Adaptive Simon Two-Stage Designs

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Adaptive Simon two-stage designs let a prespecified rule change the number of stage 2 patients according to the stage 1 response count, while still maintaining defined statistical operating characteristics.

One of the defining features of a conventional Simon two-stage design is that the second-stage sample size is fixed once the design is constructed. An adaptive design keeps the basic two-stage structure, but lets that second-stage sample size depend on the observed stage 1 result instead.

An adaptive design retains the usual stage 1 futility decision, but the complete design — including the futility rule, every possible second-stage sample size, and the final decision rule — must be evaluated jointly to verify its Type I error, power, expected sample size, and maximum sample size. This creates additional flexibility, and it makes the design search considerably more complex.

The Conventional Simon Two-Stage Design

A Simon two-stage design for a binary response endpoint specifies an unacceptable response rate p0, a desirable response rate p1, a Type I error rate alpha, a target power, a first-stage sample size n1, a first-stage futility boundary r1, a maximum total sample size n, and a final rejection boundary r.

The trial enrolls n1 patients and observes x1 responses. If x1 is at or below r1, the trial stops for futility. If x1 is above r1, the trial continues to stage 2.

In a conventional design, the number of additional patients is fixed: n2 = n - n1. It does not matter whether the trial barely cleared the continuation threshold or comfortably exceeded it; the same number of additional patients is planned either way.

What Makes a Design Adaptive?

An adaptive Simon two-stage design allows the second-stage sample size to depend on x1, the number of responses observed among the first n1 patients. Instead of a single fixed n2, the design specifies a function n2(x1).

The final decision rule can also depend on x1. Rather than a single rejection boundary r shared by every continuation path, the design may specify a boundary r(x1): reject the null hypothesis if x1 plus the stage 2 response count reaches or exceeds r(x1). Whether the construction lets only n2 vary, or lets both n2(x1) and r(x1) vary, depends on the specific methodology being used.

The design therefore contains an adaptation rule mapping every possible first-stage outcome to a second-stage sample size (and, where applicable, its own rejection boundary). The table below is illustrative only and does not represent a validated design.

The adaptation rule must be defined and evaluated during design construction. It should not be treated as an informal decision made after looking at the trial data.

Why Adapt the Second Stage?

At the end of stage 1, investigators know more than they did at the start of the trial. A conventional design uses that information mainly to decide whether to stop or continue. An adaptive design can use more of it.

Consider two trials that both clear the continuation threshold: Trial A observes only slightly more responses than required, while Trial B observes substantially more. A conventional design treats both identically for stage 2 enrollment. An adaptive design can plan different stage 2 sizes for these different outcomes.

The goal is not simply to make the trial smaller. It is to construct a design whose operating characteristics satisfy prespecified statistical requirements while making more flexible use of the information available after stage 1.

Adaptive Versus Conventional, Side by Side

Both design families have two stages and an early-stopping rule for futility. The difference is entirely in what happens after continuation is decided.

Not Simply a Smaller Simon Design

It is tempting to describe an adaptive design as a Simon design with fewer patients. That description is not adequate.

An adaptive design is defined by its complete set of possible trial paths: one path for stopping at stage 1, and one path for every continuation outcome, each with its own n2(x1). The design must be evaluated across the whole set of paths to determine whether its overall Type I error, power, and expected sample size meet the intended requirements.

Different Approaches to Adaptive Two-Stage Designs

Adaptive two-stage designs are not built from one universal methodology. Different research groups have used different statistical objectives (see references below).

Banerjee and Tsiatis (2006) developed an adaptive design using a Bayesian decision-theoretic framework, minimizing expected sample size under the null hypothesis.

Shan, Zhang, and Jiang (2016) developed a minimax adaptive design using a branch-and-bound algorithm based on conditional error functions, and separately described admissible adaptive designs from a Bayesian perspective.

Shan and colleagues also published a complementary 2016 paper on optimal adaptive two-stage designs, minimizing expected sample size under the null while maintaining specified error rates and a monotonicity property for the second-stage sample size.

The term "adaptive Simon design" therefore does not identify one unique construction. The precise design depends on the statistical framework, optimization criterion, and adaptation rule being used.

Expected Sample Size

Expected sample size is especially important when comparing two-stage designs. Let X1 be Binomial(n1, p), where p is the true response probability. Total enrollment is n1 alone on a stopping path, and n1 + n2(x1) on a continuation path.

For an adaptive design, expected total sample size can be written as: E[N | p] = n1 + the sum, over every continuation outcome x1 from r1+1 to n1, of P(X1 = x1) multiplied by n2(x1). In words: start from n1, then add each continuation outcome's second-stage size weighted by how likely that outcome is under p.

For a conventional design, n2(x1) is the same value for every continuation outcome, so this simplifies to: E[N | p] = n1 + P(X1 > r1 | p) times n2. The adaptive formulation is more flexible because different first-stage outcomes can contribute different second-stage sample sizes, which can make it possible to optimize expected sample size under a chosen criterion while still meeting the required error rates.

Minimum, Expected, and Maximum Sample Size

Adaptive designs make it especially important to distinguish three quantities.

