Time-to-Event Endpoints: Event Targets and Sample Size

Why Plan Around Events?

For a time-to-event endpoint, the information in a comparison depends substantially on how many participants experience the event, not just on how many are enrolled. A design may have a large enrollment but few observed events if the event is uncommon, follow-up is short, or censoring is frequent. Conversely, an event-driven design can continue follow-up until a planned event target is reached, subject to operational and ethical limits.

The standard event-target calculation is a useful starting point for a two-arm log-rank comparison when hazards are proportional. Translating that target into enrollment requires additional assumptions about the event-time distribution, accrual, follow-up, and censoring. These planning assumptions should be explicit: the resulting enrollment is conditional on them, not a universal conversion from events to participants.

Event Target Under Proportional Hazards

Consider a two-arm randomized comparison analyzed with a conventional log-rank test. Let \(D\) be the total number of observed events, \(q\) the fraction assigned to treatment, and \(1-q\) the fraction assigned to control. Assume a constant hazard ratio \(HR\) over time, independent censoring, and a sufficiently large sample for the usual large-sample approximation. The approximate variance of the estimated log hazard ratio is \(1/[Dq(1-q)]\). Thus the magnitude of the standardized treatment effect grows approximately as \(|\log(HR)|\sqrt{Dq(1-q)}\).

For a two-sided type I error rate \(\alpha\) and target power \(1-\beta\), the event target follows by setting that standardized effect equal to the sum of the relevant standard-normal quantiles:

This approximation assumes a single meaningful hazard ratio throughout follow-up. The direction of benefit does not change the squared effect term, but the planned \(HR\) must reflect the intended clinical contrast. A hazard ratio close to 1 requires many events; increasing the event target or changing allocation cannot compensate arbitrarily for a very small treatment effect.

From Events to Enrollment

For a simple conversion, suppose participants accrue uniformly over \(A\) months and the trial continues for an additional \(F\) months after accrual ends. A participant enrolled \(u\) months after accrual begins has \(F+A-u\) months of potential follow-up. If the event hazard in arm \(j\) is constant at \(\lambda_j\), and there is no loss to follow-up apart from study-end censoring, the event probability for that participant is \(1-\exp[-\lambda_j(F+A-u)]\). Averaging over uniform enrollment gives:

The integral explains why using only the minimum follow-up \(F\) understates event yield: participants enrolled earlier have longer follow-up. Under a planned treatment allocation fraction \(q\), the average event probability across randomized participants is \(\bar p=(1-q)p_0+qp_1\). A first approximation to total enrollment is \(N=D/\bar p\), rounded up to a number that preserves the allocation ratio and provides at least the target event count in expectation. Because actual event counts vary, a design may need a buffer or an event-driven stopping rule.

Dropout, Censoring, and Allocation

Independent dropout reduces event yield even when the event hazard itself is unchanged. For constant event hazard \(\lambda_j\) and independent exponential dropout hazard \(\gamma\), define \(r_j=\lambda_j+\gamma\). A participant followed for \(t\) months has probability \(\{\lambda_j/r_j\}[1-\exp(-r_jt)]\) of having the event before dropout. Averaging over uniform accrual and administrative follow-up gives:

This expression requires independent event and dropout times, constant hazards, and uniform accrual. It should not be used as though all censoring were independent when dropout is related to prognosis or treatment response. Also distinguish dropout from a competing event: if a different event prevents the endpoint from occurring, simply treating that event as independent censoring does not in general give the endpoint's cumulative incidence. The event process and estimand must determine how such events are handled.

Allocation affects both event yield and statistical information. For a fixed total \(D\), the information factor \(q(1-q)\) is largest at \(q=0.5\). A 2:1 ratio may be justified for treatment exposure or other design goals, but it requires more events than equal allocation for the same power under this approximation. Enrollment also changes with allocation because the two arms may have different event probabilities.

Departures From Proportional Hazards

The event-target formula compresses the treatment effect into one constant \(HR\). It can misrepresent power when the relative hazards change over time, as may occur with delayed onset, waning benefit, or crossing hazards. In such settings, the log-rank statistic combines information across event times, and the effective effect depends on when events occur as well as on the treatment contrast at each time. A single average hazard ratio may conceal clinically important timing or even opposing effects at different times.

Before relying on the simple formula, check whether the proposed mechanism and available natural-history information make a constant relative effect plausible over the planned follow-up. If not, specify time-varying hazards or another model consistent with the intended estimand, then evaluate power and enrollment under that model, commonly by trial simulation. Compare assumptions such as delayed effect, waning effect, and alternative event rates; report how often each design reaches the event target by its planned end. The event target remains a useful information benchmark, but it does not by itself guarantee the intended power under nonproportional hazards.

Planning should also examine event-rate misspecification, slower or faster accrual, less follow-up, and greater dropout. A design that reaches its target only under optimistic assumptions may be operationally fragile. Monitoring the observed event accrual rate can inform feasibility, but any changes to enrollment or follow-up must be consistent with prespecified design rules.

Common Pitfalls and Practical Checks

• Equating participants with events. The event target is a count of observed endpoint events. Convert it to enrollment using arm-specific event probabilities and the planned allocation.
• Using the follow-up duration as if it applied to everyone. Under staggered accrual, participants have different potential follow-up. Average over the accrual pattern, rather than assigning every participant the minimum follow-up.
• Ignoring the assumptions behind a hazard ratio. A constant \(HR\) supports the simple event-target approximation. If effects vary over time, evaluate the proposed design under time-varying effects rather than treating one ratio as definitive.
• Counting competing events as ordinary independent censoring. A competing event can change the probability of the endpoint. Use a model and analysis aligned with the endpoint and estimand.
• Reporting an expected event count as a guarantee. Event counts are random. Consider the probability of reaching the target by the planned end, especially when event rates or accrual are uncertain.

A useful alternative to a fixed enrollment calculation is an event-driven design that continues until the planned number of events is observed. This can reduce dependence on the calendar-time event-rate forecast, but it does not remove uncertainty about recruitment, follow-up duration, dropout, or whether the proportional-hazards model is appropriate. A fixed-time design instead fixes the analysis time and plans for the distribution of events by that date. The choice depends on operational constraints and the scientific estimand.