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

机译：卡方检验对结构方程模型中非线性错误指定的不敏感性

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摘要

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.
机译：在本文中，我们表明对于某些结构方程模型（SEM），经典的卡方拟合优度检验无法检测模型中非线性项的存在。例如，我们考虑具有潜在变量和交互作用项的回归模型。当假定的模型中存在严重的交互作用术语错误指定时，不仅模型测试对这种类型的错误指定具有零功效，而且即使是理论（卡方）分布也不会失真。我们通过利用结构方程模型中渐近鲁棒性的结果来解释这种现象。本文的重要性是要对以下结论提出警告：如果根据卡方拟合优度检验，所提出的线性模型可以很好地拟合数据，则基础模型的确是线性的。结果表明，底层模型实际上可能是严重非线性的。此外，本文表明，这种对非线性项的不敏感性只是更普遍问题的一个特殊情况，即经典卡方拟合优度测试无法检测外生回归变量之间的零相关偏差（当模型的结构部分刚刚饱和时（可以观察到或潜在）。

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