Multivariate Test Statistics and Their Approximations: Some Problems

Abstract

This Monte Carlo study examines whether, given various numbers of variables, treatments, and sample sizes, in a one-way multivariate analysis of variance, Type I error rates of the test approximations provided by the BMDP program, the Statistical Analysis System (SAS), and the Statistical Package for the Social Sciences (SPSS) for Roy's largest root, Hotelling's trace, Wilks' likelihood ratio, and P. C. S. Pillai's trace meet the stringent criterion of J. V. Bradley (1978) for robustness. For each of 85 conditions, the Type I error rate for each of the above statistical tests was estimated based on 100,000 random samples per situation. Results indicate that in multivariate analysis of variance studies with relatively small numbers of subjects of around 15 per treatment level, or fewer, the current probability levels reported by SAS and SPSS are conservative for the F approximations based on Pillai's trace and liberal for the F approximations based on Hotelling's trace. The BMDP 4V program does not report Pillai's trace and reports accurate probability values for Hotelling's trace. All of these programs report accurate values for the F approximations based on Wilks' likelihood ratio and for Roy's largest root (SPSS does not report Roy's largest root). Recommendations are made for specific conditions. Two tables and five figures present analysis results. (SLD)