Understanding Admissible Designs

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When designing a Simon's two-stage phase II trial, many widely used software packages present only the optimal and minimax designs, even though several equally valid admissible alternatives may exist.

Between these two extremes lies a family of admissible designs that often provide a better balance between patient safety and trial efficiency.

In this article, we will explain what admissible designs are, why they exist, and when you should consider choosing one.

Why Another Design?

If you have already learned about Simon's two-stage design, you have probably encountered the two classic choices: the optimal design and the minimax design.

The optimal design minimizes expected sample size under the null hypothesis. The minimax design minimizes the maximum total sample size.

Both designs satisfy the same statistical requirements: null response rate p0, target response rate p1, Type I error alpha, and power. The difference lies in what they optimize.

In practice, neither extreme always matches the priorities of the study team. The optimal design may require a relatively large maximum sample size, while the minimax design may expose more patients on average to an ineffective treatment.

The Trade-off

The two objectives naturally compete. Suppose two candidate designs satisfy the same error-rate constraints.

The optimal design saves patients on average if the treatment is ineffective. The minimax design guarantees a smaller upper limit on total enrollment.

Neither design is universally better. The practical question is how much maximum sample size you are willing to accept in exchange for a lower expected sample size under the null.

What Is an Admissible Design?

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If another valid design has both a smaller maximum sample size and a smaller expected sample size, the original design is dominated and can be discarded.

The remaining nondominated designs form the admissible set. These are the designs worth comparing when both ethical and operational objectives matter.

Thinking in Terms of a Pareto Frontier

A useful way to understand admissible designs is through the Pareto frontier. Imagine plotting every feasible Simon design with expected sample size on one axis and maximum sample size on the other.

A Pareto frontier is a set of solutions where improving one objective necessarily makes another objective worse.

Many designs sit inside the cloud because another design performs better on both measures. Those inferior designs are removed.

The remaining frontier points are admissible designs. Moving along the frontier means trading one objective against the other: reducing maximum sample size usually increases expected sample size, and reducing expected sample size usually requires a larger maximum sample size.

Optimal and Minimax Are Endpoints

One useful result from Jung and colleagues is that Simon's optimal and minimax designs can be understood as endpoint designs on the admissible frontier.

The optimal endpoint prioritizes the lowest expected sample size under p0. The minimax endpoint prioritizes the smallest possible maximum sample size.

Everything between them is also statistically valid, provided it satisfies the same Type I error and power requirements. That middle region is often where clinically practical designs live.

How Admissible Designs Are Selected

Jung and colleagues introduced a decision-theoretic way to select among admissible designs using a weighted loss function.

For a candidate design, the loss can be written as L = q x n + (1 - q) x EN, where n is maximum sample size, EN is expected sample size under the null, and q is a weight between 0 and 1.

A lower q prioritizes expected sample size. A higher q prioritizes maximum sample size. As q changes, different points on the admissible frontier can become preferred.

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An Intuitive Example

For example, BioStatHub can enumerate admissible candidates for a Simon design with p0 = 0.20, p1 = 0.40, alpha = 0.05, and power = 80%. A representative output might look like this.

Every step toward a smaller maximum sample size costs only a modest increase in expected sample size. Many investigators find these intermediate designs more attractive than either extreme.

When to Choose an Admissible Design

Choose the optimal design when patient exposure under an ineffective therapy is the highest priority, the treatment is expected to fail frequently, and minimizing average enrollment under p0 matters most.

Choose the minimax design when budget, recruitment, drug supply, or protocol constraints impose a hard upper bound on total enrollment.

Choose an admissible design when both objectives matter and neither endpoint fits the study's clinical and operational priorities.

In practice, admissible designs are often the preferred choice when neither minimizing the expected sample size nor minimizing the maximum sample size alone reflects the study's priorities.

Advantages of Admissible Designs

Admissible designs offer a better balance between ethics and logistics than choosing only between the two endpoint designs.

They often require only a small sacrifice in one objective for a meaningful gain in the other.

They also make the design choice easier to justify, because the selected design can be described as a specific point on the trade-off frontier rather than a default choice.

How BioStatHub Helps

BioStatHub computes feasible Simon two-stage designs, highlights the optimal and minimax solutions, and shows admissible trade-offs between maximum sample size and expected sample size.

BioStatHub lets you explore the complete admissible frontier rather than comparing only the two traditional endpoints.

The design overview table and frontier chart help you compare operating characteristics side by side before setting the final design.

Instead of selecting a design blindly, you can choose the design that best matches the clinical, ethical, and operational priorities of the study.

Key Takeaways

Admissible designs extend Simon's original two-stage methodology by making the compromise between maximum sample size and expected sample size explicit.

The admissible set forms the Pareto frontier of statistically valid designs.

The optimal and minimax designs are endpoint choices on that frontier.

Intermediate admissible designs often provide the most practical option for real-world phase II clinical trials.

The important insight is that optimal and minimax designs are not competing methodologies. They are two endpoints of a broader family of admissible designs. Viewing the full admissible set allows investigators to select the compromise that best fits their scientific, ethical, and operational priorities.

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