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.
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The authors would like to thank the reviewers for their useful comments for the revision of the manuscript. 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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Yung, YF., Thissen, D. & McLeod, L.D. On the relationship between the higher-order factor model and the hierarchical factor model. Psychometrika 64, 113–128 (1999). https://doi.org/10.1007/BF02294531
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DOI: https://doi.org/10.1007/BF02294531