Introduction of a new measure for detecting poor fit due to omitted nonlinear terms in SEM
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The model chi-square that is used in linear structural equation modeling compares the fitted covariance matrix of a target model to an unstructured covariance matrix to assess global fit. For models with nonlinear terms, i.e., interaction or quadratic terms, this comparison is very problematic because these models are not nested within the saturated model that is represented by the unstructured covariance matrix. We propose a novel measure that quantifies the heteroscedasticity of residuals in structural equation models. It is based on a comparison of the likelihood for the residuals under the assumption of heteroscedasticity with the likelihood under the assumption of homoscedasticity. The measure is designed to respond to omitted nonlinear terms in the structural part of the model that result in heteroscedastic residual scores. In a small Monte Carlo study, we demonstrate that the measure appears to detect omitted nonlinear terms reliably when falsely a linear model is analyzed and the omitted nonlinear terms account for substantial nonlinear effects. The results also indicate that the measure did not respond when the correct model or an overparameterized model were used.
KeywordsHeteroscedasticity Monte Carlo study Nonlinear model Structural equation modeling Interaction effect Quadratic effect
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