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On Nonequivalence of Several Procedures of Structural Equation Modeling

Published online by Cambridge University Press:  01 January 2025

Ke-Hai Yuan  and
Wai Chan
Ke-Hai Yuan*
Affiliation:
University of Notre Dame
Wai Chan
Affiliation:
The Chinese University of Hong Kong
*
Requests for reprints should be sent to Ke-Hai Yuan, Department of Psychology, University of Notre Dame, Notre Dame, IN 46556, USA. Email: kyuan@nd.edu
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Abstract

The normal theory based maximum likelihood procedure is widely used in structural equation modeling. Three alternatives are: the normal theory based generalized least squares, the normal theory based iteratively reweighted least squares, and the asymptotically distribution-free procedure. When data are normally distributed and the model structure is correctly specified, the four procedures are asymptotically equivalent. However, this equivalence is often used when models are not correctly specified. This short paper clarifies conditions under which these procedures are not asymptotically equivalent. Analytical results indicate that, when a model is not correct, two factors contribute to the nonequivalence of the different procedures. One is that the estimated covariance matrices by different procedures are different, the other is that they use different scales to measure the distance between the sample covariance matrix and the estimated covariance matrix. The results are illustrated using real as well as simulated data. Implication of the results to model fit indices is also discussed using the comparative fit index as an example.

Keywords

generalized least squaresincorrect modelincremental fit indexmaximum likelihoodnoncentrality parameter
Type
Notes and Comments
Information
Psychometrika , Volume 70 , Issue 4 , December 2005 , pp. 791 - 798
Copyright
Copyright © 2005 The Psychometric Society

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Footnotes

The work described in this paper was supported by a grant from the Research Grants Council of Hong Kong Special Administrative Region (Project No. CUHK 4170/99M) and by NSF grant DMS04-37167.

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