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Direct Schmid–Leiman Transformations and Rank-Deficient Loadings Matrices

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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.

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Notes

  1. Google Scholar reports 895 citations of the Schmid Leiman, 1957, paper as of May 17, 2017.

  2. Although the restricted bifactor model that is described in this paper is typically attributed to Schmid & Leiman, 1957, the underlying rationale for the two-level S–L transformation had been discussed previously by Thompson, (1939/1948), Thurstone, (1947), and others. Wherry (1959) also claimed to have independently derived the transformation with the assistance of B. J. Winer.

  3. Actual S–L solutions may differ from (2) due to design choices that are described later.

  4. S–L solutions with three or more levels are theoretically possible but are not considered in this paper.

  5. The supplement also contains code that will reproduce all analyses that are reported in this article.

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  1. Department of Psychology, University of Minnesota, 75 East River Road, Minneapolis, MN, 55455, USA

    Niels G. Waller

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  1. Niels G. Waller
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Correspondence to Niels G. Waller.

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Waller, N.G. Direct Schmid–Leiman Transformations and Rank-Deficient Loadings Matrices. Psychometrika 83, 858–870 (2018). https://doi.org/10.1007/s11336-017-9599-0

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  • DOI: https://doi.org/10.1007/s11336-017-9599-0

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