A Two-Stage Model for Selecting One of Two Treatments

In a previous paper (Colton 1963) I considered a cost function approach to the design of a clinical trial for the comparison of two medical treatments. I examined two types of plans, one with a fixed sample size and one with random pairing and sequential, one-pairat-a-time, observations. Anscombe 1963 considered a similar formulation, dealing with the sequential case. The results of these investigations indicated that the optimal sequential plan led to an overall smaller cost (or, alternatively, an overall greater gain) than the corresponding optimal fixed sample size plan. Numerical results showed that the overall net gain with the optimal sequential plan could be as much as 25 percent more than that for the corresponding optimal fixed sample size plan. Here I consider intermediate plans: in particular, two possible two-stage plans. The appropriate cost functions are determined, the optimal plans are obtained, and the corresponding net gains are compared with each other and with the previously reported fixed sample size and sequential results. The results show that although the twostage plans proposed are quite different, their optimal net gains are similar. The overall net gain of the two-stage optimal plans is at most 13 percent more than that of the corresponding optimal fixed sample size plan. However, the optimal two-stage plan can account for as much as 50 percent of the difference in overall net gain between the fixed sample size and sequential plans. Hence, within this cost formulation it appears that by going from the one extreme of a fixed sample size plan to a two-stage plan one can achieve about half of the gain that is attainable by the opposite extreme of a pair-by-pair fully sequential plan.