Survival trial sample size
Plan the event count and recruitment needed for a two-arm time-to-event comparison.
Concepts and reasoning
Follow-up, rather than enrollment alone, creates information
A survival endpoint records the time until an event such as relapse. At the analysis date, some participants have not yet experienced the event. They still contribute information while under observation, but they cannot be counted as if their eventual event times were known. A trial with 600 enrolled patients and 60 events may be much less informative about a hazard ratio than one with 400 enrolled patients and 240 events.
This is why planning often begins with an event target. Enrollment is then chosen to make that target plausible under assumptions about baseline survival, the treatment effect, recruitment, loss to follow-up, and the analysis date. Faster recruitment may increase enrollment without giving the last patients much observation time. Extending follow-up may add events without adding participants.
A hazard ratio is a planning model
The hazard is the instantaneous event rate among people who remain event-free. A hazard ratio of 0.7 means that, under proportional hazards, the treatment event rate is 70% of the control rate at each time. It does not mean that every patient lives 30% longer or that the absolute event probability falls by 30 percentage points.
Proportional hazards makes a single log hazard ratio a useful planning target. Delayed treatment effects, crossing hazards, competing events, or a restricted-mean survival endpoint require different planning choices. The trial’s estimand and proposed test should be specified before choosing a convenient formula. The gsDesign survival-design overview connects event targets to enrollment and calendar time.
Mathematical foundations
The event requirement
A widely used large-sample approximation for a two-sided log-rank comparison is
Here
Events must be observable
For a constant control hazard
This follows by integrating the event hazard multiplied by the probability of remaining event-free and not dropped out. Staggered recruitment requires averaging this probability over available follow-up durations; treatment and control may have different event probabilities. Simply substituting the longest follow-up for every participant overestimates expected information.
Worked example
Planning an event-driven hypothetical trial
Take two-sided alpha 0.05, power 0.80, equal allocation, and a planned hazard ratio of 0.70. The quantiles sum to approximately 2.802 and
Suppose the average probability of an observed event by the planned analysis date is 0.60 across arms, after accounting for accrual and censoring. A rough enrollment conversion is
| Average observed-event probability | Approximate enrollment for 247 events |
|---|---|
| 0.400 | 618 |
| 0.600 | 412 |
| 0.800 | 309 |
The treatment can reduce the event rate while increasing the number of patients needed to observe the target events within a fixed calendar window. That is not a contradiction: the desired clinical benefit itself reduces the flow of endpoint information.
For a simple follow-up illustration, a constant hazard of 0.1 per year over five years without dropout gives event probability
Interpreting the results
Events supply the information
The required event count is the central target for a time-to-event comparison. The participant count is then estimated from the chance that each participant experiences an observed event before follow-up ends or dropout occurs. Two studies with the same enrollment can have different power because their follow-up and event rates differ.
A treatment/control hazard ratio below one indicates a lower event hazard when the event is adverse. It is not the proportion of events prevented, a ratio of median survival times in general, or a direct survival-probability difference. Interpret the target on its specified event and follow-up scale.
A longer recruitment period can increase the average observed follow-up if additional follow-up is held fixed, while later recruits still have less observation than early recruits. Report required events, group enrollment, recruitment duration, and additional follow-up together. Expected events are a model average, not a guaranteed realized count.
Assumptions and limitations
Connect the event target to the survival model
The usual event-based approximation assumes a proportional-hazards treatment contrast and a planned allocation ratio. Delayed effects, crossing hazards, or treatment effects that fade may require a different design criterion. A single hazard ratio cannot fully describe such patterns.
Translating events into enrollment additionally requires assumptions about baseline survival, entry times, censoring, and dropout. Exponential survival implies a constant event hazard; uniform accrual implies an even recruitment rate. Real recruitment delays and changing hazards can make the projected event count optimistic or pessimistic.
Dropout should be modeled on the same time scale as the event process, and independent censoring is needed for the intended analysis. Death is not ordinary loss to follow-up when it competes with a different event of interest. Define the endpoint and handling of competing events before selecting the calculation.
Diagnostics and sensitivity analysis
Separate uncertain treatment effects from uncertain follow-up
Vary the hazard ratio and baseline median independently: the first changes the required information, while the second changes how quickly events accrue. Then examine slower recruitment, shorter additional follow-up, and greater dropout. This reveals whether feasibility is driven by treatment effect or by event availability.
Check whether the survival distribution used for planning resembles historical event-time data over the proposed observation window. A median alone does not identify the entire survival curve. Recompute enrollment under plausible alternatives when late events are important.
During a trial, compare observed recruitment and pooled event accrual with the planning assumptions through an appropriate blinded monitoring process. An event-driven stopping plan still needs prespecified follow-up and analysis rules. The gsDesign survival overview provides further details on event-based planning.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- gsDesign: overview of survival endpoint design — Author documentation explaining event-driven planning and enrollment.