On the relationship between the higher-order factor model and the hierarchical factor model (original) (raw)
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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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.
Google 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.
Google Scholar - Humphreys, L. G. (1981). The primary mental ability. In M. P. Friedman, J. P. Das, & N. O'Connor (Eds.):Intelligence and learning (pp. 87–102). NY: Plenum.
Google 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.
Google Scholar - Holzinger, K. J., & Swineford, F. (1937). The bi-factor method.Psychometrika, 2, 41–54.
Google 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.
- McLeod, L. D., & Thissen, D. (1997). Models for composite tests. (submitted for review)
- Mulaik, S. A., & Quartetti, D. A. (1997). First order or higher order general factor.Structural Equation Modeling, 4, 193–211.
Google Scholar - SAS. (1990)SAS/IML (Version 6). Cary, NC: Author.
Google Scholar - Schmid, J., & Leiman J. M. (1957). The development of hierarchical factor solutions.Psychometrika, 22, 83–90.
Google Scholar - Tucker, L. (1940). The role of correlated factors in factor analysis.Psychometrika, 5, 141–152.
Google Scholar - Wherry, R. J. (1959). Hierarchical factor solutions without rotation.Psychometrika, 24, 45–51.
Google Scholar
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Author notes
- Yiu-Fai Yung
Present address: SAS Institute, Inc., R52, Multivariate & Num. R&D, SAS Campus Drive, 27513, Cary, NC
Authors and Affiliations
- L. L. Thurstone Psychometric Laboratory, University of North Carolina at Chapel Hill, USA
Yiu-Fai Yung, David Thissen & Lori D. McLeod
Authors
- Yiu-Fai Yung
- David Thissen
- Lori D. McLeod
Additional information
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
- Received: 22 December 1997
- Revised: 03 July 1998
- Issue date: June 1999
- DOI: https://doi.org/10.1007/BF02294531