How large should a clinical trial be?
One of the most important questions in the planning of medical experiments to assess the performance of new drugs or treatments, is how big to make the trial. The problem, in its statistical formulation, is to determine the optimal size of a trial. The most frequently used methods of determining sample size in clinical trials is based on the required p-value, and the required power of the trial for a specified treatment effect. In contrast to the Bayesian decision theoretic approach there is no explicit balancing of the cost of a possible increase in the size of the trial against the benefit of the more accurate information which it would give. In this work we consider a fully Bayesian (or decision theoretic) approach to sample size determination in which the number of subsequent users of the therapy under investigation, and hence also the total benefit resulting from the trial, depend on the strength of the evidence provided by the trial. Our procedure differs from the usual Bayesian decision theory methodology, which assumes a single decision maker, by recognizing the existence of three decision makers, namely: the pharmaceutical company conducting the trial, which decides on its size; the regulator, whose approval is necessary for the drug to be licenced for sale; and the public at large, who determine the ultimate usage. Moreover, we model the subsequent usage by plausible assumptions for actual behaviour, rather than assuming that this represents decisions which are in some sense optimal. For this reason the procedure may be called "Behavioural Bayes" (or BeBay for short), the word Bayes referring to the optimization of the sample size. In the BeBay methodology the total expected benefit from carrying out the trial minus the cost of the trial is maximized. For any additional sales to occur as a result of the trial it must provide sufficient evidence both to convince the regulator to issue the necessary licence and to convince potential users that they should use the new treatment. The necessary evidence is in the form of a high probability after the trial that the new treatment achieves a clinically relevant improvement compared to the alternative treatment. The regulator is assumed to start from a more sceptical and less well-informed view of the likely performance of the treatment than the company carrying out the trial. The total benefit from a conclusively favourable trial is assessed on the basis of the size of the potential market and aggregated over the anticipated life-time of the product, using appropriate discounting for future years.