Recruitment Consequences of Arm Removal

Why Arm Removal Changes More Than Randomization

Removing an arm from an ongoing platform changes the set of treatments available to future participants. That can affect recruitment at several distinct points: who remains eligible, whether an eligible person finds an acceptable option, whether that person consents, and how enrolled participants are assigned. These effects matter for operational planning and for interpreting changes in the enrolled population. A lower enrollment rate after removal, for example, could reflect loss of a group that was eligible only for the removed arm, reduced willingness among people who remain eligible, or both.

The decision to remove an arm is not the same as the recruitment consequences that follow it. Here, treat the removal decision as authorized and focus on its effects on the recruitment flow. Participants already enrolled are not made ineligible retroactively; their follow-up and assigned treatment continue according to the trial protocol. Eligibility rules, consent information, and recruitment procedures for future participants must reflect the arms that are actually open.

Definition: A recruitment funnel describes the successive groups of people who are screened, meet the current trial eligibility criteria, are willing to participate, consent, and are enrolled. After an arm is removed, loss of eligibility means a person no longer meets the criteria for any open arm. A choice-related recruitment change occurs when someone remains eligible but is less willing to enroll because the available treatment options have changed.

The distinction is operationally important. A person who qualifies only for the removed arm is not an eligible candidate who declined; that person is no longer eligible for the platform as currently constituted. By contrast, a person who qualifies for an open arm but refuses participation because a preferred option is gone remains eligible but does not enroll. Combining these groups into a single “not recruited” count conceals why accrual changed.

A Recruitment-Flow Model

Suppose screened candidates arrive at an average rate of \(\lambda\) people per week. Let \(j\) index mutually exclusive candidate groups, let \(\pi_j\) be the proportion of screened candidates in group \(j\), and let \(q_j\) be the probability that a candidate in that group enrolls under the current set of open arms. The \(\pi_j\) values sum to 1. In a simple flow model, the expected enrollment rate is

R = \lambda \sum_j \pi_j q_j.

This rate depends both on the mix of people screened and on the probability of enrollment within each group. Removal can change which groups meet any open-arm eligibility criteria, and it can change enrollment probabilities among people who remain eligible. In practice, \(q_j\) can combine several steps, including a candidate being eligible, willing to consider the available options, and consenting. Where possible, record those steps separately rather than treating \(q_j\) as an unexplained single number.

Worked Example: Separating Eligibility Loss From Reduced Willingness

Consider a hypothetical platform with open arms A, B, and C. The screening team assesses 30 candidates per week. Group X, 40% of candidates, is eligible for A, B, or C; Group Y, 40%, is eligible for A or B; and Group Z, 20%, is eligible only for C. Before C is removed, the respective probabilities of enrollment are 0.75, 0.65, and 0.70. After C is removed, X candidates remain eligible but their enrollment probability is assumed to fall to 0.60; Y candidates remain eligible with probability 0.65; Z candidates are no longer eligible for any open arm.

Before removal. The expected weekly enrollments are \(30(0.40)(0.75)=9.0\) from X, \(30(0.40)(0.65)=7.8\) from Y, and \(30(0.20)(0.70)=4.2\) from Z. Thus \(R=9.0+7.8+4.2=21.0\) participants per week.

After removal. X contributes \(30(0.40)(0.60)=7.2\) expected enrollments per week, and Y contributes \(30(0.40)(0.65)=7.8\). Z contributes zero because it no longer meets the eligibility criteria for an open arm. The resulting expected rate is \(R=7.2+7.8=15.0\) participants per week, a decrease of 6.0 participants per week, or \(6/21=0.2857\), about 28.6%.

The six-per-week decrease has two modeled sources. Z accounts for 4.2 expected enrollments per week before removal and is now ineligible. X’s expected enrollment falls from 9.0 to 7.2 per week, a reduction of 1.8, even though X remains eligible. Y’s modeled enrollment is unchanged. These figures are consequences of the stated hypothetical assumptions, not universal effects of removing an arm.

Patient Choice and Assignment Are Different Steps

The available arms can affect whether a person is willing to enter a trial, but willingness to enroll is not the same as choosing a treatment after enrollment. In a randomized platform, an eligible person may decide whether to consent with knowledge of the available options; once enrolled, assignment follows the protocol’s randomization rule. Recruitment records should preserve that distinction. They should not describe a patient’s preference as if it determined the randomized assignment.

