Worked Example: Event-Driven Survival Re-Estimation

From an Event Target to a Calendar Forecast

In an event-driven survival trial, the primary analysis is scheduled after a prespecified number of primary events has accrued. The required event count is determined by the treatment contrast the design aims to detect, together with the type I error rate, power, and allocation. The event rate and censoring rate answer a different question: how many participants, and how much follow-up, are likely to be needed to observe that many events?

This distinction matters at an interim review. A blinded estimate of pooled event and censoring rates can refresh the projected event yield under the enrollment and follow-up schedule. By itself, however, it does not alter the event-count requirement calculated for a specified effect, power, and significance level. A protocol may prespecify a separate change to those design inputs, but that is not the same operation as estimating event yield.

Assumptions for the Event-Count Calculation

For a two-group logrank design under proportional hazards, a commonly used large-sample approximation for the required number of events is

Here \(HR\) is the prespecified hazard ratio, \(q\) is the proportion assigned to one group, \(1-\beta\) is the target power, and \(z_a\) is the \(a\)-quantile of the standard normal distribution. The approximation assumes a two-sided logrank test at level \(\alpha\), proportional hazards with the stated hazard ratio, independent participants, and a sufficiently large number of events for the normal approximation to be reasonable. It also presumes the specified allocation is achieved. The calculation targets events, not total enrollment.

The hazard ratio and power are design inputs here. A blinded estimate of the pooled event rate does not estimate the treatment hazard ratio. If the protocol changes its target effect or power, the event target must be recalculated transparently using those revised inputs; the resulting change should not be attributed to the pooled rate estimate.

Projecting Event Yield Under Uniform Accrual

To translate an event target into follow-up, consider a simple model. Let \(\lambda_E\) be the constant event hazard and \(\lambda_C\) the constant hazard of non-event censoring. Assume these competing times are independent, censoring is non-informative, and the event and censoring hazards are the same in both randomized groups for purposes of the pooled forecast. Write \(r=\lambda_E+\lambda_C\). For a participant followed for \(t\) months, the probability of a primary event by \(t\) is

The integrand is the event density at time \(u\): remaining free of either event or censoring until \(u\) has probability \(e^{-ru}\), and an event occurs in the next small interval at rate \(\lambda_E\). This makes explicit why censoring matters: it reduces the fraction of participants who remain observable long enough to have an event.

Suppose accrual is uniform over \(A\) months and the trial continues for an additional \(F\) months after accrual ends. Individual follow-up times at the analysis range from \(F\) to \(F+A\). Averaging the individual event probability over those follow-up times gives

If \(N\) participants are enrolled under this schedule, the expected number of events is \(N p_{\mathrm{event}}(A,F)\). This expectation is a planning summary, not a guarantee: the realized count varies around it. The integral also assumes a fixed administrative analysis time and no additional sources of loss beyond the modeled censoring.

Translating the Target Into Follow-Up

A useful calculation is the follow-up duration at which the projected expected event count equals the target. This is a planning benchmark rather than a probability guarantee. With the rates and accrual schedule in the preceding example, the event probability simplifies to

For 400 participants to have 247 expected events, the required event probability is \(247/400=0.6175\). Solving the equation gives

So approximately 23.9 months of post-accrual follow-up gives an expected count of 247 under this model. A practical protocol may choose a scheduled review near that time, but the realized event count still determines whether the event-driven trigger has been met.

What Can Change the Event Target?

The event-count calculation can be revisited when a protocol has prespecified a design change to the target effect, power, or testing level. For example, with equal allocation, two-sided \(\alpha=0.05\), 90% power, and a target \(HR=0.75\), the same large-sample approximation gives

The corresponding upward-rounded target is 508 events. This change follows from the different target effect and power, not from the event or censoring rate estimates. Under a fixed enrollment of 400 and the first scenario's projected event probability of 0.6188, the expected count is only about 247.5, so that schedule would not be expected to deliver 508 events. A new accrual or follow-up plan would be needed; recalculating the calendar forecast does not by itself make the target attainable.

Diagnostics, Alternatives, and Limitations

Before relying on an interim forecast, check that the inputs describe the planned analysis population and primary-event definition, and that person-time is calculated consistently. Compare projected and observed event accumulation across interim reviews. Large discrepancies may indicate changing event rates, uneven accrual, differing follow-up, data delays, or a mismatch between the assumed censoring model and actual losses. If rates vary over time or by important groups, a single constant pooled rate can obscure the pattern; simulation using a more realistic accrual and survival model may be preferable.

The exponential calculation assumes constant hazards. If hazards are not constant, the event probability should instead be derived from an appropriate survival model and integrated over the accrual schedule. Informative censoring also invalidates the simple competing-risk interpretation: if withdrawal is related to a participant's unobserved event risk, the projected probability can be biased. In addition, interim rate estimates are uncertain, especially when person-time or event counts are limited. Reporting a point forecast alone can create false precision; a prespecified range or simulation-based predictive assessment can show the consequences of uncertainty.

An event-driven design is the main alternative to a fixed-calendar analysis date: it continues until the required event count is observed, so it is less dependent on a precise event-rate forecast, but calendar duration and cost remain uncertain. A fixed-duration design instead fixes follow-up time and forecasts how many events will be available; it can be operationally simpler, but has a risk of fewer events and lower power than planned. Either approach still requires a valid event target and clear assumptions. In a re-estimation procedure, the operational rule should state what is updated, which interim data are used, how uncertainty is handled, and what event count triggers analysis.

Common Mistake

Wrong reasoning: “The pooled interim event rate is lower than expected, so the required event target must be increased by the same proportion.” The event target is not a direct function of the baseline event rate in the logrank approximation. The rate affects how long it may take, or how many participants may be needed, to accumulate that target. Correction: calculate the target from the prespecified treatment contrast, allocation, type I error, and power; use the interim event and censoring estimates to revise the forecast under the accrual and follow-up schedule. Keep those two calculations distinct unless a separate, prespecified rule changes the design inputs.