Using historical data for Bayesian sample size determination

We consider the sample size determination (SSD) problem, which is a basic yet extremely important aspect of experimental design. Specifically, we deal with the Bayesian approach to SSD, which gives researchers the possibility of taking into account pre-experimental information and uncertainty on unknown parameters. At the design stage, this fact offers the advantage of removing or mitigating typical drawbacks of classical methods, which might lead to serious miscalculation of the sample size. In this context, the leading idea is to choose the minimal sample size that guarantees a probabilistic control on the performance of quantities that are derived from the posterior distribution and used for inference on parameters of interest. We are concerned with the use of historical data-i.e. observations from previous similar studies-for SSD. We illustrate how the class of power priors can be fruitfully employed to deal with lack of homogeneity between historical data and observations of the upcoming experiment. This problem, in fact, determines the necessity of discounting prior information and of evaluating the effect of heterogeneity on the optimal sample size. Some of the most popular Bayesian SSD methods are reviewed and their use, in concert with power priors, is illustrated in several medical experimental contexts.