The removal also changes the assignment opportunities for candidates eligible for multiple arms. For example, if X candidates were randomized equally among A, B, and C before removal, then, conditional on enrollment, each assignment probability was \(1/3\). With C closed and equal allocation between A and B, each probability becomes \(1/2\). This changes the distribution of assignments among future enrollees; it does not mean that the probability of enrollment itself is \(1/2\), nor does it establish that the remaining arms are more effective.

Worked Example: Following Eligibility and Assignment Through the Funnel

Use the previous hypothetical candidate mix, but express it as 100 screened candidates in a week. Before removal, 40 are in X, 40 in Y, and 20 in Z. The model predicts 30 X enrollments, 26 Y enrollments, and 14 Z enrollments: 70 enrollments in total. Suppose assignments are equal among the arms available to each enrolled group. X’s 30 assignments divide as 10 each to A, B, and C; Y’s 26 divide as 13 each to A and B; Z’s 14 go to C. The expected totals are therefore 23 to A, 23 to B, and 24 to C.

After C is removed, the 20 Z candidates are no longer eligible for the platform. The 40 X candidates remain eligible, with an expected 24 enrollments at the post-removal probability of 0.60. The 40 Y candidates contribute 26 expected enrollments. There are now 80 candidates eligible for at least one open arm and 50 expected enrollments, so the modeled enrollment proportion among those eligible is \(50/80=0.625\), or 62.5%. If X and Y are each randomized equally between A and B, the expected assignments are 12 from X plus 13 from Y to each arm: 25 to A and 25 to B.

The screened-to-enrolled proportion falls from \(70/100=70\%\) to \(50/100=50\%\). That change combines the loss of 20 candidates from the eligible pool with lower enrollment among the remaining X candidates. Reporting only that 50 of 100 screened candidates enrolled would not reveal those separate mechanisms.

Estimating Time to a Recruitment Target

A rate can support a time estimate, but the assumptions behind that estimate must be stated. If enrollment follows a constant-rate Poisson process with rate \(R\) participants per week, then the expected time to the next \(m\) enrollments is \(m/R\) weeks. This result does not follow merely from observing or assuming a constant average rate; it uses the Poisson-process model. That model assumes a stable rate over the period considered and independent increments in enrollment counts over disjoint time intervals. If these assumptions are implausible, \(m/R\) should be described as a rate-based approximation, not an exact expected waiting time.

E[T_m] = \sum_{i=1}^{m} E[W_i] = \sum_{i=1}^{m} \frac{1}{R} = \frac{m}{R}.

Here, \(W_i\) is the waiting time between the \((i-1)\)th and \(i\)th enrollments. Under a constant-rate Poisson process, each \(W_i\) has an exponential distribution with mean \(1/R\), which justifies the calculation. If the screening rate, candidate mix, consent probability, or site capacity changes over time, a single constant \(R\) may not describe the recruitment process.

Worked Example: Expected Accrual Time After Removal

Suppose the hypothetical rates from the first example apply: enrollment is a constant-rate Poisson process at 21 participants per week before C is removed and 15 per week afterward. At the removal date, 60 participants have been enrolled toward a target of 120. The expected additional time to recruit the remaining 60 is

E[T_{60}] = \frac{60}{15} = 4.00 \text{ weeks}.

For comparison, under the same Poisson assumption, a counterfactual with no removal and a constant rate of 21 per week would have expected total time \(120/21=5.714\) weeks. If the first 60 were accrued at 21 per week and removal occurred at that point, the expected total time under the changed rates is \(60/21+60/15=2.857+4.00=6.857\) weeks. The modeled increase relative to the no-removal counterfactual is \(6.857-5.714=1.143\) weeks. These are expected times under the stated model, not guarantees about the calendar date on which the target will be reached.

If the actual post-removal rate varies by site or calendar time, or recruitment pauses while procedures change, the constant-rate calculation may be misleading. In that case, use a time-varying rate model or present scenario-based projections and state their assumptions.

Monitoring the Consequences in Practice

A useful operational review compares recruitment before and after removal using consistent definitions and denominators. At a minimum, monitor the numbers screened, eligible for at least one open arm, eligible only for the removed arm before closure (where documented), invited or offered participation, consenting, and enrolled. Also record the reasons eligible candidates do not enroll, using categories that distinguish treatment preference from other reasons. Site, calendar time, referral source, and relevant eligibility group can help identify whether a change is concentrated in particular parts of the recruitment process.

