Modified Simon's minimax and optimal two-stage designs for single-arm phase II cancer clinical trials

Abstract

Simon's two-stage design and the admissible two-stage design have been commonly used in practice for single-arm phase II clinical trials when the primary endpoint is binary. The ethical benefit of the two-stage design over the single-stage design is attained by the early termination of the trial when the treatment seems to be inactive. While Simon's optimal design is the two-stage design that minimizes the expected number of subjects under the null hypothesis, the probability of falsely declaring futility after the first stage frequently seems undesirably high. In Simon's minimax design, however, it is often the case that a high proportion of the total planned subjects are evaluated in the first stage, and thus the ethical benefit may not be achieved. In this paper, we propose modified minimax and optimal two-stage designs which guarantee not only type I and II error rates but also reasonable sample size proportions in the first stage, while maintaining the probability of falsely declaring futility under a pre-selected level. The characteristics of the modified two-stage design will be compared with those of Simon's and the admissible two-stage design. The modified minimax design yields a design that requires modest increase in 29% of cases, while the modified optimal design saves 1 to 13 subjects in 81% of cases for β = 0.2. The modified design approach provides investigators with an alternative when the sample sizes of Simon's designs are severely unbalanced or the Type II error is unacceptably high after the first stage.

Keywords: Simon’s two-stage design; admissible two-stage design; binary endpoint; modified two-stage design; single-arm phase ii clinical trials.

Figure 1

Comparisons of total sample sizes between modified designs with γ₁ = 1/3, γ₂ = 2/3, and ε = 0.1 and Simon’s designs for p₁ – p₀ = 0.2 (A) and p₁ – p₀ = 0.15 (B). The top panels of A and B show the number of additional subjects required for the modified minimax designs while the bottom panels indicate those for modified optimal designs.

Figure 2

Simon’s minimax (top two panels) and Simon’s optimal designs (bottom two panels) for p₁ - p₀ = 0.15 and 0.2: ratio of n₁ to n (left panels) and the type II error spent in the first stage (PET₁, right panels) for (α, β) = (0.05, 0.2).