Averaging Expected Sample Size Across Scenarios

Why Average Across Scenarios?

A MAMS design can have different expected enrollment under different treatment-effect configurations. For example, a design may tend to stop early when no treatment is active, but continue more often when several treatments are effective. Reporting expected enrollment under one configuration answers a useful question, but it does not summarize enrollment across a stated range of plausible configurations.

A scenario-weighted expected enrollment combines scenario-specific expectations using probabilities assigned to those configurations. The result depends on both ingredients: the expected enrollment within each scenario and the weights placed on the scenarios. It is therefore important to show those ingredients, rather than present the weighted mean as if it were a property of the design alone.

The weights represent a probability distribution over the configurations for the purpose of the summary. They are not the probabilities of rejecting hypotheses, the probabilities that an individual treatment works, or the continuation probabilities at an interim analysis. Nor are scenario probabilities automatically supplied by a MAMS design. They must be chosen or specified for the planning question being answered.

From Continuation Probabilities to a Weighted Mean

Suppose a design has \(J\) stages, and \(d_j\) participants are enrolled at stage \(j\) if that stage is reached. Here \(d_j\) is an incremental stage enrollment, not the cumulative enrollment through stage \(j\). Let \(\pi_{j,s}\) be the probability, under scenario \(s\), that stage \(j\) is reached. The first stage is always reached, so \(\pi_{1,s}=1\). If \(I_j\) equals 1 when stage \(j\) is reached and 0 otherwise, then \(N=\sum_{j=1}^J d_jI_j\). Taking the expectation under scenario \(s\) gives:

The expected enrollment across scenarios follows by averaging these scenario-specific expectations with their weights. Equivalently, the probability of reaching stage \(j\) in the scenario mixture is \(\sum_s w_s\pi_{j,s}\), so:

This identity is a direct application of the law of total expectation: the scenarios form a mutually exclusive and exhaustive partition, and the probability of each scenario is \(w_s\). It also shows two equivalent calculation routes. First calculate \(E[N\mid S=s]\) for each scenario and then weight those values, or first average the stage-reach probabilities and then calculate expected enrollment. Both routes must give the same result when they use the same stage sizes, scenarios, and weights.

Direct Weighting When Scenario Expectations Are Available

Sometimes the scenario-specific expected enrollments have already been calculated, for example from the continuation probabilities under each treatment-effect configuration. In that case, the averaging step is a weighted sum. Check the scenario definitions and probabilities before doing the arithmetic: a correct multiplication cannot repair weights that do not represent the intended scenario mixture.

Interpretation, Checks, and Alternatives

A scenario-weighted expectation is useful when a single planning summary is needed and the scenario probabilities have a defensible interpretation. For example, the weights might encode a specified distribution over plausible treatment-effect configurations. If the weights are instead chosen merely to summarize a scenario grid, the result should be labelled as a weighted summary under those chosen weights, not as a predictive expectation based on known probabilities.

A weighted mean does not replace the scenario-specific results. Two designs can have the same weighted expected enrollment but behave quite differently across scenarios: one may have similar expectations in every scenario, while another may have very low enrollment in one and very high enrollment in another. Reporting the weights and scenario-specific values makes that variation visible.

Other summaries answer different questions. A single expected enrollment under the global null or global alternative describes that particular configuration, not a mixture. An unweighted mean gives every listed scenario equal influence regardless of its stated probability. A maximum-enrollment figure describes the design’s cap, not its average under a scenario distribution. These summaries are not interchangeable; choose the one that matches the planning question, and show the scenario-specific results when the spread matters.

A frequent mistake is to list several plausible scenarios, assign equal weights without explanation, and call the arithmetic mean the expected enrollment. The correction is to state what the weights mean. Equal weights can be appropriate when the aim is an explicitly equal-weight summary of the listed scenarios; they do not become scenario probabilities merely because they sum to 1. Similarly, assigning probabilities to scenarios is a modelling or planning choice, not evidence that treatment effects are randomly drawn from those configurations in the actual trial.

The weighted mean also has limits. It compresses a range of possible scenario-specific behavior into one number, depends on the chosen scenarios and weights, and can conceal sensitivity to either choice. A useful report therefore gives the weighted expectation together with the scenario definitions, weights, and conditional expectations. If those inputs are uncertain, show how the summary changes under reasonable alternative weights rather than implying unwarranted precision.