  In small sample studies with binary outcome data, use of a normal
approximation for hypothesis testing can lead to substantial inflation of the
type-I error-rate. Consequently, exact statistical methods are necessitated,
and accordingly, much research has been conducted to facilitate this. Recently,
this has included methodology for the design of two-stage multi-arm studies
utilising exact binomial tests. These designs were demonstrated to carry
substantial efficiency advantages over a fixed sample design, but generally
suffered from strong conservatism. An alternative classical means of small
sample inference with dichotomous data is Fisher's exact test. However, this
method is limited to single-stage designs when there are multiple arms.
Therefore, here, we propose a two-stage version of Fisher's exact test, with
the potential to stop early to accept or reject null hypotheses, which is
applicable to multi-arm studies. In particular, we provide precise formulae
describing the requirements for achieving weak or strong control of the
familywise error-rate with this design. Following this, we describe how the
design parameters may be optimised to confer desirable operating
characteristics. For a motivating example based on a phase II clinical trial,
we demonstrate that on average our approach is less conservative than
corresponding optimal designs based on exact binomial tests.
