Selecting the optimal longitudinal cluster randomized design with a continuous outcome: Parallel-arm, crossover, or stepped-wedge

The optimal designs (ODs) for parallel-arm longitudinal cluster randomized trials, multiple-period cluster randomized crossover (CRXO) trials, and stepped wedge cluster randomized trials (SW-CRTs), including closed-cohort and repeat cross-sectional designs, have been studied separately under a cost-efficiency framework based on generalized estimating equations (GEEs). However, whether a global OD exists across longitudinal designs and randomization schedules remains unknown. Therefore, this research addresses a critical gap by comparing OD feature across complete longitudinal cluster randomized trial designs with two treatment conditions and continuous outcomes. We define the OD as the design with either the lowest cost to obtain a desired level of power or the largest power given a fixed budget. For each of these ODs, we obtain the optimal number of clusters and the optimal cluster-period size (number of participants per cluster per period). To ensure equitable comparisons, we consider the GEE treatment effect estimator with the same block exchangeable correlation structure and develop OD algorithms with the lowest cost for each of six study designs. To obtain OD with the largest power, we summarize the previous and propose new OD algorithms and formulae. We suggest using the number of treatment sequences $L=T-1$, where T is the number of time-periods, in both the optimal closed-cohort and repeated cross-sectional SW-CRTs to have the lowest cost. This is consistent with our previous findings for ODs with the largest power in SW-CRTs. Comparing all six ODs, we conclude that optimal closed-cohort CRXO trials are global ODs, yielding both the lowest cost and largest power.