Pattern Matrix and Structure Matrix Definition in SPSS FACTOR output
I am using the SPSS FACTOR procedure and have requested direct oblimin or promax rotation. In the output for a rotated solution, I see both a pattern matrix and a structure matrix. What is the difference between these two matrices? I thought that the pattern matrix held the usual factor loadings. If so, then what is the structure matrix?
Resolving the problem
When the rotation is orthogonal (i.e. the factors are uncorrelated; orthogonal and uncorrelated are synonymous with centered variables), then the rotated factor matrix represents both the loadings and the correlations between the variables and factors. For oblique rotations, where the factors are allowed to correlate (oblimin or promax in SPSS), then the loadings and correlations are distinct. The pattern matrix holds the loadings. Each row of the pattern matrix is essentially a regression equation where the standardized observed variable is expressed as a function of the factors. The loadings are the regression coefficients. The structure matrix holds the correlations between the variables and the factors.
Interpretation of a set of oblique factors involves both the pattern and structure matrices, as well as the factor correlation matrix. The latter matrix contains the correlations among all pairs of factors in the solution. It is automatically printed for an oblique solution when the rotated factor matrix is printed.
Gorsuch (1983) and Thompson (1983) describe concepts and procedures for interpreting the factors with these matrices.
Gorsuch, R.L. (1983). Factor Analysis (2nd Ed.). Hillsdale NJ: Erlbaum.
Thompson, B. (2004). Exploratory and Confirmatory Factor Analysis. Washington DC: American Psychological Association.