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On Insensitivity of the Chi-Square Model Test to Nonlinear Misspecification in Structural Equation Models

Published online by Cambridge University Press:  01 January 2025

Ab Mooijaart  and
Albert Satorra
Ab Mooijaart*
Affiliation:
Leiden University, Leiden
Albert Satorra
Affiliation:
Universitat Pompeu Fabra, Barcelona
*
Requests for reprints should be sent to Ab Mooijaart, Department of Psychology, Leiden University, P.O. Box 9555, 2300 RB Leiden, The Netherlands. E-mail: Mooijaart@Fsw.LeidenUniv.nl
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Abstract

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In this paper, we show that for some structural equation models (SEM), the classical chi-square goodness-of-fit test is unable to detect the presence of nonlinear terms in the model. As an example, we consider a regression model with latent variables and interactions terms. Not only the model test has zero power against that type of misspecifications, but even the theoretical (chi-square) distribution of the test is not distorted when severe interaction term misspecification is present in the postulated model. We explain this phenomenon by exploiting results on asymptotic robustness in structural equation models. The importance of this paper is to warn against the conclusion that if a proposed linear model fits the data well according to the chi-quare goodness-of-fit test, then the underlying model is linear indeed; it will be shown that the underlying model may, in fact, be severely nonlinear. In addition, the present paper shows that such insensitivity to nonlinear terms is only a particular instance of a more general problem, namely, the incapacity of the classical chi-square goodness-of-fit test to detect deviations from zero correlation among exogenous regressors (either being them observable, or latent) when the structural part of the model is just saturated.

Keywords

structural equation modelingtesting model fitnonlinear relationsinteraction termsequivalent modelsasymptotic robustnesssaturated model
Type
Theory and Methods
Information
Psychometrika , Volume 74 , Issue 3 , September 2009 , pp. 443 - 455
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NC
This article distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Copyright
Copyright © 2009 The Psychometric Society

Footnotes

Research of the second author is supported by the grants SEJ2006-13537 and PR2007-0221 from the Spanish Ministry of Science and Technology.

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