Optimal Allocation of Observations in Stepped-Wedge and Other Cluster Studies With Correlated Cluster-Period Effects

Stepped-wedge studies usually entail regular sampling of clusters over time. Yet the precision of the treatment effect estimator can sometimes be improved if the regular sampling scheme is replaced by one with preferential allocation of observations to particular time-epochs within each cluster. We present some exact results for optimizing the allocation for a general experimental layout under a mixed effects model with a time-varying cluster-autocorrelation structure, together with an algorithm for generating optimal allocations. An index of cluster variation is introduced, an increasing function of both the intra-class correlation and the total sample size, which encapsulates the influence of cluster-level variation on the optimal allocation. For any specified layout there is a sampling scheme (the 'best natural allocation') that solves the optimization problem for all values of this index up to a threshold value which depends only on the cluster autocorrelations. Under such a scheme the treatment effect estimator is equal to a simple difference between the means of the treated and control observations. Best natural allocations stand alongside conventional parallel and cross-over designs in giving equal weight to observations from all participants, even under stepped-wedge layouts with irreversible interventions. When applied to a recent study of primary care training programmes in low- and middle- income countries (The REaCH study), the results lead to substantial reductions in total sample size, without loss of precision. For stepped-wedge layouts with block-exchangeable or time-decaying cluster autocorrelations, we present explicit conditions for the optimality of staircase-type sampling schemes, which can arise as best natural allocations in such cases.