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.
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