Conditional and Unconditional Expected Enrollment

Two Expected Enrollments, Two Different Questions

A MAMS design may stop early, so the number enrolled is not necessarily the maximum planned number. An expected enrollment calculated across all trials from the point of launch answers a planning question about average recruitment burden. An expected enrollment conditional on reaching an interim stage answers a different question: among trials that have already reached that stage, how many participants will have been enrolled by the time enrollment ends, on average?

The distinction matters operationally. A recruitment team planning the overall budget needs the unconditional expectation, while a team deciding what resources to have available after an interim may find the conditional expectation more relevant. Neither is a substitute for the other, and the conditional quantity is generally larger because it is calculated among trials known to have continued far enough to enroll at least one later stage.

The calculations below assume a common trial-level continuation decision at each interim. They do not cover designs in which treatment arms stop independently at different stages.

Notation and Assumptions

Definition: Let \(m_j\) be the fixed number of additional participants enrolled in stage \(j\), if that stage is reached. Let \(R_j\) denote the event that the trial reaches stage \(j\), with \(R_1\) certain. For each interim after stage \(j\), let \(q_j=P(R_{j+1}\mid R_j)\) be the probability of continuing to the next stage, under a specified planning scenario. The total enrollment \(N\) is the sum of the stage increments actually enrolled.

For these quantities to answer the intended question, assume that stage increments are fixed and known, each participant is counted once, and the continuation probabilities refer to the same design and scenario as the enrollment calculation. A stage is considered reached after its planned participants have been enrolled and its interim decision can be made. Any chance of stopping before the next stage is included in \(q_j\). If recruitment varies within a stage, participants are lost from the enrollment count, or stage sizes are random, the simple fixed-increment formulas need modification.

The probability of reaching a later stage is the product of the continuation probabilities along the path. For example, reaching stage 3 requires continuing after both stages 1 and 2.

P(R_j)=\prod_{\ell=1}^{j-1}q_\ell,\qquad P(R_1)=1.

An empty product, as in the expression for stage 1, is defined to be 1. The probabilities \(q_j\) are conditional probabilities, not the probabilities of reaching stage \(j+1\) from trial start.

Unconditional Expected Enrollment

From trial start, stage \(j\)'s participants are enrolled only if the trial reaches that stage. Thus the expected contribution from stage \(j\) is its enrollment increment multiplied by its probability of being enrolled. Summing those contributions gives the unconditional expected enrollment.

Result: Under the fixed-increment assumptions, expected enrollment from trial start is

E(N)=\sum_{j=1}^{K}m_jP(R_j) =\sum_{j=1}^{K}m_j\prod_{\ell=1}^{j-1}q_\ell,

where \(K\) is the final planned stage.

This expression includes stage 1 with probability 1, even if the trial stops at its first interim. It is not the maximum enrollment, which is \(\sum_{j=1}^{K}m_j\), and it is not an estimate conditional on observing a particular interim decision.

Worked Example: Expected Enrollment From Trial Start

Consider a hypothetical three-stage design with stage increments of 100, 80, and 80 participants. Under one planning scenario, the continuation probabilities are \(q_1=0.65\) and \(q_2=0.40\). Find the probability of reaching each stage and the unconditional expected enrollment.

Stage 1 is certain, so \(P(R_1)=1\). The probability of reaching stage 2 is \(P(R_2)=q_1=0.65\). Reaching stage 3 requires both continuation decisions, giving \(P(R_3)=q_1q_2=(0.65)(0.40)=0.26\).

The expected enrollment is therefore

E(N)=(100)(1)+(80)(0.65)+(80)(0.26) =100+52+20.8=172.8\text{ participants}.

The maximum enrollment is \(100+80+80=260\) participants. The unconditional expectation of 172.8 is lower because some hypothetical trials stop after stage 1 or stage 2. It is an average over trials starting at launch under the specified continuation probabilities; it does not mean that an individual trial enrolls a fractional participant.

Expected Enrollment Conditional on Reaching a Stage

Once a trial has reached stage \(k\), the participants in stages 1 through \(k\) have already been enrolled. The uncertainty concerns only stages after \(k\). Conditional on \(R_k\), the probability of reaching stage \(j>k\) is the product of the continuation probabilities from stage \(k\) through stage \(j-1\). Hence the expected total enrollment, conditional on reaching stage \(k\), is the enrollment already incurred plus the conditional expected future increments.

