A Worked Projection With Changing Recruitment

Why Changing Recruitment Requires Cohort-Specific Follow-Up

A single average recruitment rate can obscure how much follow-up participants will have had by a planned analysis date. People enrolled early may have been observed for a long time, while those recruited later may contribute little follow-up. When recruitment changes over calendar time, an event-date projection should retain those differences rather than treat every participant as if they entered on the same date.

This tutorial works through a hypothetical projection with recruitment occurring at different rates in successive periods. The calculation estimates the mean cumulative number of events by each calendar time, identifies when that mean reaches a target, and translates the result to a calendar date. It is a planning projection, not a guarantee that the observed trial will reach the target on that date.

Definition: Let \(s\) denote a participant’s enrollment time, measured in months from the start of recruitment, and let \(T\) be a later calendar time. The participant’s follow-up by time \(T\) is \(T-s\) months if \(s\leq T\). A piecewise recruitment rate \(r(s)\) is the expected number of participants enrolled per month at time \(s\). The expected cumulative event count by \(T\) is obtained by adding the event probabilities for all participants enrolled by \(T\), accounting for each participant’s follow-up time.

Model and Calculation

For the main projection, assume recruitment begins at time 0 and occurs at a known, piecewise constant rate. Conditional on enrollment, each participant has an independent event time with a constant hazard \(\lambda\) per month. Thus the event time follows an exponential distribution, and the probability that a participant has an event within \(u\) months is \(1-e^{-\lambda u}\). Assume no loss to follow-up or competing event before the administrative projection date. Participants who have not had an event by \(T\) are event-free at that projection time.

These assumptions make the calculation transparent: recruitment determines how many participants enter each interval, and the event model determines each entrant’s probability of having an event by \(T\). The constant-hazard assumption is a simplification; it should be clinically defensible over the follow-up range being projected. The independent participant-level event times are used to obtain the expected count, but the projected mean does not describe all the uncertainty in the realized count.

Suppose the recruitment rate is \(r\) participants per month over an interval from \(a\) to \(b\), and the projection time \(T\) is at or after \(b\). Participants enrolled in that interval have follow-up ranging from \(T-a\) to \(T-b\). Under the continuous-rate convention, the interval contributes \(r\int_a^b [1-e^{-\lambda(T-s)}]\,ds\) expected events. Evaluating the integral gives the formula below.

E_{[a,b]}(T) = r\left[ (b-a) - \frac{e^{-\lambda(T-b)}-e^{-\lambda(T-a)}}{\lambda} \right].

The first term, \(r(b-a)\), is the expected number recruited in the interval. The subtracted term is the expected number of those participants who remain event-free at time \(T\). If \(T\) falls inside a recruitment interval, use \(T\) in place of the interval’s end \(b\), because participants enrolled after \(T\) have not yet entered the projection. Add the contributions from all recruitment intervals to obtain the full expected event count \(E(T)\).

Justification: For entrants at time \(s\), the event probability by \(T\) is \(1-e^{-\lambda(T-s)}\). A rate of \(r\) participants per month contributes \(r\,ds\) expected entrants in a short interval \(ds\), so their expected event contribution is \(r[1-e^{-\lambda(T-s)}]ds\). Integrating over the recruitment interval and evaluating the two terms yields the stated expression. Linearity of expectation allows the interval contributions to be added, without requiring event counts from different intervals to be independent.

Worked Projection With Three Recruitment Periods

Consider a hypothetical study starting recruitment on January 1, 2027. For calculation, use 30-day months and measure time from the recruitment start. Recruitment is planned at 20 participants per month during months 0–6, 30 per month during months 6–12, and 25 per month during months 12–18; recruitment then stops. Assume a constant event hazard of \(\lambda=0.04\) per month, no dropout, and a target of 200 expected events.

Worked Example: Finding the Target Date Under Changing Recruitment

First check the planned enrollment total. The three periods contribute \(20(6)=120\), \(30(6)=180\), and \(25(6)=150\) participants, for 450 expected participants in total. This is an expected total under the rate convention, not a claim that each interval must enroll exactly that integer count.

