Optimum biased coin designs for sequential clinical trials with prognostic factors

Patients in a clinical trial arrive sequentially and are assigned to one of t treatments. This assignment should maintain a balance between the numbers receiving each treatment, yet should be sufficiently random to avoid any suspicion of conscious or unconscious cheating. To balance these requirements Efron (1971) introduced biased coin designs for the comparison of two treatments in which allocation of the treatment is determined probabilistically, but with a bias towards the underrepresented treatment. One disadvantage of Efron's scheme is that it does not include balance over covariates or prognostic factors which may affect the response of the patient to the treatment. Biased coin schemes which do force balance over both treatments and prognostic factors are given by Pocock & Simon (1975) and Efron (1980). The properties of the designs have been elucidated by numerical studies and they are now being increasingly used in clinical trials. Reviews of the literature and discussions of the practical implications of the designs are given by Pocock (1979) and Simon (1979). However, the designs suffer from the disadvantage that they rely on arbitrary functions to achieve the desired balance. The procedures thus lack a firm theoretical framework. An alternative approach in the presence of prognostic factors (Begg & Iglewicz, 1980) uses optimum design theory to suggest a deterministic design criterion, which is then modified for computational convenience. In this paper I use optimum design theory to provide a procedure of the biased coin type for an arbitrary number of treatments in the presence, or absence, of prognostic factors. This has the theoretical advantage of obviating dependence on a series of arbitrary functions. The necessary optimum design theory is presented in ? 2 and, in ? 3, applied to biased coin experiments. The simplest case, that of two treatments in the absence of prognostic factors, is studied in ? 4. The extension to three or more treatments is in ? 5, followed, in ? 6, by the allowance for prognostic factors. Although, in the biased coin designs, allocation of treatments is made probabilistically, the results of the paper can be used for the sequential construction of DA-optimum designs in which randomization is not introduced by the experimenter. All the designs mentioned so far are sequential, but not adaptive. In the useful distinction made by Pericchi (1981), they are data-dependent but not outcomedependent: each allocation depends on the previous allocations and prognostic factors,