  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.
