Reporting MAMS Sample-Size Assumptions and Results

Why a Planning Summary Needs More Than a Sample Size

A single number such as “maximum enrollment: 300” does not describe a multi-arm multi-stage (MAMS) design. It leaves unanswered how participants are allocated at each stage, which arms may stop, what assumptions define the planning scenarios, and whether the power requirement is satisfied under those scenarios. A useful report makes these choices visible so a reader can reconstruct the design and judge whether its enrollment figures answer the planning question.

This tutorial concerns how to present those assumptions and results, not how to find an optimal design. The reporting standard is simple: identify the design that was evaluated, define the scenarios and operating-characteristic constraints, and show how the reported enrollment summaries follow from the possible paths through the stages. Where a result comes from numerical evaluation or simulation, report enough detail to distinguish the design result from computational uncertainty.

State the Design Before Reporting Its Results

Use one consistent convention for stage sizes. Let \(d_{js}\) denote the number newly enrolled to arm \(j\) at stage \(s\) if that arm is active and the design reaches that stage. Let \(N_s\) denote total enrollment after stage \(s\) along a specified path. When a control arm is shared, state its allocation explicitly: the control patients contribute to comparisons with each active treatment, but they are counted only once in total enrollment.

For each stage, report the planned number per arm, whether those numbers are new or cumulative, and what happens when one or more experimental arms stop. Also identify the analysis times and the rule that determines continuation. A schedule without its continuation rule is incomplete: the same stage sizes can yield different maximum and expected enrollment under different stopping rules.

For each design, a report should also identify the outcome and analysis scale, the treatment-control contrasts, the direction of benefit, and the testing procedure. State the significance level \(\alpha\), whether it is one- or two-sided, and how the design controls the relevant error rate across arms and stages. If the design uses stopping boundaries, give the rule or a precise reference to the boundary values and their application. A statement such as “power was 90%” is not interpretable unless it defines the event counted as success and the scenario under which that probability was evaluated.

Define Scenarios and Constraints Explicitly

A scenario is a set of assumptions under which the design’s operating characteristics are evaluated. It should specify the control outcome assumption and the treatment effects for each arm, as well as any other inputs required by the selected analysis, such as outcome variability, accrual assumptions if relevant to the target, or the handling of unusable outcomes. Label the scenario clearly—for example, “null for all arms” or “prespecified alternative for A and B”—and provide numerical values with units and a stated scale.

Power is a probability of a defined success event under a specified scenario. In a multi-arm design, that event might be declaring one particular treatment effective, or declaring both specified treatments effective. These are different requirements and can have different powers. A report should state which event is constrained and whether the target applies to every treatment comparison or to a joint event. Likewise, report the familywise error criterion under the null configuration that is relevant to the design. Do not substitute a power result under an alternative for an error-rate result under a null.

A compact scenario table can make assumptions auditable. Include the scenario label, control value, treatment effects by arm, and the relevant power or error-rate outputs. If several scenarios are used, show results scenario by scenario rather than reporting only the most favorable one or an average across assumptions. A design that meets its power target in one scenario may not meet it in another; the constraint must be checked against the scenarios specified for planning.

Maximum and Expected Enrollment

Maximum enrollment is the greatest total enrollment possible under the stated stage sizes and continuation rules. It is a design bound, not a prediction that the study will enroll that many participants. Expected enrollment is scenario-specific: it averages the enrollment over the possible stopping and continuation paths under that scenario. For a design with path-dependent stage increments, compute the expected number added at each stage using the probability of reaching that stage and the number enrolled on the path.

Here \(\theta\) denotes a specified scenario, \(D_s\) is the number newly enrolled at stage \(s\) when that stage is reached, and \(I(R_s)\) equals 1 if the design reaches stage \(s\) and 0 otherwise. If the increment is fixed whenever stage \(s\) is reached, this reduces to \(\sum_s D_sP(R_s\mid\theta)\). If the number enrolled at a stage depends on which arms remain active, use the expectation of the path-dependent increment; multiplying a single stage size by the chance of reaching that stage would then be incorrect.

The justification follows by writing total enrollment as the sum of stage increments. On each realized path, \(N=\sum_s D_sI(R_s)\), because an unreached stage contributes zero. Taking expectations under scenario \(\theta\) and using linearity of expectation gives the displayed expression. This argument does not require stage-reach events to be independent; they are generally related because reaching a later stage requires reaching earlier stages.

A useful summary reports maximum enrollment and scenario-specific expected enrollment side by side, with the units “participants.” The expected-to-maximum ratio can help describe how much lower the average is than the design bound, but it is not a probability of stopping early and does not replace either enrollment number. The earlier material on interpreting expected enrollment relative to the maximum provides the context for that ratio.

Report Simulation Precision and Check the Arithmetic

When operating characteristics or expected enrollment are estimated by simulation, give the number of replicates, the estimated quantity, and its Monte Carlo standard error (SE) or another appropriate measure of simulation uncertainty. Monte Carlo uncertainty describes numerical error from using a finite simulation run; it is not uncertainty about the true treatment effect or about the assumptions themselves. For an estimated probability \(\hat{p}\) based on \(M\) independent simulated trials, the usual estimated Monte Carlo standard error is \(\sqrt{\hat{p}(1-\hat{p})/M}\).

Before finalizing a report, verify that the maximum corresponds to the largest permitted path, that the expected enrollment is no greater than that maximum, and that stage totals count shared-control participants only once. Check that scenario probabilities for mutually exclusive paths sum to 1, and that all reported power and error rates refer to the stated analysis and decision rules. When results rely on a numerical search, report the final rounded stage sizes actually evaluated, not just the unrounded candidate: rounding can change both enrollment and operating characteristics.

Common Pitfalls and Reporting Alternatives

The main alternative to a detailed summary is to report only a single headline sample size or expected total. That is brief but conceals the assumptions and path structure needed to interpret the number. A stage-by-stage table paired with scenario-specific results is usually clearer. A narrative-only description can explain unusual rules, but tables are often better for checking allocation totals and comparing scenarios; both can be used when the design is complex.

• Giving a maximum without its path. State which arms must continue and which stage increments are included to attain the maximum.
• Giving power without its event or scenario. Identify the treatment effects, endpoint scale, and exact success event that defines power.
• Reporting expected enrollment as though it were fixed. Label it by scenario and distinguish it from maximum enrollment.
• Omitting simulation precision. A simulation estimate without its replicate count and Monte Carlo error gives no indication of numerical stability.
• Mixing inputs and outputs. Clearly separate assumptions (such as a control response rate) from calculated results (such as power or expected enrollment).

No report can make planning assumptions certain. Scenario-based results show how the evaluated design behaves under the stated inputs; they do not guarantee the future outcome or recruitment total. A concise, complete summary makes those limits visible while allowing another analyst to understand exactly which design was assessed.