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Experimental design represents the typical context in which the interplay between Bayesian and frequentist methodology is natural and useful. Before the data are observed, it is licit and unavoidable even for a Bayesian statistician to take into account sample variability for the evaluation of statistical procedures and for decision making. At the same time, design planning fatally involves a number of pre-experimental choices that even a frequentist statistician is forced to make, preferably by exploiting external sources of knowledge. In this paper we discuss this mutual exchange between Bayesian and frequentist methodology, with specific focus on the primary crucial aspect of experimental designs, that is sample size determination (SSD). We review this topic by highlighting how the interplay between two prior distributions helps in managing the close relationship between the two approaches. Although the distinction between decisional and performance-based methods for Bayesian SSD is discussed, the main interest of this article is on the latter. We propose a general framework that includes several performance-based methods as special cases and thus makes the comparison of their characteristics easier. Finally, we extend the overview to robust methods for Bayesian SSD that allow to deal with the critical issue of sensitivity to prior elicitation. Illustrative examples are provided for normal models.

In sequential experiments the sample size is not planned in advance. Data are progressively collected and a stopping rule based on the observed results is defined in order to terminate the study. In a Bayesian framework, it is straightforward to monitor an ongoing experiment looking at the posterior probability that a parameter of interest $$\theta $$ , belongs to a given set. Specifically, in this paper we focus on the context of phase II clinical trials, where $$\theta $$ represents treatment efficacy. The Bayesian stopping rule we adopt involves the posterior probability that $$\theta $$ exceeds a clinically relevant threshold. Moreover, we introduce a robust version of this criterion by replacing the single prior distribution with a class of prior distributions. A simulation study is performed to compare the average sample sizes of the robust sequential approach both with the sample sizes of the non robust approach and of the non sequential approach. An interesting result is that, when the class of prior distributions is sufficiently narrow, the average sample sizes of the robust sequential approach can be smaller than the non sequential sample sizes.