Creating Misspecified Models in Moment Structure Analysis
Authors: Lai Keke 
Published: January 2019

To understand how SEM methods perform in practice where models always have misfit, simulation studies often involve incorrect models. To create a wrong model, traditionally one specifies a perfect model first and then removes some paths. This approach becomes difficult or even impossible to implement in moment structure analysis and fails to control the amounts of misfit separately and precisely for the mean and covariance parts. Most importantly, this approach assumes a perfect model exists and wrong models can eventually be made perfect, whereas in practice models are all implausible if taken literally and at best provide approximations of the real world. To improve the traditional approach, we propose a more realistic and flexible way to create model misfit for multiple group moment structure analysis. Given (a) the model μ(·)\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{{{\upmu }}} (\cdot ) $$\end{document} and Σ(·)\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{{\Sigma }} (\cdot ) $$\end{document}, (b) population model parameters θ0\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document}, and (c) F1\documentclass[12pt]{minimal}
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\begin{document}$$F_1$$\end{document} and F2\documentclass[12pt]{minimal}
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\begin{document}$$F_2$$\end{document} specified by the researcher, our method creates μ∗\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and Σ∗\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} to simultaneously satisfy (a) θ0=argminF[μ∗,Σ∗;μ(·),Σ(·)]\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{{{\uptheta }}} _0 = \arg \min F[\varvec{{{\upmu }}} ^*, \varvec{{\Sigma }} ^*; \varvec{{{\upmu }}} (\cdot ), \varvec{{\Sigma }} (\cdot )]$$\end{document}, (b) the mean structure’s misfit equals F1\documentclass[12pt]{minimal}
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\begin{document}$$F_1$$\end{document}, and (c) the covariance structure’s misfit equals F2\documentclass[12pt]{minimal}
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\begin{document}$$F_2$$\end{document}.