Sample size for a log-rank test

When to use it

A trial or cohort comparing time to an event, such as death or relapse, between two groups.

Not for

Showing that a new treatment is not worse. Use the non-inferiority log-rank test.
Hazards expected to cross or diverge over time, where a constant hazard ratio does not hold.

Notation

$D$: Number of events required
$N$: Number of participants
$\mathrm{HR}$: Hazard ratio, treatment against control
$k$: Allocation ratio, treatment to control
$p_1,\ p_2$: Shares of participants in control and treatment, 1/(1 + k) and k/(1 + k)
$S_c$: Survival in the control group at the end of the study
$P_E$: Probability that a participant has the event during the study
$a,\ f$: Accrual period and additional follow-up after it
$\lambda,\ \eta$: Event hazard and loss-to-follow-up hazard
$c,\ \mu$: Chance of a competing event by the end of the study, and its hazard
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$q$: Expected dropout

Formulas

Events: Schoenfeld’s formula

D = \left\lceil \frac{(z_{1-\alpha/2} + z_{1-\beta})^{2}}{p_1\,p_2\,(\ln \mathrm{HR})^{2}} \right\rceil

The log-rank test’s power depends on the number of events. With equal groups p₁p₂ = 1/4, so D = 4(z₁₋α/₂ + z₁₋β)² / (ln HR)² [1].

From events to participants

N = \left\lceil \frac{D}{P_E} \right\rceil, \qquad P_E = p_1\,(1 - S_c) + p_2\,(1 - S_c^{\mathrm{HR}})

A survival analysis draws its power from events, not from participants, so the number to recruit is the events divided by the chance that a participant has one. Without accrual and follow-up times, S_c is taken as the control survival over the whole study.

With accrual, follow-up and losses

P_E(\lambda) = \frac{\lambda}{\lambda + \eta}\left[1 - \frac{e^{-(\lambda+\eta) f}\left(1 - e^{-(\lambda+\eta) a}\right)}{(\lambda+\eta)\,a}\right], \qquad \lambda_c = \frac{-\ln S_c}{a + f}, \quad \lambda_t = \mathrm{HR}\cdot\lambda_c

With participants entering uniformly over the accrual period and followed for at least f afterwards, each has a different time at risk. Survival is exponential, and losses to follow-up are a competing exponential risk that removes the chance of observing the event [2].

Competing risks

\mu = \frac{-\ln(1 - c)}{H}, \qquad \eta \;\leftarrow\; \eta + \mu

Optional. A competing event ends follow-up for the event of interest, so for a cause-specific hazard ratio it acts as one more censoring hazard μ over the study horizon H (the accrual plus follow-up, or one study period). The events of interest needed are unchanged; fewer participants have one, so more are needed [4]. A subdistribution (Fine–Gray) hazard ratio is a different estimand.

Dropout

n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Parameters

Hazard ratio to detect $\mathrm{HR}$

Values: Above 0, not 1

The smallest effect worth detecting; 0.7 means a 30% lower hazard on treatment. Published trials of similar treatments are the best source.

Control survival $S_c$

Values: Between 0 and 1

The proportion of the control group expected to be event-free at the end of the study, from registries or previous trials. Fewer events per participant need more participants.

Allocation ratio $k$ Optional

Values: Above 0; 1 for equal groups

Treatment participants per control participant. Equal allocation needs the fewest events.

Accrual and follow-up times $a,\ f$ Optional

Values: Both 0 or more, in the same units, not both 0

How long recruitment lasts, and how long the study continues after the last participant enters. Given together.

Loss-to-follow-up hazard $\eta$ Optional

Values: 0 or more, per the same time unit; needs accrual and follow-up

The rate at which participants are lost before their event can be seen; 0.05 per year loses about 5% a year.

Chance of a competing event $c$ Optional

Values: At least 0 and below 1; 0 by default

The share expected to have an event that ends follow-up for the event of interest by the end of the study, such as death from another cause.

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Assumptions

Proportional hazards: the hazard ratio is constant over time; check it in the analysis [3].
With accrual and follow-up given, entry is uniform and survival exponential in both groups.
Censoring is unrelated to the outcome.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

A 30% lower hazard

HR = 0.7, half the control group event-free at the end, equal groups, α = 0.05 two-sided, 80% power: 247 events. The event probability is 0.5 in control and 1 − 0.5^0.7 = 0.384 on treatment, 0.442 on average, so N = 247 / 0.442 = 559.

