  This paper develops Bayesian sample size formulae for experiments comparing
two groups. We assume the experimental data will be analysed in the Bayesian
framework, where pre-experimental information from multiple sources can be
represented into robust priors. In particular, such robust priors account for
preliminary belief about the pairwise commensurability between parameters that
underpin the historical and new experiments, to permit flexible borrowing of
information. Averaged over the probability space of the new experimental data,
appropriate sample sizes are found according to criteria that control certain
aspects of the posterior distribution, such as the coverage probability or
length of a defined density region. Our Bayesian methodology can be applied to
circumstances where the common variance in the new experiment is known or
unknown. Exact solutions are available based on most of the criteria considered
for Bayesian sample size determination, while a search procedure is described
in cases for which there are no closed-form expressions. We illustrate the
application of our Bayesian sample size formulae in the setting of designing a
clinical trial. Hypothetical data examples, motivated by a rare-disease trial
with elicitation of expert prior opinion, and a comprehensive performance
evaluation of the proposed methodology are presented.
