Tests that are robust against variance heterogeneity in k x 2 designs with unequal cell frequencies

Heterogeneity of variance produces serious bias in conventional analysis of variance tests of significance when cell frequencies are unequal. Welch in 1938 and 1947 proposed an adjusted t test for the difference between two means when cell frequencies and population variances are both unequal. This article describes two ways to use the Welch t to evaluate the significance of the main effect for two treatments across k levels of a concomitant factor in a two-way design. Monte Carlo results document the bias in conventional analysis of variance tests and the stable and appropriately conservative results from applications of the Welch t to evaluation of treatment effects in the two-way design.