Sequence balance minimisation: minimising with unequal treatment allocations

BACKGROUND Minimisation ensures excellent balance between groups for several prognostic factors, even in small samples. However, its use with unequal allocation ratios has been problematic. This paper describes a new minimisation scheme named sequence balance minimisation for unequal treatment allocations. METHODS Treatment- and factor-balancing properties were assessed in simulation studies for two- and three-arm trials with 1:2 and 1:2:3 allocation ratios. Sample sizes were set 30, 60 and 120. The number of prognostic factors on which to achieve balance was ranged from zero (treatment totals only) to ten with two levels occurring in equal probabilities. Random elements were set at 0.95, 0.9, 0.85, 0.80, 0.7, 0.6 and 0.5. Characteristics of the randomisation distributions and the impact of changing the block size while maintaining the allocation ratio were also examined. RESULTS Sequence balance minimisation has good treatment- and factor-balancing capabilities, and the randomisation distribution was centred at zero for all scenarios. The mean and median number of allocations achieved were the same as the number expected in most scenarios, and including additional factors (up to ten) in the minimisation scheme had little impact on treatment balance. Treatment balance tended to depart from the target as the random element was lowered. The variability in allocations achieved increased slightly as the number of factors increased, as the random element was decreased and as the sample size increased. The mean and median factor imbalance remained tightly around zero even when the chosen factor was not included in the minimisation scheme, though the variability was greater. The variability in factor imbalance increased slightly as the random element decreased, as well as when the number of prognostic factors and sample size increased. Increasing block size while maintaining the allocation ratio improved treatment balance notably with little impact on factor imbalance. CONCLUSIONS Sequence balance minimisation has good treatment- and factor-balancing properties and is particularly useful for small trials seeking to achieve balance across several prognostic factors.