Diagnosing Common Expected-Sample-Size Errors

Why Expected-Enrollment Calculations Go Wrong

In a multi-arm multi-stage (MAMS) trial, the total number of participants enrolled is not fixed when the trial can stop early. A stage’s enrollment occurs only if the trial reaches that stage, so expected enrollment must account for both the number enrolled at each stage and the probability that the stage is reached. Two further details complicate the calculation: a shared control group contributes participants to several comparisons but only once to trial enrollment, and a reported probability of continuing may be conditional on reaching the stage where the decision is made.

This tutorial focuses on three audit questions: Are unique participants being counted? Are the probabilities unconditional reach probabilities? Are participants assigned only to stages the trial actually reaches? A calculation can be arithmetically neat and still answer the wrong question if any of these checks fails.

The formula is about new enrollment at each stage. If a design gives cumulative sample sizes instead, those must first be converted to increments. For example, cumulative targets of 90 and 150 mean 90 participants at stage 1 and 60 additional participants at stage 2—not 90 and then another 150.

Count Unique Participants, Not Comparison Appearances

In a trial with two experimental arms and one shared control arm, each control participant may contribute to both treatment-control comparisons. That does not make the participant two people for the purpose of total enrollment. The number needed for a comparison and the total number enrolled in the trial are different quantities.

Suppose stage \(j\) enrolls \(n_{C,j}\) participants in the shared control and \(n_{k,j}\) in experimental arm \(k\). If there are \(K\) experimental arms, the stage’s unique enrollment increment is

The control count appears once in this total. Adding the sample sizes of each treatment-control comparison instead would count the shared control repeatedly. That may be appropriate for describing comparison-specific sample sizes, but not for calculating total trial enrollment.

Conditional Continuation Is Not an Unconditional Reach Probability

Continuation probabilities are often reported conditional on the trial having reached the decision point. Let \(q_j=P(R_{j+1}\mid R_j)\) be the probability of continuing from stage \(j\) to stage \(j+1\), given that stage \(j\) is reached. Let \(r_j=P(R_j)\) be the unconditional probability that stage \(j\) is reached. Because stage \(j+1\) can be reached only if stage \(j\) is reached,

Consequently, reaching a later stage requires surviving every earlier stopping decision. For example, \(P(R_3)=q_1q_2\), not just \(q_2\), when stage 3 requires continuation at both preceding decisions. More generally, \(r_j=\prod_{i=1}^{j-1}q_i\), provided the stages are sequential and the \(q_i\) are the applicable conditional continuation probabilities.

Use Stage Increments and Respect the Stopping Path

A participant is counted in expected enrollment only if the corresponding stage is reached. The indicator \(I(R_j)\) in the definition enforces this rule: if the trial stops before stage \(j\), then \(I(R_j)=0\), and that stage contributes no participants to that trial’s total. This is why planned enrollment at every possible stage cannot simply be added together and called expected enrollment.

When stage sizes are reported cumulatively, the conversion to increments is essential. If \(C_j\) is the cumulative target through stage \(j\), define \(C_0=0\). Then the new enrollment at stage \(j\) is \(\Delta N_j=C_j-C_{j-1}\). The expectation uses \(\Delta N_j\), weighted by \(P(R_j)\), rather than \(C_j\), weighted by that probability. Otherwise, participants enrolled at earlier stages are counted again.

A Short Justification and a Practical Audit

The expected-enrollment formula follows directly from the total-enrollment definition. For fixed stage increments, expectation is additive, and the indicator of reaching a stage has expectation equal to its probability:

This argument does not require the stage-reach events to be independent. In a sequential design they are nested: reaching a later stage entails reaching each earlier stage. The calculation does require that \(\Delta N_j\) is the applicable fixed increment whenever stage \(j\) is reached. If the increment varies by the observed path—for example, because different arms continue with different allocations—then the simple fixed-increment expression is not enough. One must calculate the expected enrollment contribution by path, or use \(E[\Delta N_j I(R_j)]\) with the path-dependent increment explicitly represented.

A reliable audit follows the enrollment through the design rather than starting with a single total. For each stage, write down the new unique participants, identify the event required to reach that stage, and use the probability of that event from the start of the trial. If probabilities are supplied as conditional continuation probabilities, multiply them along the path to obtain the unconditional reach probabilities.

Diagnostics, Alternatives, and Limitations

Several checks can expose errors before a result is used for planning:

• Compare the control count. If the design has one shared control cohort, check that the trial-level total includes it once at each reached stage, even though the control informs multiple contrasts.
• Check probability direction. For sequential stages, unconditional reach probabilities cannot increase: \(1=P(R_1)\geq P(R_2)\geq\cdots\geq P(R_J)\). A later-stage probability larger than an earlier-stage probability signals a likely mismatch or mistake.
• Check the stopping paths. The probabilities of mutually exclusive stopping outcomes should sum to 1. If not, a path may be missing, counted twice, or assigned the wrong conditional probability.
• Check units and bounds. Expected enrollment is measured in participants and must lie between the smallest and largest possible total enrollment. A result exceeding the all-stages total, or below the minimum enrollment, is inconsistent with the stated design.
• Check for cumulative double-counting. If a later-stage target includes participants from earlier stages, do not treat that target as a new increment.

A spreadsheet or direct algebraic calculation is suitable when stage increments and reach probabilities are clearly defined. For more complicated path-dependent designs, exact path enumeration can sum each possible total multiplied by its path probability; simulation can estimate the mean across simulated trial paths. These are useful independent checks, but neither fixes incorrectly specified inputs. In particular, simulation will reproduce a shared-control double count or an incorrect conditional probability if those errors are built into the calculation.

The fixed-increment formula also has limits. It does not by itself model recruitment delays, screen failures, attrition, or differences between enrolled and evaluable participants. Those require the relevant enrollment definitions and assumptions to be handled explicitly. Likewise, reach probabilities depend on the assumed treatment effects, outcome distributions, and stopping rules. An expectation is therefore a design-based average under specified scenarios, not a promise that a particular trial will enroll that number.

Common Mistakes

• Wrong: Add the sample sizes for each treatment-control comparison to get trial enrollment. Correction: When controls are shared, count each enrolled control participant once in the unique trial total.
• Wrong: Use \(P(R_{j+1}\mid R_j)\) as the chance of reaching stage \(j+1\) from trial start. Correction: Multiply the conditional continuation probabilities along the required path to obtain \(P(R_{j+1})\).
• Wrong: Include all planned participants at later stages whether or not the trial continues. Correction: Weight each stage’s new enrollment by its reach probability; a stage not reached contributes zero.
• Wrong: Apply reach probabilities to cumulative targets. Correction: First convert cumulative targets to stage-specific enrollment increments.