Robustness of some estimators for the analysis of covariance structures.

A sampling experiment was designed to evaluate the robustness of some estimators used in the analysis of covariance structures to misspecification of the discrepancy function in finite samples. The estimators were studied under four distribution (a multivariate normal, an elliptical, a non-elliptically symmetric, and an asymmetric multivariate distribution, and eight sample size (75, 150, 300, 600, 1200, 2400, 4800, 9600) conditions. Parameters of the composite direct product model for a 6-variable MTMM matrix were estimated using the Maximum Wishart Likelihood (MWL), generalized least squares (GLS), asymptotically distribution free (ADF), and approximate ADF (DADF) estimators. Browne's elliptical correction (CMWL) to the test statistic and standard errors for scale-invariant models was also made for each sample. Each of the 32 distribution-sample size conditions was replicated 300 times. The accuracy of the parameter estimates and estimated standard errors and the distribution of the test statistic were studied. The theoretically predicted importance of information contained in higher-order moments of arbitrary distributions was confirmed. Robustness of the MWL and normal theory GLS estimators cannot be taken for granted. Elliptical corrections may be a practical solution to some data analytic problems.