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On the Relationship between the Higher-Order Factor Model and the Hierarchical Factor Model

Published online by Cambridge University Press:  01 January 2025

Yiu-Fai Yung,
David Thissen and
Lori D. McLeod
Yiu-Fai Yung*
Affiliation:
L. L. Thurstone Psychometric Laboratory, University of North Carolina at Chapel Hill
David Thissen
Affiliation:
L. L. Thurstone Psychometric Laboratory, University of North Carolina at Chapel Hill
Lori D. McLeod
Affiliation:
L. L. Thurstone Psychometric Laboratory, University of North Carolina at Chapel Hill
*
Requests for reprints should be sent to Yiu-Fai Yung, R52, Multivariate & Num. R&D, SAS Campus Drive, SAS Institute, Inc. Cary NC 27513.
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Abstract

The relationship between the higher-order factor model and the hierarchical factor model is explored formally. We show that the Schmid-Leiman transformation produces constrained hierarchical factor solutions. Using a generalized Schmid-Leiman transformation and its inverse, we show that for any unconstrained hierarchical factor model there is an equivalent higher-order factor model with direct effects (loadings) on the manifest variables from the higher-order factors. Therefore, the class of higher-order factor models (without direct effects of higher-order factors) is nested within the class of unconstrained hierarchical factor models. In light of these formal results, we discuss some implications for testing the higher-order factor model and the issue of general factor. An interesting aspect concerning the efficient fitting of the higher-order factor model with direct effects is noted.

Keywords

factor analysishigher-order factor modelshierarchical factor modelsbi-factor solutionsgeneral factormodel equivalence

Information

Type
Original Paper
Information
Psychometrika , Volume 64 , Issue 2 , June 1999, pp. 113 - 128
Copyright
Copyright © 1999 The Psychometric Society

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Footnotes

Yiu-Fai Yung is now at the SAS Institute, Inc. The authors would like to thank the reviewers for their useful comments for the revision of the manuscript.

References

Bentler, P. M. (1990). EQS: Structural equations program manual, Los Angeles: BMDP Statistical Software.Google Scholar
Ferguson, T. S. (1996). A course in large sample theory, London: Chapman and Hall.CrossRefGoogle Scholar
Gorsuch, (1983). Factor analysis 2nd ed.,, Hillsdale, NJ: Lawrence Earlbaum Associates.Google Scholar
Gustafsson, J., & Balke, G. (1993). General and specific abilities as predictors of school achievement. Multivariate Behavioral Research, 28, 407–434.CrossRefGoogle ScholarPubMed
Humphreys, L. G. (1981). The primary mental ability. In Friedman, M. P., Das, J. P., & O'Connor, N. (Eds.), Intelligence and learning (pp. 87–102). NY: Plenum.CrossRefGoogle Scholar
Humphreys, L. G., Tucker, L. R., & Dachler, P. (1970). Evaluating the importance of factors in any given order of factoring. Multivariate Behavioral Research, 5, 209–215.CrossRefGoogle ScholarPubMed
Holzinger, K. J., & Swineford, F. (1937). The bi-factor method. Psychometrika, 2, 41–54.CrossRefGoogle Scholar
Joreskög, K. G., & Sörbom, D. (1993). LISREL 8, Chicago: Scientific Software International, Inc..Google Scholar
McDonald, R. P. (1985). Factor analysis and related methods, Hillsdale, NJ: Lawrence Earlbaum Associates.Google Scholar
McDonald, R. P. (in press). Test theory: A unified treatment. Mahwah, NJ: Lawrence Earlbaum Associates.CrossRefGoogle Scholar
McLeod, L. D., & Thissen, D. (1997). Models for composite tests. (submitted for review)Google Scholar
Mulaik, S. A., & Quartetti, D. A. (1997). First order or higher order general factor. Structural Equation Modeling, 4, 193–211.CrossRefGoogle Scholar
SAS (1990). SAS/IML, Cary, NC: Author.Google Scholar
Schmid, J., & Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 83–90.CrossRefGoogle Scholar
Tucker, L. (1940). The role of correlated factors in factor analysis. Psychometrika, 5, 141–152.CrossRefGoogle Scholar
Wherry, R. J. (1959). Hierarchical factor solutions without rotation. Psychometrika, 24, 45–51.CrossRefGoogle Scholar