Direct Schmid-Leiman Transformations and Rank-Deficient Loadings Matrices

The Schmid-Leiman (S-L; Psychometrika 22: 53-61, 1957) transformation is a popular method for conducting exploratory bifactor analysis that has been used in hundreds of studies of individual differences variables. To perform a two-level S-L transformation, it is generally believed that two separate factor analyses are required: a first-level analysis in which k obliquely rotated factors are extracted from an observed-variable correlation matrix, and a second-level analysis in which a general factor is extracted from the correlations of the first-level factors. In this article, I demonstrate that the S-L loadings matrix is necessarily rank deficient. I then show how this feature of the S-L transformation can be used to obtain a direct S-L solution from an unrotated first-level factor structure. Next, I reanalyze two examples from Mansolf and Reise (Multivar Behav Res 51: 698-717, 2016) to illustrate the utility of 'best-fitting' S-L rotations when gauging the ability of hierarchical factor models to recover known bifactor structures. Finally, I show how to compute direct bifactor solutions for non-hierarchical bifactor structures. An online supplement includes R code to reproduce all of the analyses that are reported in the article.