Evaluate the projection at month 24, which is after recruitment ends. For months 0–6, substitute \(r=20\), \(a=0\), and \(b=6\):

E_{[0,6]}(24) = 20\left[ 6-\frac{e^{-0.04(18)}-e^{-0.04(24)}}{0.04} \right] = 20(6-2.5965) = 68.07.

For months 6–12, the follow-up endpoints are \(24-12=12\) and \(24-6=18\) months:

E_{[6,12]}(24) = 30\left[ 6-\frac{e^{-0.04(12)}-e^{-0.04(18)}}{0.04} \right] = 30(6-3.3008) = 80.98.

For months 12–18:

E_{[12,18]}(24) = 25\left[ 6-\frac{e^{-0.04(6)}-e^{-0.04(12)}}{0.04} \right] = 25(6-4.1961) = 45.10.

Therefore the unrounded total is approximately \(194.144\), which rounds to \(194.14\) expected events. The target of 200 has not yet been reached by the projected mean at month 24.

Because recruitment has stopped, the derivative of the expected event count after month 18 is \(E'(T)=\lambda\int_0^{18}r(s)e^{-\lambda(T-s)}ds\). Consequently, \(E'(T)=E'(24)e^{-0.04(T-24)}\). From the three intervals, \(E'(24)=20(0.103859)+30(0.132031)+25(0.167845)=10.234\) expected events per month. Integrating this derivative from 24 to \(T\) and setting the result equal to the remaining \(200-194.144=5.856\) events gives:

5.856 = \frac{10.234}{0.04}\left[1-e^{-0.04(T-24)}\right], \qquad T = 24-\frac{1}{0.04} \ln\left(1-\frac{0.04(5.856)}{10.234}\right) \approx 24.579\text{ months}.

This result is after month 18, as required for using the post-recruitment formula. Under the stated 30-day-month convention, \(24.579(30)\approx737.4\) elapsed days after January 1, 2027. There are 731 elapsed days from that start to January 1, 2029 (365 in 2027 and 366 in leap year 2028); a further 6.4 days gives approximately January 7, 2029. This is the date when the projected mean count reaches 200 under the assumptions, not a certain date for the observed trial count.

Check the Participant Follow-Up Behind a Projection

An aggregate projection can be checked by isolating a recruitment period and asking what follow-up its participants receive. The event probability for an individual depends on that follow-up, not on the date at which the projection is reported alone.

Worked Example: Contribution From a Late Recruitment Period

Suppose 25 participants per month are recruited from month 12 through month 18. At month 21, this group has follow-up ranging from 3 to 9 months. With \(\lambda=0.04\) per month and no dropout, calculate its expected event contribution.

There are \(25(18-12)=150\) expected participants. Apply the interval formula with \(T=21\):

E_{[12,18]}(21) = 25\left[ 6-\frac{e^{-0.04(3)}-e^{-0.04(9)}}{0.04} \right] = 25\left[ 6-\frac{0.886920-0.697676}{0.04} \right] = 25(1.2689) \approx31.72.

The model therefore predicts about 31.72 events from this period by month 21. Dividing by 150 gives an average event probability of about \(0.2115\), consistent with follow-up ranging from 3 to 9 months. The fractional count is appropriate for an expectation; an observed count must be an integer.

What Changes When Recruitment Changes Again?

A forecast should be recalculated using the recruitment pattern that is actually assumed. The event hazard and target alone do not determine the date: the timing of enrollment affects the follow-up available to participants. The next example changes the final recruitment rate while keeping the earlier periods and event model fixed.

Worked Example: A Lower Final Recruitment Rate

Suppose the first two periods remain at 20 participants per month during months 0–6 and 30 per month during months 6–12, but recruitment during months 12–18 is 15 per month rather than 25. The target is now 180 expected events. At month 24, the first two contributions remain 68.07 and 80.98. The last contribution is:

E_{[12,18]}(24) = 15\left[ 6-\frac{e^{-0.24}-e^{-0.48}}{0.04} \right] = 15(1.8039) = 27.06.

