Interpreting Expected Enrollment Relative to the Maximum

Why Expected Enrollment and Maximum Enrollment Answer Different Questions

A multi-arm multi-stage (MAMS) design can stop early when interim results meet a stopping rule, so its realized enrollment may be below the amount needed to complete every stage. This creates two useful but distinct quantities. Expected enrollment summarizes average enrollment under a specified scenario and design. Maximum enrollment describes the largest enrollment the design may require if the trial proceeds through all planned stages. The expectation can help compare the likely resource use of designs; the maximum indicates the enrollment capacity that may still be needed.

Expected enrollment is scenario-specific because the probabilities of stopping or continuing depend on the assumed treatment effects, variability, stopping boundaries, allocation, and other design features. It is not a promise about the number of participants a particular trial will enroll. Nor does a low expectation change the design’s stopping rules or guarantee that recruitment will end early.

Calculating Expected Enrollment From Stage-Reach Probabilities

First consider a design with fixed total stage increments. Let \(n_k\) be the number of participants newly enrolled at stage \(k\), across all arms, and let \(R_k\) equal 1 if stage \(k\) is reached and 0 otherwise. The first stage is always reached, so \(R_1=1\). Total enrollment is the sum of the increments for stages that are reached. Since \(E_s[R_k]=P_s(R_k=1)\), taking expectations gives the stage-based calculation:

The maximum for this fixed-increment design is \(N_{\max}=\sum_{k=1}^{K} n_k\). The stage-reach probabilities are unconditional probabilities: \(P_s(R_k=1)\) is the probability, under scenario \(s\), that the trial reaches stage \(k\), including all the decisions required to continue through earlier stages. They therefore cannot increase with stage number. In particular, \(P_s(R_1=1)=1\), and \(P_s(R_{k+1}=1)\leq P_s(R_k=1)\).

If the number enrolled at a stage depends on the path—for example, because the allocation or number of active arms changes—the fixed-increment formula needs to reflect that. In general, list the possible paths \(j\), let \(N_j\) be total enrollment on path \(j\), and let \(P_s(j)\) be its probability under scenario \(s\). Then \(E_s[N]=\sum_j P_s(j)N_j\), and the maximum is the largest \(N_j\) among paths allowed by the design. This path-based definition avoids treating a path-dependent stage size as if it were fixed.

Worked Examples

Interpreting the Comparison and Checking the Calculation

The ratio \(E_s[N]/N_{\max}\) is a convenient descriptive summary of expected enrollment relative to the design cap. For example, a ratio of 0.70 means the scenario-specific expectation is 70% of the maximum. It does not mean that every trial will enroll 70% of the maximum, nor does it describe the probability of stopping at any particular stage. A single average cannot show the full distribution of possible enrollment.

For a nonnegative enrollment variable bounded above by \(N_{\max}\), \(E_s[N]\geq N_{\max}P_s(N=N_{\max})\). Consequently, \(P_s(N=N_{\max})\leq E_s[N]/N_{\max}\). This is only an upper bound on the probability of using the maximum, not a value for that probability. For instance, an expectation-to-maximum ratio of 0.48 does not tell us that there is a 48% chance of reaching the maximum.

Useful checks include confirming that the maximum matches the planned all-stages enrollment, that each stage’s reach probability is between 0 and 1, and that later-stage reach probabilities do not exceed earlier-stage probabilities. The expected enrollment should be no less than the mandatory first-stage enrollment and no greater than the maximum. If the trial must enroll all participants in stage 1, a calculated expectation below that stage’s size signals an error in the inputs or arithmetic. If expected enrollment exceeds the maximum, check for a duplicated stage, inconsistent units, or probabilities applied to the wrong increments.

The reach probabilities must also correspond to the scenario being interpreted. Changing the assumed effects or other scenario inputs can change the probabilities of continuing, and therefore the expectation, without changing the design’s maximum. If expected enrollment is estimated by simulation, the estimate also has Monte Carlo error: apparent small differences between scenarios may partly reflect simulation noise. That uncertainty in the estimate is distinct from variation in the enrollment path of an actual trial.

Why the Maximum Still Matters

The maximum is an operational requirement, not another estimate of average use. Recruitment systems, sites, funding, and follow-up arrangements may need to remain workable if the design continues through every planned stage. Planning only around a low expected enrollment can leave the trial unable to continue when an uncommon but valid path requires more participants. Early stopping is governed by the design’s rules and observed interim data; it is not guaranteed by a favorable expected-enrollment calculation.

Expected enrollment and maximum enrollment are therefore complementary, not competing, summaries. Expected enrollment is useful when comparing anticipated resource use across designs or scenarios. Maximum enrollment provides the all-stages bound for feasibility and contingency planning. If a decision specifically concerns a chance of exceeding a resource threshold, the relevant quantity is a probability or other feature of the enrollment distribution, not the expectation alone.

Common Mistakes and Limitations

• Treating expected enrollment as a promised sample size. The expectation averages over possible paths. A particular trial follows one path and may enroll substantially fewer or more participants than the expectation.
• Using the expectation as the recruitment cap. A low average does not rule out the all-stages path. If continuing to the final stage is possible, the maximum remains relevant to feasibility.
• Comparing scenarios as if they were different designs. If stage sizes and stopping rules are unchanged, a lower scenario-specific expectation does not imply a lower maximum.
• Confusing conditional and unconditional reach probabilities. The formula uses the probability of reaching a stage from the start of the trial. A conditional probability of continuing given that the previous stage was reached must be converted to an unconditional probability before weighting that stage’s enrollment.
• Assuming the expected enrollment is the sample size used for design guarantees. The design’s operating characteristics are tied to its planned stages and stopping rules, not to replacing the design with a fixed trial of size \(E_s[N]\).

These summaries also have limits. The maximum and expectation alone do not describe the distribution of enrollment across paths, the probability of reaching each individual stage, or uncertainty in an estimated expectation. A scenario-specific expectation is only as informative as the assumptions used to calculate its path probabilities. It should not be interpreted as a guarantee that the assumed scenario is true.