On construction of single-arm two-stage designs with consideration of both response and toxicity

When establishing a treatment in clinical trials, it is important to evaluate both effectiveness and toxicity. In phase II clinical trials, multinomial data are collected in m-stage designs, especially in two-stage ( $$ ) design. Exact tests on two proportions, $$ for the response rate and $$ for the nontoxicity rate, should be employed due to limited sample sizes. However, existing tests use certain parameter configurations at the boundary of null hypothesis space to determine rejection regions without showing that the maximum Type I error rate is achieved at the boundary of null hypothesis. In this paper, we show that the power function for each test in a large family of tests is nondecreasing in both $$ and $$ ; identify the parameter configurations at which the maximum Type I error rate and the minimum power are achieved and derive level-α tests; provide optimal two-stage designs with the least expected total sample size and the optimization algorithm; and extend the results to the case of $$ . Some R-codes are given in the Supporting Information.