Result: For a trial that reaches stage \(k\), the expected total enrollment conditional on reaching that stage is

E(N\mid R_k) =\sum_{j=1}^{k}m_j +\sum_{j=k+1}^{K}m_j\prod_{\ell=k}^{j-1}q_\ell.

The expected enrollment still to come after stage \(k\), conditional on reaching it, is the second sum alone.

The wording “conditional on reaching stage \(k\)” is important: it includes the enrollment in stage \(k\). If a question instead asks for expected additional enrollment after the stage-\(k\) interim, omit the first sum. State which quantity is being reported and give its units.

Worked Example: Conditioning on Reaching Stage 3

A hypothetical four-stage trial plans increments of 72, 48, 60, and 40 participants. Its continuation probabilities are \(q_1=0.75\), \(q_2=0.60\), and \(q_3=0.50\). Calculate expected total enrollment conditional on reaching stage 3, and compare it with the unconditional expectation.

By the time stage 3 is reached, the first three increments have been enrolled: \(72+48+60=180\) participants. Conditional on reaching stage 3, the trial continues to stage 4 with probability \(q_3=0.50\). Thus expected additional enrollment after the stage-3 decision is \((40)(0.50)=20\) participants, and

E(N\mid R_3)=180+20=200\text{ participants}.

From trial start, the probabilities of reaching stages 2, 3, and 4 are \(0.75\), \((0.75)(0.60)=0.45\), and \((0.75)(0.60)(0.50)=0.225\), respectively. Therefore,

E(N)=72+(48)(0.75)+(60)(0.45)+(40)(0.225) =72+36+27+9=144\text{ participants}.

The conditional expectation is 200 participants, compared with 144 unconditionally. These are not competing estimates of the same quantity: the first is among trials that have reached stage 3, while the second averages over every trial from launch. The expected additional enrollment after reaching stage 3 is 20 participants, not 200.

Why the Conditional Formula Works

A short indicator argument gives both formulas and clarifies the conditioning. Define \(I_j\) to be 1 if the trial reaches stage \(j\), and 0 otherwise. Total enrollment can be written as \(N=\sum_{j=1}^{K}m_jI_j\). By linearity of expectation, \(E(N)=\sum_jm_jE(I_j)\), and \(E(I_j)=P(R_j)\). This proves the unconditional expression.

For the conditional expression, if \(R_k\) occurs then \(I_j=1\) for every \(j\leq k\). For a later stage \(j>k\), the chance of reaching it given \(R_k\) is the product of the remaining continuation probabilities. Applying linearity of conditional expectation gives

E(N\mid R_k) =\sum_{j=1}^{k}m_j +\sum_{j=k+1}^{K}m_jE(I_j\mid R_k) =\sum_{j=1}^{k}m_j +\sum_{j=k+1}^{K}m_j\prod_{\ell=k}^{j-1}q_\ell.

This argument relies on the continuation probabilities being defined for the relevant stage and scenario. It does not require the continuation decisions at different stages to be unconditionally independent: the product follows by applying the conditional probabilities successively along the path.

Worked Example: Conditioning Changes Scenario Weights

Suppose a hypothetical planning exercise uses two scenarios. Scenario A has weight 0.60 and continuation probabilities \(q_1=0.80\), \(q_2=0.50\). Scenario B has weight 0.40 and probabilities \(q_1=0.40\), \(q_2=0.25\). There are three stages, with increments of 100, 60, and 60 participants. Find unconditional expected enrollment averaged over the scenarios, then find expected enrollment conditional on reaching stage 2 in this scenario mixture.

First calculate the unconditional expectation within each scenario. For A, it is \(100+(60)(0.80)+(60)(0.80)(0.50)=172\) participants. For B, it is \(100+(60)(0.40)+(60)(0.40)(0.25)=130\) participants. Averaging with the stated scenario weights gives

(0.60)(172)+(0.40)(130)=103.2+52=155.2\text{ participants}.

For the conditional calculation, reaching stage 2 is more likely under A than B. The probability of reaching stage 2 in the mixture is \((0.60)(0.80)+(0.40)(0.40)=0.64\). By Bayes' rule, the scenario weights among trials that reach stage 2 are

P(A\mid R_2)=\frac{(0.60)(0.80)}{0.64}=0.75,\qquad P(B\mid R_2)=\frac{(0.40)(0.40)}{0.64}=0.25.

