On Chi-Square Difference and z Tests in Mean and Covariance Structure Analysis when the Base Model is Misspecified

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Abstract
In mean and covariance structure analysis, the chi-square difference test is often applied to evaluate the number of factors, cross-group constraints, and other nested model comparisons. Let model M a be the base model within which model M b is nested. In practice, this test is commonly used to justify M b even when M a is misspecified. The authors study the behavior of the chi-square difference test in such a circumstance. Monte Carlo results indicate that a nonsignificant chi-square difference cannot be used to justify the constraints in M b. They also show that when the base model is misspecified, the z test for the statistical significance of a parameter estimate can also be misleading. For specific models, the analysis further shows that the intercept and slope parameters in growth curve models can be estimated consistently even when the covariance structure is misspecified, but only in linear growth models. Similarly, with misspecified covariance structures, the mean parameters in multiple group models can be estimated consistently under null conditions.