As was discussed in Section 6.6, in phase II cancer trials, it is undesirable to stop a study early when the treatment appears to be effective but desirable when the treatment seems to be ineffective. For this purpose, a multiple-stage design is often employed to determine whether an experimental treatment holds sufficient promise to warrant further testing (see, e.g., Fleming, 1982; Simon, 1989; Chang et al., 1987; Therneau et al., 1990). The concept of a multiple-stage design is to permit early stopping when a moderately long sequence of initial failures occurs. Chow et al. (2003b) provided tables for sample size calculation for two-stage designs such as minimax design, Simon's optimal two-stage design, and flexible two-stage design and three-stage designs such as optimal three-stage design and flexible design for single-arm phase II cancer trials.

Example 11.9.1 Suppose an investigator is interested in planning a clinical trial in patients with a specific carcinoma where standard therapy has a 20% response rate. Suppose further that the test treatment is a combination of the standard therapy and a new agent. The investigator would like to determine whether the test treatment will achieve a response rate of 40%. Since the test treatment may not be effective, it is desirable to warrant early study termination. In this case, the optimal three-stage design described above is useful. We consider the optimal three-stage design for testing

H0: p < 0.20 vs. Hj: p > 0.40 with a = ยก3 = 0.10 which result in the following sample size allocation at each stage:

That is, at stage 1, 8 patients are to be tested. We would terminate the trial if no response is observed in the eight patients. If there are one or more responses, we continue to the next stage. At the second stage, eight more patients are treated. We stop the trial if fewer than three responses are observed in the sixteen patients; otherwise continue to stage 3. At stage 3, 26 more patients are treated. We conclude the test treatment is effective if there are more than 11 responses in the 42 patients.

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