Optimal Sample Allocation for Three-Level Multisite Cluster-Randomized Trials

Optimal sampling frameworks attempt to identify the most efficient sampling plans to achieve an adequate statistical power. Although such calculations are theoretical in nature, they are critical to the judicious and wise use of funding because they serve as important starting points that guide practical discussions around sampling tradeoffs and requirements. Conventional optimal sampling frameworks, however, often identify sub-optimal designs because they typically presume the costs of sampling units are equal across treatment conditions. In this study, we develop a more flexible framework that allows costs to differ by treatment conditions and derive the optimal sample size formulas for three-level multisite cluster-randomized trials. We find that the proposed optimal sampling schemes are driven by the differences in costs between treatment conditions, cross-level sampling cost ratios and cross-level variance decomposition ratios. We illustrate the utility of the proposed framework by comparing it to a conventional framework and find that the proposed framework frequently identifies more efficient designs. The proposed optimal sampling framework has been implemented in the R package odr.