All extracted factors are initially orthogonal (Thompson, 2004), but remain so only as long as the rotation is orthogonal (we discussed this briefly in the section above). In practice, this results in interpretation of one matrix (a pattern matrix) when conducting an orthogonal rotation, and two matrices (a pattern and structure matrix) when conducting an oblique rotation. To further complicate interpretation, an unrotated factor matrix is always output, and a reference structure matrix is output when an oblique rotation is done —neither of which is traditionally interpreted. The presence of multiple matrices, whether they are to be interpreted or not, is a source of much confusion in practice.[6] Let’s start with a few gentle conceptual definitions of each:

Unrotated factor matrix coefficients are generally reported by most statistical software regardless of rotation. They represent the unrotated factor loadings, and are generally not of interest.[7] In PROC FACTOR, they appear toward the top of the output in a table entitled “Factor Pattern.” Despite the title, this should not be confused with the pattern matrix that is described below.

Pattern matrix coefficients are essentially a series of standardized regression coefficients (betas or βs in the regression world) expressing the variable as a function of factor loadings. These are also output for all rotation methods. You can think of these as the list of ingredients in the recipe (e.g., to make Item 13, add 0.70 of factor 1, 0.13 of factor 2, 0.02 of factor 3, etc. Mix well… delicious!). Like regression coefficients, they hold all other variables (factors) in the equation constant when estimating the pattern matrix coefficients. So, if factors are uncorrelated, pattern and structure coefficients are identical. As factors become more strongly correlated, the two types of coefficients will become less alike. Thus, think of pattern matrix coefficients as “row” statistics, describing the individual item’s unique relationships to each factor. In PROC FACTOR, these appear toward the bottom of the output in a table entitled “Rotated Factor Pattern.”

Reference structure matrix coefficients are a series of semipartial correlations between the individual variables and the overall factors. These are output by SAS when conducting an oblique rotation and are not commonly output in other statistical software. This matrix provides the same information as the pattern matrix, only in a different metric: total variance vs units of standard deviation. Although the different scale can be useful to those who think in terms of correlations, this matrix is not traditionally analyzed.

Structure matrix coefficients are simple correlations between an individual variable and the composite or latent variable (factor). These are output only when conducting an oblique rotation. In multiple regression, these would be similar to correlations between an individual variable and the predicted score derived from the regression equation. The difference between structure and pattern coefficients is the difference (again, returning to regression) between simple correlations and semipartial (unique relationship only) correlations. In PROC FACTOR, these appear after the pattern matrix in a table entitled “Factor Structure.”

Individuals are often confused by the various matrices that are output as well as the different matrices output by orthogonal and oblique rotation. As we mentioned above, an orthogonal rotation will output 1) an unrotated factor matrix and 2) a pattern matrix, but an oblique rotation will output 1) an unrotated factor matrix, 2) a pattern matrix, 3) a reference structure matrix, and 4) a structure matrix.

The pattern matrix is generally interpreted when conducting an orthogonal rotation, and both the pattern and structure matrix are interpreted when conducting an oblique rotation. The structure matrix is provided only during an oblique rotation because it provides information about the items under the assumption of correlated factors. If all factors were perfectly uncorrelated with each other, the pattern and structure coefficients would be identical. In this case, there is no effect of holding other factors constant when computing the pattern matrix, and the structure and pattern coefficients would be the same, just like simple and semipartial correlations in multiple regression with perfectly uncorrelated predictors. Thus, the structure matrix is useful only when interpreting an oblique rotation.

Thompson (2004) and others (Gorsuch, 1983; Nunnally & Bernstein, 1994) have argued that it is essential to interpret both pattern and structure coefficients in an oblique rotation in order to correctly and fully interpret the results of an EFA. In practice, few do. Further, when rotations are oblique and factors are correlated, they argue it is important to report the intercorrelations of factors also. We will highlight this process when appropriate.