Constant-Rate Event Accrual

Why Recruitment Timing Changes the Event Projection

An event target is reached through both enrollment and follow-up. Even if the same number of participants is enrolled, the projected event count at a given cutoff can differ depending on how long recruitment takes: early enrollees have more time to experience the event than late enrollees. A constant-rate model gives a direct way to account for that difference when event hazards are also constant.

The calculation below is an expectation, not a promise that the observed event count will equal the projected value. It is useful for planning a likely event yield under specified assumptions, and for seeing how recruitment duration and post-recruitment follow-up affect that yield. Its simplicity is also its main limitation: real enrollment and event patterns may not remain constant.

Model and Notation

Let recruitment start at calendar time 0 and continue at a constant rate \(r\) participants per unit of time for \(R\) time units. If recruitment ends before the analysis cutoff, let \(F\) be the additional follow-up time after recruitment ends, so the cutoff is at \(T=R+F\). Let \(\lambda\) denote the constant event hazard per participant per unit of time. The model assumes that each participant is at risk from enrollment, has the same constant hazard, and has no loss to follow-up or competing event that prevents the event from being observed. For the basic formula, event times are independent across participants.

The exponential survival expression follows from the definition of a hazard. If the hazard is constant, the survival function satisfies \(S'(s)=-\lambda S(s)\) with \(S(0)=1\); solving this differential equation gives \(S(s)=e^{-\lambda s}\). The time unit used for \(\lambda\), \(R\), \(F\), and \(T\) must be consistent. For example, a hazard per month must be paired with durations in months.

We use a continuous enrollment-rate approximation. If a fixed total of \(N\) participants is enrolled uniformly over \(R\) time units, then \(r=N/R\). The continuous calculation is often a convenient planning approximation to discrete enrollment.

Deriving the Expected Event Count

At a cutoff \(T\), a participant enrolled at time \(u\) has had \(T-u\) time units of follow-up. Enrollment occurs only from time 0 to \(R\), and only participants enrolled by the cutoff can contribute. Therefore, the expected number of events is the enrollment rate multiplied by the integral of the event probability over eligible enrollment times:

This expression covers a cutoff during recruitment as well as one after recruitment. Evaluating the integral gives two useful forms.

For a cutoff after recruitment, write \(F=T-R\) and \(N=rR\). The same result can be expressed in terms of the final sample size and the average event probability over enrollment times:

The fraction in this formula is the average event-free survival probability across enrollment times: participants have follow-up durations ranging from \(F\) to \(F+R\). The complement is the average event probability. This averaging is over the enrollment schedule; it does not mean every participant has the same follow-up duration.

How Recruitment Duration and Follow-Up Affect the Projection

When comparing recruitment plans, distinguish the effect of recruitment duration from the effect of changing the total number enrolled. The next example holds the total sample size and post-recruitment follow-up fixed. It isolates how the enrollment schedule changes the amount of time participants have been followed at the cutoff.

A cutoff can also occur before recruitment finishes. In that case, the expected number enrolled by the cutoff is \(rT\), not \(rR\), and their follow-up durations range from nearly zero to \(T\). The first branch of the formula accounts for this directly.

Checks, Comparisons, and Failure Modes

A useful comparison is the shortcut that assigns everyone the mean follow-up time. When recruitment lasts \(R\) months and an additional \(F\) months of follow-up are observed, mean follow-up under uniform enrollment is \(F+R/2\). The shortcut estimates events as \(N[1-e^{-\lambda(F+R/2)}]\), but this is not the exact formula. The event probability \(1-e^{-\lambda s}\) is a concave function of follow-up \(s\), since its second derivative is \(-\lambda^2e^{-\lambda s}<0\). Therefore the event probability at the mean follow-up is at least as large as the average event probability across participants. The shortcut generally overestimates the expected count when follow-up varies.

The all-at-start calculation, \(N(1-e^{-\lambda F})\), is also inappropriate when participants enrolled over time: it gives everyone only the post-recruitment follow-up \(F\), ignoring the additional follow-up experienced by earlier entrants. A valid projection must match the actual enrollment window and the cutoff definition.

Check a computed projection against basic limits. It cannot exceed the number enrolled by the cutoff. Increasing \(\lambda\) while keeping enrollment and follow-up fixed cannot decrease the projected count. If \(\lambda=0\), the expected count is zero. If the cutoff is exactly at the end of recruitment, set \(F=0\); events can still have occurred among early enrollees. As a check on the formula, when \(F=0\) and recruitment lasts \(R\), the expected count is \(r[R-(1-e^{-\lambda R})/\lambda]\), which is positive for \(\lambda>0\).

Common warning signs include a hazard expressed per year paired with follow-up in months, a cutoff time measured from recruitment completion when the formula treats it as time from recruitment start, or an event projection larger than the number enrolled. Recheck units, the time origin, and which participants are eligible at the cutoff before interpreting a result.