Relative efficiency of unequal cluster sizes in stepped wedge and other trial designs under longitudinal or cross-sectional sampling

Background: A cluster trial with unequal cluster sizes often has lower precision than one with equal clusters, with a corresponding inflation of the design effect. For parallel group trials, adjustments to the design effect are available under sampling models with a single intracluster correlation. Design effects for equal clusters under more complex scenarios have appeared recently (including stepped wedge trials under cross-sectional or longitudinal sampling). We investigate the impact of unequal cluster size in these more general settings. Results: Assuming a linear mixed model with an exchangeable correlation structure that incorporates cluster and subject autocorrelation, we compute the relative efficiency (RE) of a trial with clusters of unequal size under a size-stratified randomization scheme, as compared to an equal cluster trial with the same total number of observations. If there are no within-cluster time effects, the RE exceeds that for a parallel trial. In general, the RE is a weighted average of the RE for a parallel trial and the RE for a crossover trial in the same clusters. Existing approximations for parallel designs are extended to the general setting. Increasing the cluster size by the factor (1 + CV2), where CV is the coefficient of variation of cluster size, leads to conservative sample sizes, as in a popular method for parallel trials. Conclusion: Methods to assess experimental precision for single-period parallel trials with unequal cluster sizes can be extended to stepped wedge and other complete layouts under longitudinal or cross-sectional sampling. In practice, the loss of precision due to unequal cluster sizes is unlikely to exceed 12%.