The conditional probability of continuing from stage 2 to stage 3 in this group is \((0.75)(0.50)+(0.25)(0.25)=0.4375\). Since 160 participants have been enrolled by stage 2, the conditional expected total is

E(N\mid R_2)=160+(60)(0.4375)=186.25\text{ participants}.

Using the original weights 0.60 and 0.40 to average conditional expectations would be wrong: trials that reach stage 2 are disproportionately from scenario A. Conditioning changes the scenario composition of the trials being averaged.

Interpretation, Diagnostics, and Alternatives

Expected enrollment is a planning summary, not a guarantee about an individual trial. A useful check is to compare the result with the possible enrollment range. With fixed increments and certain stage 1, the unconditional expectation must lie between \(m_1\) and the maximum \(\sum_jm_j\). Conditional on reaching stage \(k\), the expectation must lie between the cumulative enrollment through stage \(k\) and the maximum. The probabilities of reaching successive stages must also be nonincreasing and between 0 and 1.

Before reporting a result, check that each \(q_j\) is a continuation probability conditional on reaching the relevant interim, that products use the correct sequence of stages, and that the requested quantity is total or additional enrollment. If probabilities come from a scenario model, label the scenario; an expectation under one assumed treatment-effect configuration is not automatically an overall prediction.

Quantity	Question answered	Calculation
Maximum enrollment	How many participants if every stage is reached?	Sum all stage increments
Unconditional expected enrollment	How many on average across trials launched from trial start?	Weight each increment by its reach probability
Conditional expected enrollment	How many on average among trials known to have reached stage \(k\)?	Include accrued enrollment and weight future increments by conditional reach probabilities

The maximum is useful for capacity and worst-case recruitment planning. The unconditional expectation describes average enrollment from launch. The conditional expectation describes the average total enrollment for a selected group of trials that has reached a specified stage. For budgeting, a single mean may not be enough: the distribution of enrollment or the chance of exceeding a recruitment limit may also matter.

Mistake: using a marginal reach probability as though it were a continuation probability. For example, \(P(R_3)\) is not \(P(R_3\mid R_2)\). The former includes the chance of reaching stage 2; the latter is the stage-2 continuation probability.
Mistake: calling expected additional enrollment the conditional total. The conditional total includes the participants already enrolled through stage \(k\); the additional amount does not.
Mistake: averaging conditional expectations with unchanged scenario weights. If scenarios have different probabilities of reaching the conditioning stage, the weights among reachers change. Use Bayes' rule, or calculate the mixture's conditional continuation probabilities directly.

Key takeaway: From trial start, weight every stage increment by the probability of reaching that stage. Conditional on reaching an interim, count enrollment already accrued in full and weight only later increments by the probabilities of continuing from that interim onward.

Check Your Understanding

Try each question, then compare your work with the solutions below. All values are hypothetical.

A three-stage design has increments of 90, 70, and 50 participants, with \(q_1=0.80\) and \(q_2=0.50\). What is unconditional expected enrollment?
For the design in question 1, what is expected total enrollment conditional on reaching stage 2? How much of that expectation is additional enrollment after stage 2?
A four-stage design has increments of 40 participants per stage, and each continuation probability is 0.50. What is the probability of reaching stage 4, and what is unconditional expected enrollment?
Explain why \(P(R_3)\) cannot generally be substituted for \(P(R_3\mid R_2)\) when calculating expected future enrollment conditional on reaching stage 2.

Solutions. (1) The stage-reach probabilities are \(1\), \(0.80\), and \((0.80)(0.50)=0.40\). Thus \(E(N)=90+(70)(0.80)+(50)(0.40)=166\) participants. (2) Reaching stage 2 means 160 participants have been enrolled. The probability of continuing to stage 3 is 0.50, so \(E(N\mid R_2)=160+(50)(0.50)=185\) participants; expected additional enrollment is 25 participants. (3) Reaching stage 4 requires three continuations, so \(P(R_4)=(0.50)^3=0.125\). The reach probabilities are \(1, 0.50, 0.25, 0.125\), giving \(E(N)=40(1+0.50+0.25+0.125)=75\) participants. (4) \(P(R_3)\) includes the chance of reaching stage 2, whereas \(P(R_3\mid R_2)\) assumes stage 2 has already been reached. The latter is the relevant probability for weighting stage-3 enrollment after conditioning on \(R_2\).