Sample sizes required to detect two-way and three-way interactions involving slope differences in mixed-effects linear models

Based on maximum likelihood estimates obtained from mixed-effects linear models, closed-form power functions are derived to detect two-way and three-way interactions that involve longitudinal course of outcome over time in clinical trials. Sample size estimates are shown to decrease with increasing within-subject correlations. It is further shown that when clinical trial designs are balanced in group sizes, the sample size required to detect an effect size for a three-way interaction is exactly fourfold that required to detect the same effect size of a two-way interaction. Furthermore, this fourfold relationship virtually holds for unbalanced allocations of subjects if one factor is balanced in the three-way interaction model. Simulations are presented that verify the sample size estimates for two-way and three-way interactions.