Hostname: page-component-76d6cb85b7-pn7tm Total loading time: 0 Render date: 2026-07-16T22:03:38.328Z Has data issue: false hasContentIssue false
  • English
  • Français

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: nwaller@umn.edu
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bandalos, D. L., & Kopp, J. P.(2013). The utility of exploratory bi-factor rotations in scale construction. In Paper Presented at the Annual Meeting of the American Psychological Society,Washington D.C.Google Scholar
Bernaards, C. A.,Jennrich, R. I.(2005)Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis Educational and Psychological Measurement 65,676696.CrossRefGoogle Scholar
Briggs, N. E., &MacCallum, R. C.(2003)Recovery of weak common factors by maximum likelihood and ordinary least squares estimation Multivariate Behavioral Research 38,2556.CrossRefGoogle ScholarPubMed
Browne, M. W.(1968)A comparison of factor analytic techniques Psychometrika 33,267334.CrossRefGoogle ScholarPubMed
Browne, M.(2001)An overview of analytic rotation in exploratory factor analysis Multivariate Behavioral Research 36,111150.CrossRefGoogle Scholar
Cliff, N.(1966)Orthogonal rotation to congruence Psychometrika 31,3342.CrossRefGoogle Scholar
Eckart, C.,Young, G.(1936)The approximation of one matrix by another of lower rank Psychometrika 1,211218.CrossRefGoogle Scholar
Gignac, G. E.(2016)The higher-order model imposes a proportionality constraint: That is why the bifactor model tends to fit better Intelligence 55,5768.CrossRefGoogle Scholar
Grayson, D., &Marsh, H. W.(1994)Identification with deficient rank loading matrices in confirmatory factor analysis: Multitrait-multimethod models Psychometrika 59,121134.CrossRefGoogle Scholar
Harman, H., &Jones, W.(1966)Factor analysis by minimizing residuals (minres) Psychometrika 31,3351368.CrossRefGoogle ScholarPubMed
Hendrickson, A. E., &White, P. O.(1966)A method for the rotation of higher-order factors British Journal of Mathematical and Statistical Psychology 19,97103.CrossRefGoogle ScholarPubMed
Holzinger, K. J., &Swineford, F.(1937)The bi-factor method Psychometrika 2,4154.CrossRefGoogle Scholar
Horn, R. A., &Johnson, C. R.(2012)Matrix analysis CambridgeCambridge University PressCrossRefGoogle Scholar
Jennrich, R. I., &Bentler, P. M.(2011)Exploratory bi-factor analysis Psychometrika 74,537549.CrossRefGoogle Scholar
Jennrich, R. L., &Bentler, P. M.(2012)Exploratory bi-factor analysis: The Oblique case Psychometrika 77,442454.CrossRefGoogle ScholarPubMed
Jennrich, R. I., &Sampson, P. F.(1966)Rotation for simple loadings Psychometrika 31,313323.CrossRefGoogle ScholarPubMed
Kristof, W.(1970)A theorem on the trace of certain matrix products and some applications Journal of Mathematical Psychology 7,515530.CrossRefGoogle Scholar
Mansolf, M., &Reise, S. P.(2016)Exploratory bifactor analysis: The Schmid–Leiman orthogonalization and Jennrich-Bentler analytic rotations Multivariate Behavioral Research 51,698717.CrossRefGoogle ScholarPubMed
Mansolf, M., &Reise, S. P.(2017)When and why the second-order and bifactor models are distinguishable Intelligence 61,120129.CrossRefGoogle Scholar
McDonald, R. P.(1985)Factor analysis and related methods Hillsdale, New JerseyLawrence ErlbaumGoogle Scholar
Mulaik, S. A., &Quartetti, D. A.(1997)First order or higher order general factor? Structural Equation Modeling 4,193211.CrossRefGoogle Scholar
R Core Team(2017).R: A language and environment for statistical computing.R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.Google Scholar
Reise, S. P.(2012)The rediscovery of bifactor measurement models Multivariate Behavioral Research 47,667696.CrossRefGoogle ScholarPubMed
Reise, S. P.,Moore, T. M., &Haviland, M. G.(2010)Bifactor models and rotations: Exploring the extent to which multidimensional data yield univocal scale scores Journal of Personality Assessment 92,544559.CrossRefGoogle ScholarPubMed
Reise, S.,Moore, T., &Maydeu-Olivares, A.(2011)Target rotations and assessing the impact of model violations on the parameters of unidimensional item response theory models Educational and Psychological Measurement 71,684711.CrossRefGoogle Scholar
Revelle, W.(2017). psych: Procedures for Personality and Psychological Research, Northwestern University, Evanston, Illinois, USA. https://CRAN.R-project.org/package=psychVersion=1.7.5.Google Scholar
Rindskopf, D., &Rose, T.(1988)Some theory and applications of confirmatory second- order factor analysis Multivariate Behavioral Research 23,5167.CrossRefGoogle Scholar
Schmiedek, F., &Li, S-C(2004)Toward an alternative representation for disentangling age-associated differences in general and specific cognitive abilities Psychology and Aging 19,4056.CrossRefGoogle ScholarPubMed
Schmidt, J., &Leiman, J. M.(1957)The development of hierarchical factor solutions Psychometrika 22,5361.CrossRefGoogle Scholar
Schönemann, P. H.(1966)A generalized solution of the orthogonal procrustes problem Psychometrika 31,110.CrossRefGoogle Scholar
Schönemann, P. H.(1985)On the formal differentiation of traces and determinants Multivariate Behavioral Research 20,113139.CrossRefGoogle ScholarPubMed
Thomson, G. H. (1939/1948). The factorial analysis of human ability. New York, New York:Houghton Mifflin.Google Scholar
Thurstone, L. L.(1947)Multiple-factor analysis Chicago, IL:University Chicago PressGoogle Scholar
Tucker, L. R.(1940)The role of correlated factors in factor analysis Psychometrika 5,141152.CrossRefGoogle Scholar
Yung, Y. F.,Thissen, D., &McLeod, L. D.(1999 On the relationship between the higher-order factor model and the hierarchical factor model Psychometrika)64,113128.CrossRefGoogle Scholar
Supplementary material: File

Electronic supplementary material

The online version of this article (https://doi.org/10.1007/s11336-017-9599-0) contains supplementary material, which is available to authorized users.
Download Electronic supplementary material(File)
File 85.5 KB