Univariate and Multivariate Omnibus Hypothesis Tests Selected to Control Type I Error Rates When Population Variances Are Not Necessarily Equal

Abstract

When independent random samples are selected from normal (multivariate normal) populations with equal variances (covariance matrices) to test the equality of population means (mean vectors), the choice at each level of the omnibus hypothesis is clear: independent samples t, ANOVA F, Hotelling’s T², or MANOVA. Population variances (covariance matrices) that are not necessarily equal, however, cloud the picture. In terms of maximizing power while adequately controlling Type I error rates over the widest variety of conditions, empirical literature suggests use of (a) the Wilcox (1992) H to test the univariate H₀: μ₁ = μ₂, (b) the Wilcox (1993a) Z to test the univariate H₀: μ₁ = μ₂ = ... = μ_G, (c) either the Kim (1992), the James (1954) second-order, or the Johansen (1980) procedure to test the multivariate H₀: μ₁ = μ_2, and (d) either the Coombs-Algina U* (Coombs & Algina, in press-a), the James (1954) second-order, or the Johansen (1980) procedure to test the multivariate H₀: μ₁ = μ₂ = ... = μ_G.