A funnel helps diagnose where a projection is failing. If screening volume is stable but the eligible proportion falls, check whether the candidate mix has changed or the post-removal eligibility rules were applied differently. If eligibility is stable but consent falls, investigate whether the available options, information provided, or other participation barriers have changed. If consent remains stable but enrollment falls, review operational steps between consent and enrollment, such as confirmation of eligibility or capacity constraints. These checks identify explanations to investigate; they do not by themselves establish why the rate changed.

Use consistent denominators. A consent percentage among eligible candidates is not comparable to an enrollment percentage among everyone screened.
Separate actual counts from projections. The scenario model’s probabilities are planning inputs; replace them with observed, clearly defined data as it accumulates.
Check for concurrent changes. Site openings, seasonal referral patterns, revised screening procedures, or recruitment pauses can coincide with arm removal and affect accrual.
Update participant-facing information. Future candidates should be told accurately which arms are open; recruitment materials that still describe a removed option can distort both choice and consent.

Alternatives, Limitations, and a Common Mistake

A simple overall-rate comparison is easy to communicate, but it can hide the mechanism of change. A stratified funnel model, as used above, makes eligibility groups and enrollment probabilities explicit. It is more informative when candidate groups differ in which arms they can receive, but it depends on having reliable group definitions and enough observations to estimate the relevant rates. Historical rates can provide a useful comparison, yet they may not predict post-removal recruitment if the population, sites, or procedures have changed. A scenario analysis can show how conclusions vary under alternative enrollment probabilities, but it remains conditional on those assumptions.

The model also has limits. It does not account automatically for competition between studies, changes in referral behavior after publicity about arm closure, site-level clustering, capacity constraints, or the time needed to revise materials and workflows. Nor does a predicted decrease in recruitment establish that removing the arm was a poor decision: recruitment consequences are one part of a broader design and ethical assessment.

Common mistake: “The removed arm had 20% of the screened candidates, so removing it must reduce recruitment by exactly 20%.” This reasoning treats the share of screened candidates as the share of enrollments lost. The correction is to identify who remains eligible for open arms and how enrollment probabilities change within each group. In the worked scenario, the 20% in Z accounted for 4.2 of 21 expected weekly enrollments before removal, while the decrease among still-eligible X candidates added another 1.8; together these produced a 28.6% reduction in the modeled enrollment rate.

A second frequent error is to divide the remaining target by a reported average rate and call the result an expected time without stating a process model. Under a constant-rate Poisson assumption, that division gives the expected waiting time. Without an appropriate stochastic model, it is safer to call the result a rate-based planning approximation and to acknowledge uncertainty in the rate and in the time to target.

Check Your Understanding

Answer each question using the definitions and assumptions in this tutorial. Solutions follow the questions.

A group is eligible only for an arm that has closed. Is it correct to count its members as eligible candidates who refused enrollment? Explain.
With 20 screened candidates per week, 25% in a group eligible only for the removed arm, and 60% of that group expected to enroll before closure, how many expected weekly enrollments from that group are lost after removal?
After removal, 90 candidates remain eligible and 54 enroll. What is the observed enrollment proportion among eligible candidates? What does that figure not tell you by itself?
Under a constant-rate Poisson enrollment model with a rate of 12 participants per week, what is the expected time to enroll 30 additional participants?
Name two distinct explanations for a decline in enrollment after arm removal and one piece of funnel information that would help distinguish them.

Solutions. (1) No. Those candidates no longer meet eligibility criteria for any open arm; they should be counted as no longer eligible, not as eligible people who refused. (2) The group accounts for \(20(0.25)=5\) screened candidates per week, of whom \(5(0.60)=3.00\) were expected to enroll before closure. With no open eligible arm for them, the modeled loss is 3.00 expected enrollments per week. (3) The proportion is \(54/90=0.600\), or 60.0%. It does not identify the cause of non-enrollment or show how many screened candidates were excluded before reaching the eligible group. (4) \(E[T_{30}]=30/12=2.50\) weeks, conditional on the stated constant-rate Poisson model. (5) Examples include loss of candidates who were eligible only for the removed arm and reduced willingness among candidates who remain eligible. Counts of screened candidates by eligibility group, together with reasons eligible candidates did not enroll, would help distinguish these explanations.