Unbalanced 2 x 2 factorial designs and the interaction effect: A troublesome combination

Abstract

In this power study, ANOVAs of unbalanced and balanced 2 x 2 datasets are compared (N = 120). Datasets are created under the assumption that H1 of the effects is true. The effects are constructed in two ways, assuming: 1. contributions to the effects solely in the treatment groups; 2. contrasting contributions in treatment and control groups. The main question is whether the two ANOVA correction methods for imbalance (applying Sums of Squares Type II or III; SS II or SS III) offer satisfactory power in the presence of an interaction. Overall, SS II showed higher power, but results varied strongly. When compared to a balanced dataset, for some unbalanced datasets the rejection rate of H0 of main effects was undesirably higher. SS III showed consistently somewhat lower power. When the effects were constructed with equal contributions from control and treatment groups, the interaction could be re-estimated satisfactorily. When an interaction was present, SS III led consistently to somewhat lower rejection rates of H0 of main effects, compared to the rejection rates found in equivalent balanced datasets, while SS II produced strongly varying results. In data constructed with only effects in the treatment groups and no effects in the control groups, the H0 of moderate and strong interaction effects was often not rejected and SS II seemed applicable. Even then, SS III provided slightly better results when a true interaction was present. ANOVA allowed not always for a satisfactory re-estimation of the unique interaction effect. Yet, SS II worked better only when an interaction effect could be excluded, whereas SS III results were just marginally worse in that case. Overall, SS III provided consistently 1 to 5% lower rejection rates of H0 in comparison with analyses of balanced datasets, while results of SS II varied too widely for general application.