F Tests for Mean and Covariance Structure Analysis

Abstract

Covariance structure analysis is used for inference and for dimension reduction with multivariate data. When data are not normally distributed, the asymptotic distribution free (ADF) method is often used to fit a proposed model. The ADF test statistic is asymptotically distributed as a chi-square variate. Experience with real data indicates that the ADF statistic tends to reject theoretically meaningful models. Empirical simulation shows that the ADF statistic rejects correct models too often for all but impractically large sample sizes. By comparing mean and covariance structure analysis with its analogue in the multivariate linear model, we propose some modified ADF test statistics whose distributions are approximated by F distributions. Empirical studies show that the distributions of the new statistics are more closely approximated by F distributions than are the original ADF statistics when referred to chi-square distributions. Detailed analysis indicates why the ADF statistic fails on large models and why F tests and corrections give better results. Implications for power analysis and model tests in other areas are discussed.