Expected Events Under Arm Dropping

Why Arm Dropping Changes the Event Expectation

In a time-to-event MAMS design, an intermediate outcome may determine whether an arm continues recruiting. Dropping an arm can reduce its eventual event count because planned participants who would have entered later are not enrolled. But participants enrolled before the decision may still be followed, and their events may contribute to the final analysis. The expected number of final-endpoint events therefore depends both on how many participants enter each recruitment period and on the probability that the arm is still open at that period.

The calculation is an expectation over possible selection paths. It is not a promise that the trial will observe exactly that many events. In particular, if the intermediate outcome used for selection is associated with the final endpoint, the final-event probability among participants recruited after a successful selection decision may differ from its unconditional value.

Definition: A recruitment cohort is the set of participants planned to enter an arm during a specified period, such as before the first selection look or between two looks. The arm is active at a cohort’s entry if it has not been discontinued at any preceding selection look. A participant contributes a final-endpoint event if that event has occurred and is counted by the final analysis cutoff.

Let \(a\) index an arm, and let \(j=0,\ldots,K\) index its recruitment cohorts, where \(K\) planned selection looks can create up to \(K+1\) cohorts. Write \(n_{aj}\) for the number of participants planned for cohort \(j\) if arm \(a\) is active when that cohort would enter. For a participant \(i\) in that potential cohort, let \(Y_{aji}=1\) if the participant has a counted final-endpoint event by the final analysis cutoff, and \(Y_{aji}=0\) otherwise. Let \(A_{aj}=1\) if arm \(a\) is active at cohort \(j\)'s entry, and \(A_{aj}=0\) otherwise.

Key result: If a cohort is enrolled only when its arm is active, the expected number of final-endpoint events in arm \(a\) is

E(D_a)=\sum_{j=0}^{K}\sum_{i=1}^{n_{aj}}P(A_{aj}=1,\;Y_{aji}=1)

This expression is exact when the variables describe the potential enrollment and event outcomes under the design. For a cohort with the same conditional event probability for each participant, define \(q_{aj}=P(Y_{aji}=1\mid A_{aj}=1)\). Then

E(D_a)=\sum_{j=0}^{K} n_{aj}P(A_{aj}=1)q_{aj}.

The factor \(q_{aj}\) is conditional on the arm being active at that cohort’s entry. It can reflect the cohort’s time in the study, administrative censoring, and any association between the selection history and the final endpoint. If planned cohort size or participant-level event probabilities vary, use the exact sum of joint probabilities rather than treating \(n_{aj}\) and \(q_{aj}\) as common constants.

Retention Across One or More Looks

Let \(R_{ak}=1\) if arm \(a\) passes the selection rule at look \(k\), and \(R_{ak}=0\) if it is discontinued there. For the first cohort, which enters before any selection look, \(A_{a0}=1\). For a cohort entering after look \(k\), the arm must have passed every preceding look. Thus its active indicator is the product of the preceding retention indicators:

A_{aj}=\prod_{k=1}^{j}R_{ak}, \qquad j=1,\ldots,K.

Consequently, \(P(A_{aj}=1)\) is the probability of passing all selection looks before cohort \(j\). With multiple looks, this is a joint probability; it is not generally the product of marginal pass probabilities. Multiplying conditional pass probabilities is valid when they are defined in sequence, with each probability conditional on having passed earlier looks.

A useful derivation follows directly from the event count. The total is the sum of event indicators for all participants who could enter under the design, with each indicator counted only if the arm is active:

D_a=\sum_{j=0}^{K}\sum_{i=1}^{n_{aj}}A_{aj}Y_{aji}, \qquad E(D_a)=\sum_{j=0}^{K}\sum_{i=1}^{n_{aj}}E(A_{aj}Y_{aji}) =\sum_{j=0}^{K}\sum_{i=1}^{n_{aj}}P(A_{aj}=1,Y_{aji}=1).

The second equality uses linearity of expectation; it does not require event indicators from different participants to be independent. If the event probability is constant within a cohort conditional on activation, the joint probability factors as \(P(A_{aj}=1)P(Y_{aji}=1\mid A_{aj}=1)\), giving the simplified formula. It does not factor into activation probability times an unconditional event probability unless the relevant independence condition holds.

Worked Calculations

Worked Example: One Selection Look

Suppose a hypothetical treatment arm plans to enroll 80 participants before one intermediate-outcome look and another 60 afterward if the arm passes. The probability of passing is 0.70. By the final analysis cutoff, 35% of the first cohort and, conditional on passing, 25% of the potential second cohort are expected to have a final-endpoint event. Assume the first-cohort event probability is 0.35 regardless of the selection result.

The first cohort is enrolled in every path, so its expected event contribution is \(80(0.35)=28\) events.

The second cohort enters only if the arm passes. Its expected contribution is \(60(0.70)(0.25)=10.5\) events.

Therefore, \(E(D_a)=28+10.5=38.5\) events by the final analysis cutoff. This is an average across hypothetical repetitions of the design; an observed event count must be a whole number and need not equal 38.5.

Worked Example: Two Sequential Selection Looks

Consider a hypothetical arm with 100 participants planned before look 1, 50 between looks 1 and 2 if the arm passes look 1, and 40 after look 2 if it passes both looks. The probability of passing look 1 is 0.80. Among paths that pass look 1, the conditional probability of passing look 2 is 0.75. By the final cutoff, the event probabilities are 0.40 for the first cohort, 0.30 for the second cohort conditional on passing look 1, and 0.20 for the third cohort conditional on passing both looks. Assume the first cohort’s event probability is 0.40 regardless of later decisions.

