Abstract
There are sometimes cost, scientific, or logistical reasons to allocate individuals unequally in an individually randomized trial. In cluster randomized trials we can allocate clusters unequally and/or allow different cluster size between trial arms. We consider parallel group designs with a continuous outcome, and optimal designs that require the smallest number of individuals to be measured given the number of clusters. Previous authors have derived the optimal allocation ratio for clusters under different variance and/or intracluster correlations (ICCs) between arms, allowing different but prespecified cluster sizes by arm. We derive closed-form expressions to identify the optimal proportions of clusters and of individuals measured for each arm, thereby defining optimal cluster sizes, when cluster size can be chosen freely. When ICCs differ between arms but the variance is equal, the optimal design allocates more than half the clusters to the arm with the higher ICC, but (typically only slightly) less than half the individuals and hence a smaller cluster size. We also describe optimal design under constraints on the number of clusters or cluster size in one or both arms. This methodology allows trialists to consider a range for the number of clusters in the trial and for each to identify the optimal design. Except if there is clear prior evidence for the ICC and variance by arm, a range of values will need to be considered. Researchers should choose a design with adequate power across the range, while also keeping enough clusters in each arm to permit the intended analysis method.
Highlights
In individually randomized trials it has long been known that it is more efficient to randomize to arms unequally when the variance of the outcome differs between arms.[1]
We extend our example based on the PA4E1 trial to consider how best to design a trial if there is uncertainty in the values of intracluster correlations (ICCs) expected for each trial arm, as there could well be in the PA4E1 trial
Our research has revealed important novel and simple findings concerning optimal trial design where cluster size can be chosen freely by the researchers, identifying designs with unequal allocation when the ICC or variance differ between arms
Summary
In individually randomized trials it has long been known that it is more efficient to randomize to arms unequally when the variance of the outcome differs between arms.[1]. Our methodology may still be helpful in this case because it allows trialists to see how to minimize the number of individuals that need to be measured in the trial, where outcomes are not routinely collected and only a random sample of those exposed are measured. Such a design is common in community randomized trials where the number of individuals exposed per cluster may be very large as in the SNEHA-TARA trial.[7] We use the term cluster size in this article to denote the number of individuals measured in a cluster, and we assume all clusters provide some measurements. We show how to calculate sample size, and illustrate our approach through considering how the PA4E1 trial might have been designed under a range of hypothetical (but realistic) alternative scenarios leading to unequal allocation
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