Numerical Search for MAMS Stage Sizes

Why Search Over Integer Stage Sizes?

A MAMS design must satisfy statistical requirements and operational limits at the same time. Stage sizes are integer counts, and a change to one stage can affect the precision of interim comparisons, the chance that an arm continues, power at later analyses, and expected enrollment. Consequently, a sample size that meets a target in one calculation is not automatically a feasible MAMS design.

Numerical search makes this joint problem explicit. Specify a finite set of candidate stage-size vectors, evaluate each candidate under the same design model and prespecified scenarios, and retain only those meeting every required constraint. The search does not replace the assumptions behind power or enrollment calculations; it applies them systematically to an integer design space.

Definition: Consider a design with \(J\) experimental arms and \(K\) stages. Let \(n_k\) be the number randomized to each active experimental arm at stage \(k\), and let \(\mathbf n=(n_1,\ldots,n_K)\) be the stage-size vector. In the equal-allocation, shared-control setup used here, \(n_k\) control participants are also enrolled at stage \(k\) if at least one experimental arm remains active. A candidate is feasible if it meets every prespecified statistical and enrollment constraint in every scenario to which that constraint applies.

This formulation assumes the stage sizes are planned randomized sample sizes, the stopping rules and analysis boundaries are fixed, and a specified probability model determines interim outcomes. The evaluator must account for dependence between comparisons that share control data. It must also use the same endpoint, allocation, nuisance-parameter, and attrition assumptions for every candidate. If a design uses unequal allocations or a different control-accrual rule, its enrollment calculation must reflect that rule rather than using the formula below unchanged.

What Each Candidate Must Be Checked Against

Before searching, write down the constraints and their scope. Let \(\mathcal S\) be the prespecified set of scenarios. For a scenario \(s\), the design evaluator returns quantities such as the marginal probability of rejecting the null for experimental arm \(j\), the familywise type I error for the relevant null configuration, and expected enrollment. These probabilities are calculated under the stated interim model and decision rules.

A typical set of conditions might require each arm to have power of at least \(1-\beta\) in each efficacy scenario where that arm is meant to be detectable; familywise type I error no greater than \(\alpha\) in each required null configuration; maximum enrollment no greater than a recruitment limit; and expected enrollment no greater than a specified budget in each operational scenario. The criteria must be stated before running the search. In particular, checking error control only under the global null is insufficient if the required error-control claim also covers partial-null configurations.

For each scenario, power for arm \(j\) is the probability that the design rejects that arm’s null hypothesis at some permitted analysis under that scenario. It is not the probability of continuation. Continuation probabilities are inputs to expected enrollment, whereas power depends on the full set of paths that lead to rejection.

Expected Enrollment with Shared Control

Enrollment depends on which arms remain active. Define \(A_{jk}=1\) if arm \(j\) is active at stage \(k\), and \(A_{jk}=0\) otherwise. Define \(C_k=1\) if at least one experimental arm is active at stage \(k\), and \(C_k=0\) otherwise. Under the stated shared-control rule, stage \(k\) enrolls \(n_k\) participants for each active experimental arm and \(n_k\) controls if \(C_k=1\). Therefore, total enrollment at that stage is \(n_k(C_k+\sum_{j=1}^J A_{jk})\).

N_{\mathrm{total}}=\sum_{k=1}^{K}n_k\left(C_k+\sum_{j=1}^{J}A_{jk}\right), \qquad E_s(N_{\mathrm{total}})=\sum_{k=1}^{K}n_k\left\{P_s(C_k=1)+\sum_{j=1}^{J}P_s(A_{jk}=1)\right\}.

The expectation formula follows from linearity of expectation; it does not require the continuation indicators to be independent. In particular, \(P_s(C_k=1)\) is the probability that at least one arm continues, which is generally not the sum of the individual continuation probabilities. At stage 1, all arms are active, so \(C_1=1\) and \(A_{j1}=1\) for every arm. Maximum enrollment, attained if every arm continues at every stage, is \((J+1)\sum_{k=1}^K n_k\) for this setup.

