Power of a test that is robust against variance heterogeneity

Welch (1947) proposed an adjusted t test that can be used to correct the serious bias in Type I error protection that is otherwise present when both sample sizes and variances are unequal. The implications of the Welch adjustment for power of tests for the difference between two treatments across k levels of a concomitant factor are evaluated in this article for k x 2 designs with unequal sample sizes and unequal variances. Analyses confirm that, although Type I error is uniformly controlled, power of the Welch test of significance for the main effect of treatments remains rather seriously dependent on direction of the correlation between unequal variances and unequal sample sizes. Nevertheless, considering the fact that analysis of variance is not an acceptable option in such cases, the Welch t test appears to have an important role to play in the analysis of experimental data.