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A multi-arm multi-stage trial is a multi-arm trial which includes interim analyses - analysing the data at certain specified points, generally discontinuing treatments which are concluded to not work and proceeding with the remainder. It is possible that the advantages of multi-arm trials over single-arm trials may be enhanced further by considering the allocation ratio, R. For an R:1 allocation ratio, Rn patients are allocated to the control arm and n patients allocated to each active treatment arm. In this study, the optimal allocation ratio will be defined as the allocation ratio which results in the smallest total sample size satisfying some required power and probability of type I error. This is an intuitive definition in the context of clinical trials, as a smaller trial will in general be more ethical and less expensive than a larger one satisfying the same error rates. The purpose of this paper is to investigate the optimal allocation ratio in the case of multiple active treatment arms. The setup for a single stage trial with K active treatment arms is described in Section 2, along with a brief exposition of Dunnett's statement regarding the optimal allocation ratio in such circumstances. Equations for type I error and power are derived, and the methodology used to investigate how total sample size may be minimised using allocation ratio is described. A two-stage trial is then considered, using the same methodology. Figures and tables showing how total sample size changes with allocation ratio, for a range of type I error and power values, are given in Section 3. The possible ethical and financial benefits of changing allocation ratio, including a simple example, is also included in Section 3. The results, and what they could mean in practical terms, are discussed in Section 4.