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.