Cluster randomized trials evaluate the effect of a treatment on persons nested within clusters, with clusters being randomly assigned to treatment. The optimal sample size at the cluster and person level depends on the study cost per cluster and per person, and the outcome variance at the cluster and the person level. The variances are unknown in the design stage and can differ between treatment arms. As a solution, this paper presents a Maximin design that maximizes the minimum relative efficiency (relative to the optimal design) over the variance parameter space, for trials with two treatment arms and a quantitative outcome. This maximin relative efficiency design (MMRED) is compared with a published Maximin design which maximizes the minimum efficiency (MMED). Both designs are also compared with the optimal designs for homogeneous costs and variances (balanced design) and heterogeneous costs and homogeneous variances (cost-conscious design), for a range of variances based upon three published trials. Whereas the MMED is balanced under high uncertainty about the treatment-to-control variance ratio, the MMRED then tends towards a balanced budget allocation between arms, leading to an unbalanced sample size allocation if costs are heterogeneous, similar to the cost-conscious design. Further, the MMRED corresponds to an optimal design for an intraclass correlation (ICC) in the lower half of the assumed ICC range (optimistic), whereas the MMED is the optimal design for the maximum ICC within the ICC range (pessimistic). Attention is given to the effect of the Welch–Satterthwaite degrees of freedom for treatment effect testing on the design efficiencies.

中文翻译：

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具有异质成本和差异的集群随机试验的最大化设计

集群随机试验评估治疗对嵌套在集群中的人的影响，集群被随机分配到治疗。集群和人员级别的最佳样本量取决于每个集群和每个人的研究成本，以及集群和人员级别的结果方差。差异在设计阶段是未知的，并且在治疗组之间可能不同。作为一种解决方案，本文提出了一种 Maximin 设计，该设计可最大化方差参数空间上的最小相对效率（相对于最佳设计），用于具有两个治疗臂和定量结果的试验。将此最大最小相对效率设计 (MMRED) 与已发布的最大化最小效率 (MMED) 的最大化设计进行比较。两种设计还与基于三项已发表试验的一系列差异的同质成本和差异（平衡设计）和异质成本和同质差异（成本意识设计）的最佳设计进行了比较。虽然 MMED 在治疗与对照方差比的高度不确定性下是平衡的，但 MMRED 然后倾向于平衡臂之间的预算分配，如果成本异质，则导致样本量分配不平衡，类似于成本意识设计。此外，MMRED 对应于假设 ICC 范围下半部分的类内相关 (ICC) 的最佳设计（乐观），而 MMED 是 ICC 范围内最大 ICC 的最佳设计（悲观）。