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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
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.

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