Minimisation for the design of parallel cluster-randomised trials: An evaluation of balance in cluster-level covariates and numbers of clusters allocated to each arm

Background: Cluster-randomised trials often use some form of restricted randomisation, such as stratified- or covariate-constrained randomisation. Minimisation has the potential to balance on more covariates than blocked stratification and can be implemented sequentially unlike covariate-constrained randomisation. Yet, unlike stratification, minimisation has no inbuilt guard to maintain close to a 1:1 allocation. A departure from a 1:1 allocation can be unappealing in a setting with a small number of allocation units such as cluster randomisation which typically include about 30 clusters.

Methods: Using simulation (10,000 per scenario), we evaluate the performance of a range of minimisation procedures on the likelihood of a 1:1 allocation of clusters (10–80 clusters) to treatment arms, along with its performance on covariate imbalance. The range of minimisation procedures includes varying: the proportion of clusters allocated to the least imbalanced arm (known as the stochastic element) – between 0.7 and 1, percentage of first clusters allocated completely at random (known as the bed-in period) – between 0% and 20% and adding ‘number of clusters allocated to each arm’ as a covariate in the minimisation algorithm. We additionally include a comparison of stratifying and then minimising within key strata (such as country within a multi country cluster trial) as a potential aid to increasing balance.

Results: Minimisation is unlikely to result in an exact 1:1 allocation unless the stochastic element is set higher than 0.9. For example, with 20 clusters, 2 binary covariates and setting the stochastic element to 0.7: only 41% of the possible randomisations over the 10,000 simulations achieved a 1:1 allocation. While typical sizes of imbalance were small (a difference of two clusters per arm), allocations as extreme as of 10:10 were observed. Adding the ‘number of clusters’ into the minimisation algorithm reduces this risk slightly, but covariate imbalance increases slightly. Stratifying and then minimising within key strata improve balance within strata but increase imbalance across all clusters, both on the number of clusters and covariate imbalance.

Conclusion: In cluster trials, where there are typically about 30 allocation units, when using minimisation, unless the stochastic element is set very high, there is a high risk of not achieving a 1:1 allocation, and a small but nonetheless real risk of an extreme departure from a 1:1 allocation. Stratification with minimisation within key strata (such as country) improves the balance within strata although compromises overall balance.