Why Optimize Two Enrollment Measures?
A MAMS design can stop early when interim results meet its stopping rules, so the number of participants enrolled is not necessarily the same in every trial realization. Yet a design must also be resourced for the possibility that all planned stages are reached. This creates two distinct planning quantities: the maximum enrollment, which describes the largest planned accrual, and the expected enrollment, which averages accrual over possible stopping paths under a specified treatment-effect scenario.
The two quantities answer different questions. Maximum enrollment informs the largest recruitment demand the design must accommodate. Expected enrollment describes the average demand if the assumed scenario and stopping behavior were repeated. A design can have lower expected enrollment but a higher maximum, or the reverse. There is therefore no universal best configuration until the design team states how it will compare these objectives.
The earlier material on scenario grids supplies a finite set of plausible treatment-effect scenarios. Here, those scenarios are used to compare candidate stage-size configurations, subject to prespecified power requirements. The comparison does not replace the design’s testing procedure: power must be evaluated under the specified hypotheses, analysis, stopping rules, and multiplicity control.
Define the Candidates and the Constraints
Write \(D_j\) for the number of participants enrolled at stage \(j\) if that stage is reached, and \(I_j\) for an indicator equal to 1 when stage \(j\) is reached and 0 otherwise. If stage increments are fixed within a candidate, total enrollment is \(N=\sum_j I_jD_j\). More generally, when stage increments depend on which arms remain, calculate total enrollment along each allowed path and average over those paths.
When \(D_j\) is fixed for the candidate, \(E_k(I_jD_j)=P_k(I_j=1)D_j\), so expected enrollment is the sum of each stage’s planned increment multiplied by its probability of being reached under scenario \(k\). These reach probabilities are scenario-specific: a configuration may stop early often when treatment effects are large, but proceed further more often when effects are weak.
Let \(\pi_k\) be the design’s power under scenario \(k\), defined as the probability that the prespecified decision rule declares the relevant treatment effect successful under that scenario. If the planning requirement is at least \(\pi_{\min,k}\) in every required scenario, the feasible set is
where \(d\) represents the candidate’s stage-size choices. Power values must come from a valid calculation for the intended design; they are not inferred from expected enrollment. A candidate that saves participants on average but misses a required power target is not feasible.
Expected Enrollment From Stage-Reach Probabilities
Worked Example: Calculating Expected Enrollment for One Candidate
Suppose a hypothetical three-stage design enrolls 90 participants in stage 1, then 60 more in each later stage if reached. Under a specified treatment-effect scenario, the probabilities of reaching stages 2 and 3 are 0.70 and 0.35. Assume stage 3 can be reached only after stage 2, as required by the design’s stage order.
The maximum enrollment occurs when all three stages are reached:
Stage 1 is always reached, stage 2 is reached with probability 0.70, and stage 3 with probability 0.35. Therefore,
The candidate has an expected enrollment of 153 participants under this scenario, not a guarantee that exactly 153 will be enrolled. Its maximum remains 210 participants. The calculation assumes the stated increments apply whenever the corresponding stage is reached and that the reach probabilities are appropriate for this scenario and stopping rule.
The indicator form gives a short justification for the calculation: by linearity of expectation, \(E(\sum_j I_jD_j)=\sum_j E(I_jD_j)\). For fixed \(D_j\), the product \(I_jD_j\) equals \(D_j\) when stage \(j\) is reached and zero otherwise, so its mean is \(D_jP(I_j=1)\). No assumption that stage-reach indicators are independent is needed.
Compare Candidates Only After Checking Power
In practice, candidate configurations are generated and evaluated using the design’s prespecified methods. For each candidate, record its maximum enrollment, its power in every required scenario, and its expected enrollment under each scenario. If a single expected-enrollment summary is wanted, scenario weights \(q_k\) may be specified in advance, where \(q_k\geq0\) and \(\sum_k q_k=1\). The weighted summary is
This weighted average is a planning summary under the stated weights; it does not replace the scenario-specific results. In particular, a favorable weighted average can conceal poor performance in a scenario that has a small weight. The power constraints should therefore be checked scenario by scenario whenever the requirement applies to every scenario.
Worked Example: Screening Configurations Against Power Requirements
Consider three hypothetical configurations evaluated under two prespecified treatment-effect scenarios, \(S_1\) and \(S_2\). The required power is at least 0.80 in each scenario. The table gives outputs from a valid design calculation; power is treated as an evaluated input here, not reconstructed from enrollment alone. Suppose the planning weights are \(q_1=0.60\) and \(q_2=0.40\).
| Configuration | Maximum enrollment | Power in \(S_1\) | Power in \(S_2\) | \(E_1(N)\) | \(E_2(N)\) |
|---|---|---|---|---|---|
| A | 240 | 0.91 | 0.76 | 166 | 208 |
| B | 260 | 0.89 | 0.84 | 155 | 220 |
| C | 230 | 0.86 | 0.82 | 180 | 196 |
Configuration A fails the requirement because \(0.76<0.80\) in \(S_2\), even though its power in \(S_1\) exceeds 0.80. Configurations B and C meet both requirements. Their weighted expected enrollments are
If the stated objective is to minimize weighted expected enrollment among candidates meeting the power requirements, B is preferred: 181 is less than 186.4 participants. But B has a maximum enrollment of 260, compared with 230 for C. If a separate planning rule requires maximum enrollment no greater than 250, B is excluded and C is the feasible choice. This example shows why the objective and any additional constraints must be stated explicitly.
