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Higher-Order Approximations to the Distributions of Fit Indexes Under Fixed Alternatives in Structural Equation Models

Published online by Cambridge University Press:  01 January 2025

Haruhiko Ogasawara
Haruhiko Ogasawara*
Affiliation:
Otaru University of Commerce
*
Requests for reprints should be sent to Haruhiko Ogasawara, Department of Information and Management Science, Otaru University of Commerce, 3-5-21, Midori, Otaru 047-8501, Japan. E-mail: [email protected]
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Abstract

Higher-order approximations to the distributions of fit indexes for structural equation models under fixed alternative hypotheses are obtained in nonnormal samples as well as normal ones. The fit indexes include the normal-theory likelihood ratio chi-square statistic for a posited model, the corresponding statistic for the baseline model of uncorrelated observed variables, and various fit indexes as functions of these two statistics. The approximations are given by the Edgeworth expansions for the distributions of the fit indexes under arbitrary distributions. Numerical examples in normal and nonnormal samples with the asymptotic and simulated distributions of the fit indexes show the relative inappropriateness of the normal-theory approximation using noncentral chi-square distributions. A simulation for the confidence intervals of the fit indexes based on the normal-theory Studentized estimators under normality with a small sample size indicates an advantage for the approximation by the Cornish–Fisher expansion over those by the noncentral chi-square distribution and the asymptotic normality.

Keywords

fixed alternativesfit indexesnonnormal distributionsnoncentral chi-square distributionsEdgeworth expansionstructural equation modelingRMSEA
Type
Original Paper
Information
Psychometrika , Volume 72 , Issue 2 , June 2007 , pp. 227 - 243
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

The author is indebted to the reviewers for their comments and suggestions, which have led to the improvement of the previous versions of this paper. This work was partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science, and Technology.

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