Optimal Designs in Open-Cohort Longitudinal Cluster Randomized Trials With a Continuous Outcome

Although sample size calculation for open-cohort longitudinal cluster randomized trials (LCRTs) under a fixed design framework was developed by Kasza et al., unifying the closed-cohort and repeated cross-sectional sampling provided in Hooper et al. when a churn rate is constant, there has been no prior efforts in developing optimal open-cohort LCRTs that maximizes the design efficiency. This work assumes a prespecified number of periods $T$ and a constant number of replaced individuals at each period in open-cohort LCRTs. We propose algorithms for deriving optimal sample size under a cost-efficiency framework and arrive at the local optimal design (LOD) for fixed correlation parameters and MaxiMin optimal design for addressing uncertainty in correlation parameters. When correlation parameters are known, as the number of replaced individuals increases, for open-cohort PA-LCRTs, the optimal cluster-period size generally decreases and then increases whereas the optimal number of clusters and power under LOD first increase and then decrease. In contrast, for CRXO trials and standard SW-CRTs, the optimal cluster-period size and churn rate under LOD increase whereas the optimal number of clusters and power under LOD decrease. When correlation parameters are unknown, but the parameter space is available, with a small number of replaced individuals, there is no difference in optimal designs between PA-LCRTs and CRXO trials. The number of replaced individuals also has less impact on the optimal cluster-period size than optimal number of clusters. We demonstrate our new optimal design methods using the context of two real-world LCRTs.