Efficient treatment allocation in 2 × 2 cluster randomized trials, when costs and variances are heterogeneous.

Typically, clusters and individuals in cluster randomized trials are allocated across treatment conditions in a balanced fashion. This is optimal under homogeneous costs and outcome variances. However, both the costs and the variances may be heterogeneous. Then, an unbalanced allocation is more efficient but impractical as the outcome variance is unknown in the design stage of a study. A practical alternative to the balanced design could be a design optimal for known and possibly heterogeneous costs and homogeneous variances. However, when costs and variances are heterogeneous, both designs suffer from loss of efficiency, compared with the optimal design. Focusing on cluster randomized trials with a 2 × 2 design, the relative efficiency of the balanced design and of the design optimal for heterogeneous costs and homogeneous variances is evaluated, relative to the optimal design. We consider two heterogeneous scenarios (two treatment arms with small, and two with large, costs or variances, or one small, two intermediate, and one large costs or variances) at each design level (cluster, individual, and both). Within these scenarios, we compute the relative efficiency of the two designs as a function of the extents of heterogeneity of the costs and variances, and the congruence (the cheapest treatment has the smallest variance) and incongruence (the cheapest treatment has the largest variance) between costs and variances. We find that the design optimal for heterogeneous costs and homogeneous variances is generally more efficient than the balanced design and we illustrate this theory on a trial that examines methods to reduce radiological referrals from general practices.