Statistical power in two-level hierarchical linear models with arbitrary number of factor levels

As the US health care system undergoes unprecedented changes, the need for adequately powered studies to understand the multiple levels of main and interaction factors that influence patient and other care outcomes in hierarchical settings has taken center stage. We consider two-level models where n lower-level units are nested within each of J higher-level clusters (e.g. patients within practices and practices within networks) and where two factors may have arbitrary a and b factor levels, respectively. Both factors may represent a×b treatment combinations, or one of them may be a pretreatment covariate. Consideration of both factors at the same higher or lower hierarchical level, or one factor per hierarchical level yields a cluster (C), multisite (M) or split-plot randomized design (S). We express statistical power to detect main, interaction, or any treatment effects as a function of sample sizes (n,J), a and b factor levels, intraclass correlation ρ and effect sizes δ given each design d∈{C,M,S}. The power function given a,b,ρ, δ and d determines adequate sample sizes to achieve a minimum power requirement. Next, we compare the impact of the designs on power to facilitate selection of optimal design and sample sizes in a way that minimizes the total cost given budget and logistic constraints. Our approach enables accurate and conservative power computation with a priori knowledge of only three effect size differences regardless of how large a×b is, simplifying previously available computation methods for health services and other researches.