Optimal designs for estimating the most successful dose.

There has been much recent interest in phase I/II dose finding designs in which information on both toxicity and efficacy is used. Unlike the classic phase I dose finding design in which the aim is to identify the MTD (maximum tolerated dose corresponding to some percentile of acceptable toxicity), a phase I/II dose finding study aims to locate the most successful dose (MSD), i.e. the dose which maximizes the product of the probability of seeing no toxicity together with the probability of seeing a therapeutic response). In this work we present an abstract theoretical design for this purpose. We call this design a 'virtual' design. The virtual design, conceptually similar to that developed for phase I designs alone, is based on a bivariate response. The design has optimal properties in that the current estimates of both probability of toxicity and those for response achieve the Cramer-Rao bound for every dose level. Unhappily, the virtual design is not available for practical use but its use can be exploited in theoretical investigations in much the same way as one uses the Cramer-Rao bound for unbiased estimators, i.e. a tool which enables us to see how much room for improvement may exist for any given real design. Via examples taken from the literature on phase I/II dose finding we illustrate how this technique can provide us with further insight on the relative performance of competing designs.