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Direct Schmid–Leiman Transformations and Rank-Deficient Loadings Matrices

Published online by Cambridge University Press:  01 January 2025

Niels G. Waller
Niels G. Waller*
Affiliation:
University of Minnesota
*
Correspondence should be made to Niels G. Waller, Department of Psychology, University of Minnesota, 75 East River Road, Minneapolis, MN 55455, USA. Email: [email protected]
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Abstract

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.

Keywords

bifactorSchmid Leimanhierarchical factor analysis
Type
Original Paper
Information
Psychometrika , Volume 83 , Issue 4 , December 2018 , pp. 858 - 870
Copyright
Copyright © 2017 The Psychometric Society

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