Optimal vs. Minimax Two-Stage Design Comparison

Methodology

This calculator implements Simon's two-stage single-arm Phase II design. Patients are enrolled in Stage 1 and the trial stops early for futility when the number of responses is at or below the first-stage critical value R₁. If the trial continues, the total sample size is N. At the end, the treatment is rejected when the total number of responses is at or below R; otherwise it is declared sufficiently promising for further development.

What is being optimized?

The optimum design is the feasible design with the smallest expected sample size under P=P₀. The minimax design is the feasible design with the smallest maximum total sample size N. These are different optimization criteria, so the two designs can have different Stage 1 sizes, stopping boundaries, and expected sample sizes.

Exact probability calculations

Let X₁ be the number of responses among N₁ patients in Stage 1 and X₂ the number of additional responses among N-N₁ patients. For a candidate design (N₁, R₁, N, R), the probability of rejection at response probability p is

The achieved Type I error is α_actual = 1 - Pr(reject | P₀), while the achieved Type II error is β_actual = Pr(reject | P₁). A candidate is feasible only when α_actual ≤ α and β_actual ≤ β.

Expected sample size and early termination

Under P=P₀, the probability of early termination is PET = Pr(X₁ ≤ R₁). The expected sample size is

The calculator searches the integer design space up to the user-specified maximum N. Among feasible candidates, it reports the design minimizing E(N) as the optimum design and the design minimizing N as the minimax design. If there is a tie on the primary criterion, the design with the smaller expected sample size is used as the tie-breaker.

Worked validation example

the relevant methodological literature gives a validation example based on Simon (1989): P₀=0.05, P₁=0.25, α=0.10, and β=0.10. This yields the optimum design as N₁=9, R₁=0, N=24, R=2, with E(N)=14.55 and PET=0.630; the minimax design is N₁=13, R₁=0, N=20, R=2, with E(N)=16.41 and PET=0.513. The implementation in this page reproduces those values to rounding precision.

References

1. Simon, R. (1989). Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials, 10(1), 1–10. doi:10.1016/0197-2456(89)90015-9.
2. the software, LLC. Two-Stage Designs for Tests of One Proportion (Simon), the relevant methodological literature. The procedure defines N₁, N, R₁, R, PET, E(N), and the exact alpha/beta constraints, and states that Simon's algorithm finds the minimum-N minimax and minimum-E(N) optimum designs.
3. the software, LLC. this method 11 User's Guide 1, section “Two-Stage Phase II Clinical Trials,” Example 2 — Validation using Simon. This example reports the same optimum and minimax designs used for the validation above.