The first cohort contributes \(100(0.40)=40\) expected events. The probability the arm is active for the second cohort is 0.80, so that cohort contributes \(50(0.80)(0.30)=12\).

The probability of passing both looks is the sequential product \(0.80(0.75)=0.60\). Therefore, the third cohort contributes \(40(0.60)(0.20)=4.8\).

Adding the cohort contributions gives \(E(D_a)=40+12+4.8=56.8\) events by the final cutoff. Using 0.75 alone for the third cohort’s activation probability would be incorrect: 0.75 is conditional on having passed look 1, not the probability of passing both looks.

Worked Example: Selection and Final Events Are Associated

Suppose a hypothetical arm has a potential post-look cohort of 50 participants, enrolled only if the arm passes its selection look. The pass probability is 0.50. In this example, the event probability for a potential cohort participant is 0.60 conditional on the arm passing, but 0 conditional on failing. These probabilities describe a possible association between the intermediate selection result and final-endpoint events.

The cohort’s expected contribution is \(50(0.50)(0.60)=15\) events.

Across all potential selection paths, a participant’s unconditional potential event probability is \(0.50(0.60)+0.50(0)=0.30\). Multiplying the pass probability by this unconditional probability would give \(50(0.50)(0.30)=7.5\), which is not the correct contribution. It applies an unconditional event probability to a cohort observed only on passing paths. The joint probability of both enrollment and an event is \(0.50(0.60)=0.30\) per potential participant, giving 15 expected events for the cohort.

Assumptions, Checks, and Alternatives

The calculation must match the trial’s operational rules. In particular, specify whether discontinuation stops only new recruitment or also changes follow-up for participants already enrolled. The usual calculation above assumes that participants recruited before a drop can continue to be followed and that their observed events count through the final analysis cutoff. If follow-up stops, or if the final cutoff differs by recruitment cohort, the event probabilities \(q_{aj}\) must reflect that rule.

For a time-to-event endpoint, \(q_{aj}\) is the probability of an event by the specified cutoff under the cohort’s entry time, follow-up, censoring, and endpoint definition. A single event probability should not be applied to all cohorts when their follow-up durations differ. In a setting where death or another competing event prevents the endpoint event, \(q_{aj}\) must be the probability of the endpoint event under the specified competing-risk and censoring rules, not a probability that ignores them.

Before interpreting an expected count, check the following:

Enrollment rule: Is \(n_{aj}\) the planned cohort size conditional on the arm remaining open, or an unconditional target?
Selection history: Does each activation probability represent passing every look before that cohort enters?
Follow-up: Are events among participants enrolled before discontinuation retained in the final count?
Conditional event probability: Does \(q_{aj}\) correctly condition on the selection history when that history is associated with final outcomes?
Analysis cutoff: Are event probabilities calculated by the same final cutoff and endpoint rules used for the planned analysis?

One alternative is a fixed-enrollment calculation that assumes every planned cohort is recruited. That gives the expected event count without arm dropping, but overstates expected recruitment and may overstate events when an arm is often discontinued. Another alternative is to report the event count conditional on the arm surviving all looks. That answers a different question from the unconditional expected count across all trial paths and can be much larger. Exact enumeration of selection paths can calculate the expectation when there are few looks; averaging over simulated paths is another computational option when the joint probabilities are complex. Whichever method is used, it must preserve the dependence between selection and final events.

An expected event count is useful for planning recruitment and assessing whether a design is likely to provide enough final-endpoint information. It is not itself a measure of power, a probability of reaching a particular event target, or a guarantee that a target will be reached. Designs with the same expected count can have different variability in event counts and different probabilities of falling short.

Common Pitfall

Common mistake: Multiply the total planned enrollment by an unconditional event probability, then reduce that total by a single “drop rate.” This can miscount participants recruited at different times, omit events from participants already enrolled when an arm is dropped, and ignore association between selection and final outcomes. Instead, divide recruitment into cohorts, calculate each cohort’s probability of being enrolled, and use its event probability conditional on the relevant selection history.

Check Your Understanding

Use the cohort-based expectation and distinguish conditional from unconditional probabilities.

An arm plans 40 participants before a look and 30 afterward if it passes. The pass probability is 0.60. By the final cutoff, event probabilities are 0.25 for the first cohort regardless of the decision and 0.20 for the second cohort conditional on passing. What is the expected event count?
An arm has two looks. It passes look 1 with probability 0.90 and passes look 2 with probability 0.50 conditional on passing look 1. What is the probability that a post-look-2 cohort is enrolled?
A potential post-look cohort has 20 participants. The arm passes with probability 0.40. Conditional on passing, each potential participant has event probability 0.50 by the cutoff; conditional on failing, the event probability is 0.10. What is the cohort’s expected contribution?
Why should events among participants enrolled before an arm is dropped usually remain in the expectation if follow-up continues to the final cutoff?

Solutions. (1) The first cohort contributes \(40(0.25)=10\). The second contributes \(30(0.60)(0.20)=3.6\). The total is \(13.6\) expected events. (2) The probability of passing both looks is \(0.90(0.50)=0.45\), so the post-look-2 cohort is enrolled with probability 0.45. (3) For each potential participant, the joint probability of enrollment and an event is \(0.40(0.50)=0.20\). Across 20 participants the expected contribution is \(20(0.20)=4\) events. The failure-path event probability does not contribute because that cohort is not enrolled on those paths. (4) Dropping the arm stops future recruitment under the stated rule; it does not erase events already observed or prevent subsequent follow-up of participants already enrolled. Their event contributions therefore remain part of the final count.