Justification: Write stage-\(k\) enrollment as the sum of the control contribution \(n_kC_k\) and the experimental-arm contributions \(n_kA_{jk}\). Summing over stages gives \(N_{\mathrm{total}}\). Taking expectation under scenario \(s\), applying linearity, and using \(E_s(C_k)=P_s(C_k=1)\) and \(E_s(A_{jk})=P_s(A_{jk}=1)\) gives the expression above. No independence assumption among arms’ continuation decisions is needed for this expectation.

A Reproducible Search Procedure

Choose finite integer ranges for each stage size from defensible planning limits. These ranges define the candidates to be examined; they are not themselves evidence that any candidate is feasible. For every vector in the resulting finite set, run one consistent evaluator that calculates the relevant operating characteristics in all prespecified scenarios. Record both the input assumptions and the returned quantities so that another analyst can reproduce the feasibility decision.

Freeze the design specification.
Record the number of arms and stages, allocation and shared-control rules, interim boundaries, endpoint model, scenario set, and all required constraints.

Generate integer candidates.
Define explicit lower and upper limits for each \(n_k\), then enumerate every vector in those ranges. Make clear whether any structural restrictions exclude a vector.

Evaluate each candidate consistently.
For every scenario, calculate the required arm-specific powers, error rates, continuation probabilities, expected enrollment, and maximum enrollment under the same model.

Apply every constraint.
Mark a candidate feasible only if all applicable criteria pass in all required scenarios. Keep the individual results, not only a single pass/fail label.

Report the search record.
Report the ranges searched, assumptions, constraints, number of candidates examined, and feasible candidates. This makes the result auditable and repeatable.

Worked Examples

Worked Example: Checking One Candidate Across Scenarios

Consider a hypothetical two-stage design with two experimental arms, equal stage sizes \(n_1=n_2=8\), and one shared control group. Suppose the prespecified requirements are: power at least \(0.80\) for each arm in each relevant efficacy scenario; familywise type I error no greater than \(0.05\) in every required null configuration; maximum enrollment no greater than 50; and expected enrollment no greater than 44 in each operational scenario.

An exact evaluator under the specified model returns, for the first efficacy scenario, marginal powers \(0.82\) and \(0.80\), and continuation probabilities at stage 2 of \(P(A_{12}=1)=0.50\), \(P(A_{22}=1)=0.55\), and \(P(C_2=1)=0.75\). Its largest familywise error rate over the required null configurations is \(0.047\). The maximum enrollment is \((2+1)(8+8)=48\).

Expected enrollment in this scenario is \(3(8)+8(0.75+0.50+0.55)=24+14.4=38.4\) participants. The first term is stage 1 enrollment; the second is expected stage 2 enrollment, counting shared controls only when at least one arm continues.

Suppose a second operational scenario gives continuation probabilities \(0.65\), \(0.50\), and \(0.80\) for arms 1 and 2 and for at least one arm, respectively. Expected enrollment is \(24+8(0.80+0.65+0.50)=24+15.6=39.6\) participants. If powers in this scenario are \(0.84\) and \(0.81\), the candidate passes both power checks. Thus it satisfies all the stated constraints: its minimum reported power is \(0.80\), its largest error rate is \(0.047<0.05\), its maximum enrollment is \(48\leq50\), and its largest expected enrollment is \(39.6\leq44\).

Worked Example: Translating Decision Paths into Power and Enrollment

For the first scenario in the preceding example, suppose the evaluator reports for arm 1 a probability of \(0.46\) of rejecting at stage 1 and a probability of \(0.36\) of continuing to stage 2 and rejecting there. These events are disjoint because a stage-1 rejection ends that arm’s testing. Its power is therefore \(0.46+0.36=0.82\). For arm 2, suppose the corresponding probabilities are \(0.35\) and \(0.45\); its power is \(0.35+0.45=0.80\).

The continuation probabilities need not equal one minus the stage-1 rejection probabilities: an arm can stop for futility as well as for efficacy. Here, the probabilities of continuing, \(0.50\) for arm 1 and \(0.55\) for arm 2, are consistent with the stage-1 rejection probabilities because \(0.46+0.50=0.96\leq1\) and \(0.35+0.55=0.90\leq1\). The remaining probability in each case can correspond to stopping without rejection.

