Updating Projections With Observed Accrual

Why Update an Event-Date Projection?

A planned event date is conditional on a plan: how quickly participants enroll, how long they are followed, and how often events occur. As a trial proceeds, observed enrollment and event counts replace part of that plan. An updated projection should use those observations for the past and state separately what is assumed for the future. Keeping these pieces distinct makes the projection interpretable and helps prevent planned-but-unrealized enrollment from being counted as if it had occurred.

The goal here is an updated expected event count by a candidate date, and, when useful, the date at which that expectation reaches a prespecified event target. This is a plug-in projection under specified assumptions. It is not a guarantee that the target will be reached by that date, nor by itself a probability that the trial will meet a deadline.

Modeling the Observed and Future Pieces

Suppose event times from enrollment follow a specified distribution with survival function \(S(u)=P(\text{event time}>u)\), where \(u\) is follow-up time. Thus \(1-S(u)\) is the probability that an event has occurred by \(u\). For a participant enrolled at date \(e_i\), let \(v_i\) be their observed event-free follow-up at the cutoff. If the participant is known to be event-free through \(v_i\), and is assumed to remain under event ascertainment through \(T\), their probability of having an event by \(T\), conditional on being event-free so far, is

This follows by dividing the probability of surviving event-free through \(T-e_i\) by the probability of surviving event-free through \(v_i\). Subtracting that conditional survival probability from one gives the conditional event probability. If event times are exponential with rate \(\lambda\), then \(S(u)=e^{-\lambda u}\), and this reduces to \(1-e^{-\lambda(T-c)}\) for someone event-free and under observation at cutoff \(c\). That simplification relies on the exponential model’s constant hazard; it should not be used automatically for other event-time distributions.

Let \(D_{\text{obs}}(c)\) be the number of events observed by the cutoff. The expected contribution from participants already enrolled is \(D_{\text{obs}}(c)\), plus the conditional event probabilities for those still event-free. If future enrollment is represented by an expected accrual rate \(a(s)\) at calendar time \(s\), then future entrants contribute expected events according to their available follow-up by \(T\). Under the specified survival model, the combined projection is

Here \(R(c)\) is the set of participants enrolled by \(c\) who are event-free at their observed follow-up, and \(a(s)\) describes the assumed future enrollment schedule. The integral is an expected contribution: participants entering at \(s\) have \(T-s\) time units in which to experience an event. In a discrete enrollment plan, replace the integral with a sum over planned future cohorts, multiplying each cohort size by its event probability by \(T\).

Worked Projections

How to Make and Check an Update

For a fixed event model and nonnegative future enrollment, the expected event count will generally not decrease as \(T\) increases: participants have at least as much follow-up, and additional future entrants may contribute. This makes a target-date search interpretable, but it does not make the estimate precise. A projection can be operationally implausible even when its arithmetic is correct—for example, if the assumed recruitment rate exceeds available capacity or the event model conflicts with the observed pattern.

Useful checks include verifying that observed events are counted once, that an event-free participant is not also counted as a future entrant, and that each planned cohort receives only the follow-up it can actually have by \(T\). Confirm units as well: an event rate per month must be paired with follow-up measured in months. If the event process or accrual schedule is not adequately described by a single rate, use an appropriate time-varying model or a cohort-specific schedule rather than hiding that variation in an average.

What the Updated Date Does—and Does Not—Say

An updated projection is preferable to carrying forward a plan that has already diverged from observed enrollment or events. It also differs from a simple extrapolation of the crude event count per calendar month: that rate can ignore staggered entry, unequal follow-up, and the fact that future participants have less time to experience an event. A survival-based projection accounts for follow-up time under its stated event model. A more elaborate simulation can represent additional variation or operational features, while a prediction interval can express uncertainty around event counts; neither should be confused with the plug-in expected date calculated here.

The principal limitation is that replacing past assumptions with observed data does not guarantee that future conditions resemble the model. Changes in recruitment, event ascertainment, treatment mix, or follow-up can invalidate a projection. When follow-up is incomplete or participants may be lost, the model must represent that process. When the event-time model is uncertain, the reported date should be understood as conditional on the selected model, not as a precise forecast regardless of model choice.