Optimal Two-stage Designs With Consideration of Response and Safety and Their Correlation

In phase Ⅱ clinical trial for cancer study, the efficiency and safety of a new drug are particularly important. Under consideration of ethics and cost, the optimal multi-stage designs, especially the two-stage designs, were used to reduce the sample size. In this paper, to have a positive correlation between the efficiency and safety, a new set of hypotheses and exact tests were proposed, proving that the power function is nondecreasing in the response rate p_r and the safety rate p_t, respectively, using conditional expectation. The type Ⅰ error rate achieves its maximum at two points and the power achieves its minimum on a line. With control of maximum type Ⅰ error rate and minimum power, the optimal two-stage designs were derived to minimize the expected total sample size, thus an exact construction method was presented for the optimal design of two-stage trials. Additionally, since the alternative space in Eq.(2) is a subset of that in Eq.(1), an optimal two-stage design for Eq.(1) is also a two-stage design for Eq.(2).