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

Abstract

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.

Moment Testing for Interaction Terms in Structural Equation Modeling

“…For Theorem 1 to hold, a basic condition is that the structural part of the model is saturated; that is, using the language of Mooijaart and Satorra (2009), the degrees of freedom of the structural part of the model need to be zero. The measurement part of the model can, however, have restrictions so that the degrees of freedom for the whole model can in fact be large.…”

“…Consider a null model H 0 with only linear terms, that is, Γ 2 equal to zero. We note that, in contrast with Mooijaart and Satorra (2009), σ 3 is now present in the analysis, and that θ 3 is present or not depending on whether the model fitted is H 0 or H 1 . It holdsσ = σ 12,1σ12,3 σ 3,1σ3,3 , whereσ 12,1 andσ 3,1 are, respectively, the Jacobian of σ 12 and σ 3 with respect to θ 1 ; andσ 12,3 andσ 3,3 are, respectively, the Jacobian of σ 12 and σ 3 with respect to the interaction term parameters θ 3 .…”

“…This will guarantee that the models H 0 and H 1 are equivalent at the level of first-and second-order moments (see Mooijaart & Satorra, 2009). …”

“…The main conclusion is that the selection of third-order moments can be based on the power of a moment test, thus assessing the relevance in the analysis of different sets of third-order moments can be computationally simple. The paper gives an illustration of the method and argues for the need of refraining from adding into the analysis an excess of higher-order moments.In Mooijaart and Satorra (2009) it has been shown that, under some general conditions, the normal theory test statistics, which are based on means and covariances only, are not able to assess interactions among observable or latent variables of the model. One conclusion is that for analyzing models with interactions more information than just means and covariances need to be brought into the analysis.…”

“…In Mooijaart and Satorra (2009) it has been shown that, under some general conditions, the normal theory test statistics, which are based on means and covariances only, are not able to assess interactions among observable or latent variables of the model. One conclusion is that for analyzing models with interactions more information than just means and covariances need to be brought into the analysis.…”