Expected Sample Size Under Partial Treatment Activity

Why Partial Treatment Activity Changes Expected Enrollment

A MAMS design may begin with several experimental treatments, but the assumed benefit need not be the same for every arm. One treatment might have the effect used in the design’s power calculation, another might have no effect, and a third might have a smaller effect. These differences change the chances that individual arms pass interim continuation rules—and therefore change how many participants are enrolled at later stages.

The global null and global alternative describe useful limiting scenarios: all experimental arms have no effect, or all have their assumed beneficial effects. Partial activity asks a different planning question: what expected enrollment results from a specified pattern of effects across arms? The answer depends on both each arm’s chance of continuing and the dependence among arms’ interim statistics.

Modeling Arm-Specific Continuation

Suppose there are \(J\) experimental arms, indexed by \(j=1,\ldots,J\), and a common control. Let \(Z_{jk}\) be the interim test statistic for arm \(j\) at stage \(k\). For illustration, assume that under a specified effect scenario the vector of stage-\(k\) statistics follows a multivariate normal model:

Here \(\boldsymbol{\eta}_k\) contains the arm-specific mean shifts on the \(Z\)-statistic scale, and \(R_k\) is the correlation matrix, which can reflect the shared control. If arm \(j\) continues when \(Z_{jk}>c_{jk}\), its marginal continuation probability is

where \(\Phi\) is the standard normal cumulative distribution function. The mean shift \(\eta_{jk}\) is determined by the assumed effect for that arm and the stage’s information. A beneficial effect generally raises the continuation probability relative to a no-effect assumption, but does not guarantee that the arm continues.

These marginal probabilities do not, by themselves, determine whether any arm continues. If statistics share a control, they are typically correlated. For two arms, for example,

The joint probability must be calculated using the joint model, such as a bivariate normal probability with the appropriate correlation. Multiplying marginal probabilities as if the arms were independent is justified only when independence is a reasonable assumption for the decision statistics.

From Continuation Decisions to Expected Enrollment

Let \(A_k\) denote the set of experimental arms still active when stage \(k\) begins. The first-stage set contains all \(J\) arms; later sets depend on earlier decisions. Let \(g_k(A_k)\) be the number of participants enrolled at stage \(k\) when the active set is \(A_k\). If no arms remain, the design stops and \(g_k(\varnothing)=0\). Total enrollment is the sum of stage enrollments along the realized path.

The second expression makes the scenario dependence explicit: the probabilities of the possible active sets are determined by the arm-specific effects, the interim rules, and the dependence structure. The formula does not require equal effects or equal continuation probabilities across arms.

A common allocation rule makes the calculation especially transparent. Suppose a stage enrolls \(n_C\) control participants if at least one arm remains and \(n_T\) participants per active treatment arm. Then

The treatment-arm term follows by writing the number of active arms as a sum of indicators: \(|A_k|=\sum_j I(j\in A_k)\). Taking expectations gives \(E(|A_k|)=\sum_jP(j\in A_k)\). The control term is different: one control group is enrolled if any arm remains, so its expected contribution uses \(P(A_k\ne\varnothing)\), not the sum of marginal continuation probabilities. This distinction matters whenever control enrollment is shared.

More Than One Interim Stage

With additional stages, an arm must remain active through all earlier decisions to contribute treatment participants at a later stage. For instance, if an arm’s stagewise continuation decisions are independent across stages and its continuation probabilities are \(q_{j1}\) and \(q_{j2}\), its probability of reaching stage 3 is \(q_{j1}q_{j2}\). Independence of decisions across stages is a modeling assumption, often motivated by independent increments; it should not be imposed if the design’s statistics or decision rules make it inappropriate.

Across arms, the probability that at least one reaches a later stage still depends on their joint behavior. One useful general calculation is to enumerate possible active sets, determine each set’s probability under the multivariate interim model, and multiply by the enrollment cost for that set. For many arms, direct enumeration may be inefficient; numerical integration or simulation from the specified joint model can estimate the same expectation.

Assumptions, Diagnostics, and Comparisons

An expected-enrollment calculation is conditional on a complete set of design and scenario assumptions. Specify the effect for every arm, the information and continuation rule at each interim, the joint distribution of the test statistics, and the enrollment cost for every possible active set. If the design uses a shared control, the model should reflect the resulting correlation. If stage sizes or allocation ratios change when arms are dropped, use the actual stage-specific enrollment function \(g_k(A_k)\), not a constant cost for each stage.

Check the calculation in several ways. First, the expected enrollment must lie between the minimum enrollment and the maximum permitted by the design. Second, the probability of any arm continuing must be at least as large as each individual continuation probability and no larger than their sum (or 1, if the sum exceeds 1). Third, when one arm’s assumed effect is increased while all else is held fixed in a monotone continuation design, its marginal continuation probability should not decrease. Finally, compare a direct sum over active sets with simulation from the same joint model when practical; material disagreement can reveal an error in indexing, dependence, or stage costs.

The global-null calculation sets all arm effects to zero; the global-alternative calculation assigns the assumed beneficial effect to every arm. Partial activity lies between these cases only in an informal sense: enrollment need not be a simple numerical interpolation, because continuation probabilities and the chance of any arm remaining are nonlinear functions of the effects and their dependence. Averaging expected enrollment across several scenarios is a separate planning choice that requires scenario weights; a single partial-activity calculation does not supply those weights.

A common mistake is to calculate each arm’s expected treatment enrollment correctly, then use \(q_1+q_2\) as the probability of enrolling the shared control group. That sum is the expected number of continuing arms, not the probability that at least one continues. The correction is to use the joint probability of the active set; for two arms, \(P(A\ne\varnothing)=q_1+q_2-P(\text{both continue})\). Another failure is to assign every arm the same effect simply because the design has one common alternative hypothesis. Under partial activity, the scenario must explicitly state the effect assumed for each arm.

Expected enrollment is a planning summary, not a prediction of the realized sample size in a particular trial. It is also sensitive to uncertain effects, covariance assumptions, and operational features such as delayed outcomes or missed assessments. When these inputs are uncertain, report the scenario-specific expectations and the assumptions behind them rather than presenting one value as universally applicable.