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

University of Notre Dame, kyuan{at}nd.edu

Peter M. Bentler

University of California, Los Angeles

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 Ma be the base model within which model Mb is nested. In practice, this test is commonly used to justify Mb even when Ma 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 Mb. 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.

Key Words: chi-square difference • nested models • model misspecification • parameter bias • mean comparison • growth curves

Educational and Psychological Measurement, Vol. 64, No. 5, 737-757 (2004)
DOI: 10.1177/0013164404264853


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