Probing Categorical Moderation Under Variance Heterogeneity

Numerous psychological and educational researchers are concerned with the moderating effect of a categorical variable on the relationship between a continuous predictor and a continuous response. Despite the great interest in moderation analyses, there exist few studies that explicitly address the essential issues of effect size measure, power calculation, and sample size determination under heterogeneity of variance. This article introduces a useful effect size index for representing the moderating effects of categorical variables. Based on the extended Welch statistic, a nearly unbiased estimator and related effect size measures are presented. To facilitate power and sample size computations, different approximations are considered for the general distribution of the extended Welch test of moderating effects. Detailed numerical appraisals are conducted to examine the relative performance of the described effect size measures, power formulas, and sample size procedures under various variance structures and model configurations. The presented techniques and computer algorithms enhance the methodological development and practical value of research examining moderation.