Validating Enrollment Calculations by Enumeration

Why Enumerate Continuation Patterns?

In a multi-arm multi-stage (MAMS) design, the trial may stop early or continue with only some treatment arms. As a result, the final number enrolled is not fixed: it depends on which arms satisfy the continuation rules at each interim analysis. An analytical expected-enrollment calculation summarizes this uncertainty, but a small design can also be checked directly by listing every possible continuation pattern.

Enumeration is especially useful as a validation check. It makes explicit which outcomes of the interim decisions have been included, whether their probabilities add to one, and how many participants each outcome entails. It is not a replacement for a general calculation when the number of arms or stages is large; it is a transparent audit that can expose mistakes in a calculation or its assumptions.

The scenario matters: continuation probabilities depend on the assumed treatment effects, outcome variability, decision boundaries, and other design assumptions. Enumeration checks the arithmetic conditional on those probabilities; it does not establish that the scenario assumptions or decision rules are scientifically appropriate.

Two Equivalent Ways to Calculate the Expectation

Suppose a design has a finite set of possible continuation patterns, indexed by \(r\). Let \(N_r\) be the total enrollment under pattern \(r\), and let \(\pi_r\) be its probability under the scenario being evaluated. The patterns must be exhaustive and mutually exclusive: exactly one occurs. Therefore, \(\sum_r \pi_r=1\), and the expected enrollment is

A second calculation works stage by stage. Write total enrollment as the sum of enrollment increments at each stage. If stage \(j\) adds \(n_j\) participants whenever it is reached, and \(R_j\) is the event that it is reached, then

where \(I(R_j)\) equals 1 if stage \(j\) is reached and 0 otherwise. If the number enrolled at a stage depends on which arms remain active, the increment \(n_j\) must instead be represented by the relevant arm-specific and shared-control increments. In either form, the expected total follows by adding the expected increments; independence between stage decisions is not required.

A Short Justification of the Stage-Reach Check

For each continuation pattern \(r\), let \(I_{jr}\) equal 1 when stage \(j\) is reached under that pattern and 0 otherwise. The enrollment on pattern \(r\) is \(N_r=\sum_j n_j I_{jr}\), when increments \(n_j\) are fixed whenever a stage is reached. Substitution into the enumeration formula gives

The last equality holds because the pattern probabilities for which stage \(j\) is reached sum to \(P(R_j)\). This establishes why the enumeration and stage-reach calculations must agree when they describe the same design and scenario. If increments vary by pattern, use each pattern’s actual increment in the first calculation, or sum the corresponding arm-specific expected increments in the second.

Enumerating Arm-Specific Continuation With Shared Control

For multiple treatment arms, a pattern must record which arms continue, not merely whether the trial continues. Shared control requires particular care. If a single group of control participants serves all active treatment comparisons, those participants are counted once in total enrollment, even though their data contribute to more than one comparison.

Joint Patterns Matter: Do Not Assume Independence

An enumeration uses the joint probabilities of complete patterns. Marginal continuation probabilities alone do not determine the probability that at least one of several arms continues unless their dependence is also accounted for. Shared-control enrollment is often tied to precisely that “at least one arm” event.

How to Audit an Enumeration

A useful validation is systematic: define the pattern set, verify its probabilities, calculate each total from the actual enrollment rules, and independently reconstruct the expectation from stage or arm increments. A discrepancy should be investigated, not averaged away. Common sources include missing or duplicated patterns, confusing conditional probabilities with unconditional probabilities, and miscounting shared control.

Diagnostics should include more than checking the final sum. Confirm that the initial-stage enrollment appears in every pattern; a later stage contributes only when reached; and each participant group is counted exactly once. When probabilities are conditional on earlier continuation, multiply along the relevant branch to obtain a complete-pattern probability. For example, if stage 2 is reached with probability 0.75 and, conditional on reaching it, stage 3 is reached with probability 0.60, then \(P(\text{stage 3 reached})=(0.75)(0.60)=0.45\).

Enumeration is most practical when the number of patterns is small enough to inspect. For many arms or stages, the number of branches can become unwieldy; a stage-reach calculation or a carefully checked computational calculation is more efficient. Simulation can estimate expected enrollment when direct probabilities are difficult to obtain, but it introduces Monte Carlo error. Enumeration, when feasible and supplied with correct pattern probabilities, calculates the scenario-specific expectation exactly up to the precision of those probabilities and arithmetic.

Common Mistakes and Limitations

• Adding probabilities that describe overlapping events. “A continues” and “B continues” overlap when both continue. To calculate the probability that at least one continues, sum the mutually exclusive patterns that satisfy that condition, or use a valid joint-probability calculation.
• Using conditional probabilities as if they were pattern probabilities. A conditional probability must be multiplied by the probability of reaching the condition. The probability of two successive continuations is not generally the second conditional probability alone.
• Double-counting shared control. Count the actual control cohort specified by the design, not a separate copy for every comparison it supports.
• Checking arithmetic but not assumptions. Agreement between two calculations cannot validate incorrect continuation probabilities, an incorrectly specified enrollment rule, or an unsuitable scenario.

Enumeration is a validation tool, not a general solution to uncertainty about the design. It does not replace evaluating expected enrollment across relevant scenarios, assessing whether the assumed model is credible, or considering the uncertainty in simulation-based estimates. Its value is that, in a small example, every contribution to the expected total can be traced and checked.