A Scaled F Distribution as an Approximation to the Distribution of Test Statistics in Covariance Structure Analysis

Abstract

When the assumption of multivariate normality is violated or when a discrepancy function other than (normal theory) maximum likelihood is used in structural equation models, the null distribution of the test statistic may not be χ² distributed. Most existing methods to approximate this distribution only match up to 2 moments. In this article, we propose 2 additional approximation methods: a scaled F distribution that matches 3 moments simultaneously and a direct Monte Carlo–based weighted sum of i.i.d. χ² variates. We also conduct comprehensive simulation studies to compare the new and existing methods for both maximum likelihood and nonmaximum likelihood discrepancy functions and to separately evaluate the effect of sampling uncertainty in the estimated weights of the weighted sum on the performance of the approximation methods.

Keywords:

• diagonally weighted least square
• maximum Wishart likelihood
• robust statistics
• Satorra–Bentler corrections

ACKNOWLEDGMENTS

We would like to thank the reviewers at the Educational Testing Service for their helpful comments on this article and thank Dr. P. M. Bentler for bringing our attention to some of the important papers relevant to this work.