How many times should a cluster randomized crossover trial cross over?

Trial planning requires making efficient yet practical design choices. In a cluster randomized crossover trial, clusters of subjects cross back and forth between implementing the control and intervention conditions over the course of the trial, with each crossover marking the start of a new period. If it is possible to set up such a trial with more crossovers, a pertinent question is whether there are efficiency gains from clusters crossing over more frequently, and if these gains are substantial enough to justify the added complexity and cost of implementing more crossovers. We seek to determine the optimal number of crossovers for a fixed trial duration, and then identify other highly efficient designs by allowing the total number of clusters to vary and imposing thresholds on maximum cost and minimum statistical power. Our results pertain to trials with continuous recruitment and a continuous primary outcome, with the treatment effect estimated using a linear mixed model. To account for the similarity between subjects' outcomes within a cluster, we assume a correlation structure in which the correlation decays gradually in a continuous manner as the time between subjects' measurements increases. The optimal design is characterized by crossovers between the control and intervention conditions with each successive subject. However, this design is neither practical nor cost-efficient to implement, nor is it necessary: the gains in efficiency increase sharply in moving from a two-period to a four-period trial design, but approach an asymptote for the scenarios considered as the number of crossovers continues to increase.