Construction of an optimal sequential plan for testing a treatment for an adverse effect

Classical sequential procedures that collect a single observation at a time are often found impractical, expensive, and time consuming. Sequentially planned procedures, or simply sequential plans, extend and generalize the concepts of sequential analysis by allowing observations to be collected in groups of variable sizes. After every group, all of the previously collected data are used to determine the next course of action. An optimal (Bayes) sequential plan minimizes the (Bayes) risk function that combines the decision loss, observation (variable) cost, and group (fixed) cost. In general, determining the optimal sequential plan remains an open and challenging problem mainly because it requires risk optimization over a huge and rather unstructured set of all sequential plans. This article demonstrates how to obtain the optimal solution for a particular class of problems that may arise in testing a treatment for a rare but severe adverse effect. This solution is obtained by studying a number of properties of the Bayes sequential plan such as transitivity and monotonicity. This allows one to reduce the search to a small, manageable set of sequential plans within which the optimal plan can be calculated.