structure plots. The Kaiser Criterion (eigenvalue >1) and MAP analysis support a three-factor solution; parallel analysis recommends a four-factor solution; and the scree plot supports one-, three-, or four-factor solutions. Theory, however, suggests there are three factors with a single, second-order factor. CFA would help us test the question of factor structure; but for the sake of expediency, let us assume a three-factor structure because of the strong theoretical basis and general support among the various extraction criteria.

Next, we examine the pattern and structure coefficients, presented in First-order factor loadings from engineering data. The structure coefficients show relatively strong loadings across factors, and the factors are moderately correlated (r= 0.30 to 0.50). The moderate correlations among the factors could indicate the presence of a second-order factor or it could suggest the factors are not unique, and fewer factors should have been extracted. Note that while very low correlations between factors can indicate that there is no higher-order factor, moderate to strong correlations between factors demonstrate only the possibility of a higher-order factor.

Second-order factor analysis. In order to conduct a second-order factor analysis, we will need to output our correlation matrix from our first-order analysis and read it in as the source of data for our second-order analysis. The code to do this is presented below. We use ODS to output a data set of inter-factor correlations from our first-order analysis. We also specify the type of this data set by including type= corr in parentheses after the data set name. Setting the type allows PROC FACTOR to recognize this data set as a correlation matrix and not a raw data set. We then input this data set directly into the FACTOR procedure and perform our second-order analysis. Since we are inputting a correlation matrix and not a raw data set, we must also specify the number of observations in our data set using the NOBS option.

*First-order analysis;
ods output InterFactorCorr=first_order_corr(type=corr);
proc factor data = engdata  nfactors = 3  method = prinit  priors = SMC 
      rotate = OBLIMIN;
   var EngProb: INTERESTeng: BELONGeng: ;
run;
ods output close;

*Second-order analysis;
proc factor data = first_order_corr  nobs=372  nfactors=1 method = prinit 
      priors = SMC;
   var Factor1 Factor2 Factor3;
run;

Using the above syntax, the correlation matrix was analyzed via EFA to determine whether it might be reasonable to assume a second-order factor that incorporates all three first-order factors (keep in mind our concerns about performing this analysis at all…). Again, we used iterated PAF extraction and would have used direct oblimin rotation (authors such as Thompson recommend using oblique rotations for higher-order EFA) but, as we asked for a single factor to be extracted, there was no rotation.

The results (presented in Figure 10.2 Results of second-order factor analysis) support an argument that there is one single second-order factor, and that all three scales load moderately to strongly on it (ranging from 0.52 to 0.87). This latent variable seems to be of more interest than problem-solving and belongingness, but you would have to ask our colleague who created the scale what it all means.