A class of phase II designs with three possible outcomes

In the Phase II design of Fleming (1982, Biometrics 38, 143-151), the possible outcomes are to reject H1:p less than or equal to p1 or to reject H2: p greater than or equal to p2, where p1 less than p2. The design is constrained by specifying that Pr(reject H1[p1) less than or equal to alpha and Pr(reject H2[p2) less than or equal to beta. This can lead to some ambiguity as to the appropriate practical decision at the end of the trial, as the confidence region for p may include a substantial part of the interval between p1 and p2. We propose a class of designs wherein an allowable outcome is not to reject either H1 or H2. An additional constraint is placed on the design by specifying Pr(reject Hj[pm) less than or equal to gamma (j = 1, 2), where p1 less than pm less than p2. This type of design can provide a more realistic basis for decision making following the trial, although traditional values of alpha and/or beta need to be viewed from a somewhat different perspective in order to maintain a reasonable sample size. Some optimal single-stage and sequential multistage designs are presented.