Events: 247 · Group 1: 280 · Group 2: 279 · Total: 559

Protocol statement
To detect a hazard ratio of 0.7 with 80% power at a two-sided significance level of 0.05, using a log-rank test, and assuming 50% of the control group event-free at the end of the study, 247 events are required; with an expected event probability of 44.22%, the required sample size is 280 in the first group and 279 in the second group (559 in total). Calculated with StatWisely (statwisely.com) using: Events = [z_(1-α/2)+z_(1-β)]²/[p₁p₂(log(HR))²]. These are planning estimates; confirm the final design with a qualified statistician.

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Two years of accrual, three of follow-up

With staggered entry, participants recruited late are followed for less time, so fewer have the event: the average event probability falls to 0.372 and N = 664 for the same 247 events.

Events: 247 · Group 1: 332 · Group 2: 332 · Total: 664

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Allowing for losses to follow-up

With losses at a hazard of 0.05 a year as well: event probability 0.340, N = 728.

Events: 247 · Group 1: 364 · Group 2: 364 · Total: 728

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Two treated per control

With k = 2, p₁p₂ = 2/9 instead of 1/4, so 278 events are needed: 220 in control and 438 on treatment.

Events: 278 · Group 1: 220 · Group 2: 438 · Total: 658

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Allowing for dropout

Expecting 10% to withdraw before contributing any follow-up: 312 and 310 are recruited.

Events: 247 · Group 1: 280 · Group 2: 279 · Total: 559 · To recruit: 312 and 310

Protocol statement
To detect a hazard ratio of 0.7 with 80% power at a two-sided significance level of 0.05, using a log-rank test, and assuming 50% of the control group event-free at the end of the study, 247 events are required; with an expected event probability of 44.22%, the required sample size is 280 in the first group and 279 in the second group (559 in total). Allowing for 10% dropout, 312 in the first group and 310 in the second group (622 in total) should be recruited. Calculated with StatWisely (statwisely.com) using: Events = [z_(1-α/2)+z_(1-β)]²/[p₁p₂(log(HR))²]. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

A competing cause of death

With a 0.2 chance of dying from another cause first, treated as censoring: the same 247 events, but the event probability falls to 0.401, so N = 617.

Events: 247 · Group 1: 309 · Group 2: 308 · Total: 617

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Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.
The events agree with the closed form: 4 × (1.960 + 0.842)² / (ln 0.7)² = 246.8, rounded up to 247.
Checked by simulating the log-rank test in R (survival::survdiff), planned for 80% power at hazard ratios of 0.5, 0.7 and 0.8. With equal groups Schoenfeld’s events delivered 80%, also with staggered entry and losses. With unequal groups the approximation drifts: twice as many on treatment, the arm with the lower hazard, gave 83–84%, more than planned, while twice as many controls gave about 78% at a hazard ratio of 0.7. When the larger arm is the one with the higher hazard, plan for a little more power, or compare Freedman’s method, which gave 83% there.

How to cite

StatWisely. Sample size for a log-rank test. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/survival-logrank/

Cite the original references for the formulas your design uses as well.

References

1.Schoenfeld DA. Sample-size formula for the proportional-hazards regression model. Biometrics. 1983;39(2):499–503.
2.Lachin JM, Foulkes MA. Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification. Biometrics. 1986;42(3):507–519.
3.Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: CRC Press; 2015.
4.Pintilie M. Dealing with competing risks: testing covariates and calculating sample size. Statistics in Medicine. 2002;21(22):3317–3324.

When to use it

Not for

Notation

Formulas

Events: Schoenfeld’s formula

From events to participants

With accrual, follow-up and losses

Competing risks

Dropout

Parameters

Hazard ratio to detectHR\mathrm{HR}HR

Control survivalScS_cSc​

Allocation ratiokkkOptional

Accrual and follow-up timesa, fa,\ fa, fOptional

Loss-to-follow-up hazardη\etaηOptional

Chance of a competing eventcccOptional

Significance levelα\alphaα

Power1−β1-\beta1−β

Expected dropoutqqqOptional

Assumptions

Worked examples

A 30% lower hazard

Two years of accrual, three of follow-up

Allowing for losses to follow-up

Two treated per control

Allowing for dropout

A competing cause of death

Validation

How to cite

References

Hazard ratio to detect $\mathrm{HR}$

Control survival $S_c$

Allocation ratio $k$ Optional

Accrual and follow-up times $a,\ f$ Optional

Loss-to-follow-up hazard $\eta$ Optional

Chance of a competing event $c$ Optional

Significance level $\alpha$

Power $1-\beta$

Expected dropout $q$ Optional