Sample size determination for three-level randomized clinical trials with randomization at the first or second level

Clinical trials in the context of comparative effectiveness research (CER) are often conducted to evaluate health outcomes under real-world conditions and standard health care settings. In such settings, three-level hierarchical study designs are increasingly common. For example, patients may be nested within treating physicians, who in turn are nested within an urgent care center or hospital. While many trials randomize the third-level units (e.g., centers) to intervention, in some cases randomization may occur at lower levels of the hierarchy, such as patients or physicians. In this article, we present and verify explicit closed-form sample size and power formulas for three-level designs assuming randomization is at the first or second level. The formulas are based on maximum likelihood estimates from mixed-effect linear models and verified by simulation studies. Results indicate that even with smaller sample sizes, theoretical power derived with known variances is nearly identical to empirically estimated power for the more realistic setting when variances are unknown. In addition, we show that randomization at the second or first level of the hierarchy provides an increasingly statistically efficient alternative to third-level randomization. Power to detect a treatment effect under second-level randomization approaches that of patient-level randomization when there are few patients within each randomized second-level cluster and, most importantly, when the correlation attributable to second-level variation is a small proportion of the overall correlation between patient outcomes.