Survival analysis sample size calculator for the log-rank test

In a trial with a binary outcome you count responders in each arm, and sample size is a function of participants. In a survival trial the outcome is the time until an event, and many participants will not have had the event by the time you analyse (they are censored). The information in the data comes from the events, so the log-rank test is powered by the number of events observed, not the number of participants enrolled.

Schoenfeld showed that, under proportional hazards, the log-rank statistic has a variance that depends on the number of events. For a two-sided test at significance level alpha, power 1 - beta and 1:1 allocation, the required number of events is d = 4 x (z(1 - alpha/2) + z(1 - beta))^2 / (ln HR)^2, where HR is the hazard ratio you want to detect and ln is the natural logarithm. For alpha 0.05 and 80% power the z values are 1.960 and 0.842, so the numerator is 4 x 7.85 = 31.4.

Plug in HR = 0.70 and the denominator is (ln 0.70)^2 = 0.127, so d = 31.4 / 0.127 = 247 events, rounded up. HR = 0.50 needs only about 66 events, because a large effect shows up quickly; HR = 0.80 needs about 631. Moving to 90% power at HR = 0.70 raises the requirement to about 331 events. Notice how strongly the requirement depends on the hazard ratio: halving the effect size roughly quadruples the events.

The hazard ratio should be the smallest effect that would be clinically worthwhile, not the effect you hope for. Picking an optimistic hazard ratio gives an attractive small trial that is underpowered for a realistic effect. Look at earlier phase data, published trials with the same endpoint, and what clinicians and patients would regard as meaningful. See the glossary entry on minimal clinically important difference for how that idea is handled with other outcome types, and clinical trial endpoints for how time-to-event endpoints such as overall survival and progression-free survival differ.

The overall event probability is the second weak point. A median survival or event-free rate from the control arm of a comparable study is a starting point, and you adjust it for your planned accrual and follow-up. Be cautious: event rates in trials are often lower than in historical data, because trial populations tend to be healthier or better monitored. Plan with a range and see how the total changes.

Remember that an event-driven trial is not finished when the last participant is enrolled. The analysis happens when the target number of events has occurred, which means your data management has to count events promptly and consistently. Event definitions, adjudication and the date of the event all need clear rules in the protocol and the CRF. In Capture, you can build the event and censoring forms, add edit checks for dates, and review events across sites while the study is running.

Suppose a trial wants to detect a hazard ratio of 0.70 with a two-sided alpha of 0.05 and 90% power. The Schoenfeld formula gives about 331 events. The team expects that 55% of participants will have an event by the time of the final analysis, so it needs 331 / 0.55 = 602 participants. If the same team accepted 80% power, it would need 247 events and 450 participants, which is 152 fewer, but the chance of missing a true effect rises from 10% to 20%. Seeing the trade-off in numbers is the main use of the calculator.

Then test the assumptions that carry the result. If the event probability turns out to be 45% instead of 55%, the requirement becomes 736 participants for 90% power. If the true hazard ratio is 0.75 rather than 0.70, you need roughly 508 events instead of 331. Planning a range, with a base case and a pessimistic case, is more honest than a single number.

The calculator assumes proportional hazards (the hazard ratio is constant over time), 1:1 allocation, a two-sided test, independent censoring and a single primary comparison. Real survival data often breaks these assumptions: curves cross (as with some immunotherapies, where benefit is delayed), allocation is unequal, there are several treatment arms, interim analyses spend alpha, or the endpoint has competing risks. In each case a different method or simulation is needed, such as weighted log-rank tests, restricted mean survival time or group sequential designs.

It does not compute the accrual and follow-up schedule needed to reach the event target; it uses the event probability you provide. It also does not handle non-inferiority margins, cure fractions, stratified analyses, cluster designs or Bayesian approaches. The Bayesian clinical trial design guide explains where those differ.

Involve a biostatistician before the sample size goes into a protocol, certainly for any study that will be submitted to a regulator or ethics committee. This calculator is a demonstration and planning aid, not validated software. Teams with statisticians who work in SAS or R can see how data exports fit in at EDC for biostatisticians and SAS programmers. For the simpler binary-outcome case, use the sample size calculator and, for dropout inflation, the dropout-adjusted calculator.