Thus the unrounded total is approximately \(176.105\), leaving about \(3.895\) expected events. The month-24 rate of increase is \(E'(24)=20(0.103859)+30(0.132031)+15(0.167845)=8.556\) events per month. Since recruitment has again ended, solve using the same post-recruitment relationship:

T = 24-\frac{1}{0.04} \ln\left(1-\frac{0.04(3.895)}{8.556}\right) \approx24.459\text{ months}.

This is about \(24.459(30)=733.8\) elapsed days after January 1, 2027. Subtracting the 731 days to January 1, 2029 leaves about 2.8 days, so the approximate calendar date is January 3, 2029. Relative to the first projection, the lower final recruitment rate changes both the number enrolled and the projected timing of the target. The forecast should be interpreted under its stated assumptions rather than as an observed result.

Assumptions, Diagnostics, and Alternatives

A useful projection is not just a calculation; it is a calculation whose inputs remain plausible. Check the following before using a target date for planning:

Recruitment timing: Confirm that the intervals cover the intended enrollment period without gaps or overlaps, and that each rate uses the same time unit as the event hazard.
Follow-up limits: For every interval, verify \(T\geq b\) before applying the formula as written. If the projection time is within an interval, truncate that interval at \(T\); do not count future enrollment.
Enrollment total: Sum rate times interval length and compare with the planned or operationally possible sample size. A forecast that exceeds the recruitment capacity or study limit is not operationally coherent.
Event model: Compare the constant-hazard assumption with available clinical and historical evidence. A changing event hazard, treatment effect, or participant mix can make one exponential rate unsuitable for the full follow-up period.
Forecast behavior: Expected events should be nondecreasing as \(T\) increases. After recruitment ends, the rate of increase should decline under this constant-hazard model as fewer event-free participants remain.

This calculation is a deterministic mean projection. A random-count projection or simulation can represent variability in recruitment and event times, while scenario analysis can show how alternative rates or hazards change the forecast. Earlier material on variance and prediction intervals for event counts, Monte Carlo projection, and scenario analysis addresses those questions. These approaches complement, rather than replace, checking the recruitment intervals and follow-up assumptions in the mean calculation.

If dropout is material, the no-dropout event probability used here is not adequate. Under independent constant event and dropout hazards \(\lambda\) and \(\gamma\), respectively, the probability of an event before dropout and by follow-up \(u\) is \(\frac{\lambda}{\lambda+\gamma}[1-e^{-(\lambda+\gamma)u}]\). Use that probability in the interval integration, or use an appropriate richer model; do not treat dropout as if it were event-free follow-up with no effect on event opportunity.

Key takeaway: With changing recruitment, calculate expected events by recruitment interval so that every participant’s follow-up is represented. The target date is the solution to \(E(T)=D\), where \(D\) is the target count; it is a model-based mean projection, not a promise about the observed event date.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A group is recruited at 10 participants per month from month 0 to month 4. At month 10, how much follow-up do the earliest and latest entrants have?
Under a constant event hazard of \(0.05\) per month and no dropout, what is the probability of an event by month 6 for a participant enrolled at month 2?
Why should a recruitment interval that extends beyond projection time \(T\) be truncated at \(T\)?
After recruitment ends, what qualitative pattern does the constant-hazard model predict for the rate at which expected events accumulate?
Does a projected mean count of 200 at a calendar date mean the observed trial will have exactly 200 events on that date? Explain.

Solutions. (1) The earliest entrants have 10 months of follow-up and the latest have \(10-4=6\) months. (2) Follow-up is \(6-2=4\) months, so the probability is \(1-e^{-0.05(4)}=1-e^{-0.2}\approx0.1813\), or about 18.13%. (3) Participants enrolled after \(T\) are not yet in the study by the projection time and cannot contribute events by then. (4) The expected event count continues to rise but its rate of increase declines, because the pool of event-free participants shrinks. (5) No. It means the model’s expected cumulative count reaches 200 then; the realized count and the date it reaches 200 vary around that projection.