On sample size calculation for 2x2 fixed-effect ANOVA when variances are unknown and possibly unequal

Abstract

The factorial 2 x 2 fixed-effect ANOVA is a procedure used frequently in scientific research to test mean differences between-subjects in all of the groups. But if the assumption of homogeneity is violated, the test for the row, column, and the interaction effect might be invalid or less powerful. Therefore, for planning research in the case of unknown and possibly unequal variances, it is worth developing a sample size formula to obtain the desired power. This article suggests a simple formula to determine the sample size for 2 x 2 fixed-effect ANOVA for heterogeneous variances across groups. We use the approximate Welch t test and consider the variance ratio to derive the formula. The sample size determination requires two-step iterations but the approximate sample sizes needed for the main effect and the interaction effect can be determined separately with the specified power. The present study also provides an example and a SAS program to facilitate the calculation process.