Simon's two-stage design calculator

What the calculator does

BioStatHub searches the feasible two-stage designs for your response rates and returns the optimal, minimax and admissible candidates, each a valid design for your Type I error and power targets.

Optimal designs

The design with the smallest expected sample size under the unacceptable response rate p₀, so an ineffective treatment enrolls as few patients as possible.

Minimax designs

The design with the smallest maximum sample size, so total enrollment stays bounded even when the trial continues into stage 2.

Admissible designs

The Pareto frontier of designs that are not beaten on both maximum and expected sample size, for comparing the trade-offs in between.

Adaptive extensions (preview)

From a generated Simon design, explore an adaptive two-stage design whose second-stage sample size depends on how many responses stage 1 observes, instead of one fixed number.

Adaptive design search is currently available as a preview. Availability may change as the product evolves.

Inputs you provide, results you compare

Inputs

Unacceptable response rate (p₀)

The response rate that would make the treatment unpromising. It defines the null hypothesis and the probability of early termination is reported at this rate.

Promising response rate (p₁)

The response rate that would justify further development. It defines the alternative hypothesis the design is powered against.

Type I error (α)

The acceptable probability of calling a treatment promising when its true response rate is only p₀.

Type II error (β), where power = 1 − β

The acceptable probability of missing a treatment whose true response rate is p₁. A target power of 80% means β = 0.20.

Outputs and trade-offs

Maximum sample size (N)

The total enrollment when the trial continues past stage 1, which is also the largest possible number of patients.

Expected sample size EN(p₀)

The average enrollment once trials that stop early and trials that continue are both counted.

Probability of early termination

How often the design stops after stage 1 under p₀ — the chance an ineffective treatment is abandoned early.

Operating characteristics

The stage 1 and final decision boundaries (n₁, r₁, N, r), achieved Type I error and power, and the power, expected sample size and early-termination curves.

The trade-off is the decision. The optimal design minimises expected enrollment under p₀ but can require a larger maximum sample size; the minimax design caps enrollment at the smallest possible maximum but usually expects more patients when the treatment is inactive. Admissible designs sit between those extremes, so you can see what each alternative costs before choosing.

Generate, Compare, Decide, Document

1. Generate

   Enter p₀, p₁, α and β. BioStatHub searches feasible two-stage designs with exact binomial probabilities and returns the optimal, minimax and admissible candidates.

2. Compare

   Review maximum sample size, expected sample size under p₀, probability of early termination, decision boundaries and operating characteristics side by side, and compare an adaptive extension of the same design.

3. Decide

   Choose the design that fits your study and record the rationale, so the final decision is traceable instead of a black-box recommendation.

4. Document

   Keep the selected design and its parameters in a Final Protocol view with validation evidence, create a read-only snapshot to share with your team, and export the computed designs as JSON or CSV.

Start with your own response rates

Open the calculator, enter the unacceptable response rate, the promising response rate, Type I error and power, and generate the first candidates in the workspace.

Open the Calculator