Trade-Offs, Dominance, and the Choice Rule
A useful first comparison is to identify candidates that are dominated. Among candidates meeting all power requirements, configuration \(a\) dominates configuration \(b\) on these two enrollment measures if \(a\) has maximum enrollment no greater than \(b\), expected enrollment no greater than \(b\) under every scenario being compared, and is strictly better on at least one of those quantities. A dominated candidate offers no enrollment advantage on either measure, although other design features not shown in the comparison could still matter.
When neither candidate dominates the other, the decision depends on priorities. Common choice rules include minimizing maximum enrollment among power-feasible candidates, minimizing a prespecified weighted expected enrollment, or minimizing expected enrollment subject to a maximum-enrollment cap. These are different optimization problems, and they can select different candidates. Reporting the scenario-specific values alongside any summary makes that dependence visible.
Worked Example: Identifying a Trade-Off Rather Than a Single Winner
Suppose configurations P and Q both satisfy every required power target. Their maximum enrollments are 220 and 250 participants, respectively. Under two scenarios, their expected enrollments are \(E_1(N_P)=170\), \(E_2(N_P)=205\), \(E_1(N_Q)=155\), and \(E_2(N_Q)=215\). Let the prespecified scenario weights be \(q_1=0.70\) and \(q_2=0.30\).
The weighted expected enrollment for P is
For Q it is
P has the smaller maximum enrollment, 220 rather than 250, and the smaller expected enrollment in scenario 2, 205 rather than 215. Q has the smaller expected enrollment in scenario 1, 155 rather than 170, and the smaller weighted expected enrollment, 173 rather than 180.5. Neither dominates the other. If the priority is a lower maximum, P is preferable; if the specified criterion is lower weighted expected enrollment, Q is preferable. The arithmetic does not determine which priority is appropriate.
Diagnostics and Limitations
Before comparing a set of candidates, check that the calculations describe the intended design and that the comparisons are interpretable. Useful checks include:
- Reach probabilities are scenario-specific. Do not reuse probabilities from one effect scenario for another unless the calculation justifies it.
- Path totals agree with the design. Verify that each path respects the stage order and arm-stopping rules, and that the maximum is the largest allowed total—not merely the sum of increments on a typical path.
- Power constraints are applied to every required scenario. A weighted average of powers does not establish that each scenario meets its target.
- Units and enrollment definitions match. State whether totals count randomized participants, participants assigned to each arm, or an already inflated target accounting for attrition. Do not mix these quantities in a comparison.
- Conclusions are stable enough for the decision. If small changes in assumed effects, scenario weights, or stage sizes change the ranking, report that sensitivity rather than presenting a fragile ranking as definitive.
Expected enrollment depends on assumptions about treatment effects and the resulting stopping behavior. It is not a promise about the enrollment of a future trial, and an average over selected scenarios is not automatically a population-level expectation. Candidate comparisons also do not capture every operational consideration, such as recruitment timing or feasibility at a particular site. Those considerations may inform the final decision, but they do not alter the definitions of maximum enrollment, expected enrollment, or the power-feasibility requirement.
A Practical Comparison Sequence
State which scenarios must meet which minimum power requirements before comparing enrollment summaries.
Obtain its power by scenario, allowed path totals, maximum enrollment, and scenario-specific expected enrollment under the intended stopping rules.
A candidate with insufficient power in any required scenario is not in the feasible set, regardless of its enrollment advantage.
Report maximum and scenario-specific expected enrollment; if using a weighted expectation, state the weights and the decision rule.
This sequence separates statistical adequacy from enrollment preference: power determines which candidates are acceptable, while the chosen enrollment objective distinguishes among acceptable candidates.
Check Your Understanding
Answer each question, then check the solutions below.
- A candidate has stage increments of 80, 50, and 50 participants. The probabilities of reaching stages 2 and 3 under one scenario are 0.60 and 0.20. Find its maximum and expected enrollment under that scenario.
- Why is a candidate with weighted expected enrollment of 140 not necessarily preferable to one with weighted expected enrollment of 150?
- Two power-feasible candidates have maximum enrollments 200 and 225. Their expected enrollments in scenarios 1 and 2 are, respectively, (160, 190) and (150, 200). Do either dominate the other on these measures?
- Suppose a candidate has power 0.86 in one required scenario and 0.78 in another, where the minimum is 0.80 in both. Is it feasible? Explain.
Solutions. (1) The maximum is \(80+50+50=180\) participants. Expected enrollment is \(80+(0.60)(50)+(0.20)(50)=80+30+10=120\) participants. (2) It may fail a required power target, have a larger maximum, or depend on weights that do not reflect the planning objective; expected enrollment alone is not enough to establish preference. (3) Neither dominates: the first has smaller maximum and smaller scenario-2 expected enrollment, while the second has smaller scenario-1 expected enrollment. (4) It is not feasible, because \(0.78<0.80\) in a required scenario.