Minimax, Optimal, and Admissible Adaptive Designs

The terms minimax, optimal, and admissible describe different design objectives, in the same way they do for conventional Simon designs, but applied to the larger adaptive search space.

A minimax adaptive design minimizes the largest possible total enrollment, subject to the prespecified Type I error, power, and other design constraints. An admissible adaptive design is a nondominated compromise between the minimax and optimal objectives, in the same spirit as an admissible design in the conventional Simon setting.

In the framework discussed in this article, an optimal adaptive design minimizes expected sample size under the null response rate. This is a methodological choice, not the only possible one: other methods may minimize expected sample size under a different response rate, a weighted set of response rates, or a prior distribution, and may use a different weighting scheme.

These objectives can produce different designs even under the same trial hypotheses and error constraints, so choosing among them uses the same reasoning covered in optimal versus minimax designs and admissible designs.

Why the Adaptive Search Is Harder

Searching for a conventional Simon design already means searching over combinations of n1, r1, n, and r. An adaptive search adds another layer: for a first-stage sample size n1, there can be up to n1 + 1 possible first-stage response counts, and each continuation outcome may need its own second-stage sample size while the whole schedule still satisfies the required Type I error, power, and maximum-size constraints.

This is why adaptive-design methods rely on specialized search algorithms, such as the branch-and-bound approach over conditional error functions used by Shan and colleagues. The computational challenge is not evaluating one candidate design; it is efficiently searching a much larger collection of possible adaptation rules.

What to Examine in an Adaptive Design

An adaptive design should not be evaluated solely by its nominal Type I error and power. Useful operating characteristics include:

Adaptive Does Not Mean Ad Hoc

The word "adaptive" can create confusion. This distinguishes it from other things "adaptive" can mean in a trial: blinded or unblinded sample-size re-estimation, adaptive randomization, and adaptive enrichment are different concepts. Here, an adaptive design does not mean investigators are free to change the design once they see the data. The adaptation rule, mapping every possible x1 to a specific n2(x1) (and, where applicable, r(x1)), is part of the prespecified statistical design, and the observed first-stage result simply determines which prespecified rule applies.

Regulatory guidance generally emphasizes that adaptive features should be prespecified, justified, and appropriately controlled; the applicable requirements depend on the jurisdiction and development context. The FDA's guidance for industry on adaptive clinical trials, finalized in November 2019, discusses principles for designing, conducting, and reporting adaptive trials, including the need for prespecification (see references below).

For European development programs, ICH E20 addresses adaptive designs for confirmatory clinical trials. Status checked 2026-09-22: ICH E20 remains a draft guideline. It reached Step 2b of the ICH process in June 2025, and the overview of public comments from the consultation period was published in February 2026, but the guideline has not yet been finalized. Verify the current status directly against the official ICH and EMA pages before relying on it, since this is time-sensitive and expected to keep progressing during 2026.

Investigators should consult applicable regulatory guidance and obtain appropriate statistical and regulatory advice for their specific development program.

How BioStatHub Helps

BioStatHub's design workspace generates conventional Simon two-stage designs first. From a specific generated design, you can explore an adaptive extension of that design: BioStatHub searches for a minimax adaptive, optimal adaptive, or admissible adaptive schedule built on the methodology of Shan, Zhang, and Jiang (2016), and returns the full second-stage schedule n2(x1) together with its exact Type I error, power, expected sample size, and maximum sample size.

Because a full adaptive search over every possible first-stage size is far more expensive than evaluating one fixed design, BioStatHub anchors the adaptive search to the first-stage size of the design you are already reviewing by default. This is a genuinely different, and typically less advantageous, question than searching the unrestricted design space, so BioStatHub states that trade-off before running the search rather than presenting an anchored result as the best possible adaptive design.

Where BioStatHub's own search finds an adaptive schedule that improves on a design published in Shan et al. (2016) for the same inputs, it discloses that difference explicitly, together with the published values, rather than silently reporting a different number than the literature.

Adaptive design is currently offered as a preview feature while its scope and pricing are finalized. The conventional Simon design workflow, including the optimal, minimax, and admissible searches described elsewhere in this Knowledge Center, is unaffected either way.

It is worth being precise about what each party is responsible for: BioStatHub searches for and computes a candidate adaptive schedule against the criteria you specify; it does not independently validate that the schedule is appropriate for your protocol. As with any generated design, review the schedule and its operating characteristics with your study statistician before relying on it.

Key Takeaways

Adaptive Simon two-stage designs extend Simon's original methodology by letting the second-stage sample size depend on the observed first-stage result, through a prespecified adaptation rule n2(x1).

A conventional design fixes n2 once continuation is decided; an adaptive design can choose a different n2 for every possible first-stage outcome, at the cost of a much larger design search.

Minimax, optimal, and admissible objectives apply to adaptive designs in the same way they apply to conventional ones, and different research groups (Banerjee and Tsiatis; Shan and colleagues) have used different frameworks to construct them.

To continue the design-selection workflow, review optimal versus minimax designs, admissible designs, and expected sample size before deciding whether an adaptive extension is worth the added complexity for your study.

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