The probability that at least one arm continues is \(0.75\), not \(0.50+0.55=1.05\). Using the joint continuation calculation gives expected stage 2 enrollment of \(8(0.75+0.50+0.55)=14.4\) participants. This illustrates why arm-specific powers and continuation probabilities must be kept as distinct outputs, and why shared-control enrollment requires the joint probability that any arm continues.

Worked Example: A Small Exhaustive Search

Suppose a hypothetical planning exercise allows \(n_1,n_2\in\{6,7,8,9,10\}\) for a two-arm, two-stage design. There are \(5\times5=25\) candidate vectors. Applying the requirements in the first example to all candidates, an exact evaluator reports that \((8,8)\) and \((9,7)\) pass every constraint. For \((9,7)\), suppose the evaluator reports minimum required power \(0.803\), maximum familywise error \(0.049\), maximum enrollment \((2+1)(9+7)=48\), and largest scenario-specific expected enrollment \(40.1\). Each value meets its corresponding bound.

A reproducible search record would include all 25 vectors and their scenario-specific results, including the 23 candidates that failed and the constraint or constraints each failed. Reporting only the two passing vectors would make it harder to detect an omitted scenario or a mistaken feasibility rule. The search identifies feasible candidates; the choice among feasible candidates depends on a separately stated design preference.

Diagnostics, Alternatives, and Limits

A numerical search is only as reliable as its evaluator and its implementation. Useful checks include verifying that all stage sizes are positive integers within the stated ranges; confirming that the number of candidates equals the product of the range sizes; checking that probabilities lie between 0 and 1; and confirming that the first-stage enrollment and all-continue maximum enrollment match the design specification. Continuation probabilities should also agree with the stopping rules, and expected enrollment should not exceed maximum enrollment.

When a candidate appears unexpectedly feasible or infeasible, inspect scenario labels, arm ordering, stage-size indexing, and which error-control configurations were checked. A common implementation error is to count a full shared-control cohort at a later stage even when no experimental arm continues. Another is to check a minimum power requirement using a favorable scenario while overlooking a required efficacy scenario. Retain scenario-level outputs so these errors are visible.

Solving for sample size at a target power can be useful when a single sample-size quantity varies and the other design features are fixed. Integer enumeration is more direct when several stage sizes must jointly satisfy multiple nonlinear constraints and stopping probabilities depend on the candidate. A continuous numerical optimizer may be faster in a large search space, but it can return noninteger values or fail to examine all feasible integer candidates; it therefore needs additional checks before its result can support an integer design decision.

Exhaustive enumeration can become computationally expensive as the number of stages, arms, or allowable stage sizes grows. Its conclusions are also limited to the scenarios, model assumptions, and candidate ranges supplied. A feasible vector does not demonstrate adequate performance under unexamined scenarios, and numerical feasibility does not establish that the assumptions are clinically or operationally appropriate.

Common pitfall: Adding the individual arm-continuation probabilities to obtain the probability that any arm continues, then counting shared controls using that sum. The sum can exceed 1 and is not the union probability. Use the joint quantity \(P(C_k=1)\) for shared-control accrual, alongside the separate \(P(A_{jk}=1)\) terms for experimental-arm accrual.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

With two experimental arms, \(n_2=10\), \(P(A_{12}=1)=0.40\), \(P(A_{22}=1)=0.30\), and \(P(C_2=1)=0.55\), what is expected stage 2 enrollment under the shared-control rule?
Why can marginal continuation probabilities not generally be added to calculate the probability that at least one arm continues?
A candidate meets every power requirement and the maximum-enrollment limit, but exceeds the expected-enrollment limit in one required scenario. Is it feasible?
What must be recorded to make an integer search reproducible and auditable?

Solutions. (1) Expected stage 2 enrollment is \(10(0.55+0.40+0.30)=12.5\) participants. (2) The arm-continuation events can overlap, so adding their marginal probabilities double-counts outcomes where both arms continue; use their joint union probability. (3) No. Feasibility requires every applicable constraint to pass in every required scenario. (4) Record the candidate ranges, design and model assumptions, scenario set, constraints, evaluator outputs for each candidate, and the resulting pass/fail decisions.