Sample size for a non-inferiority log-rank test

When to use it

A trial of a treatment that is cheaper, safer or simpler, where showing it is not meaningfully worse is enough.

Not for

Showing that the new treatment is better. Use the log-rank test.

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
$\mathrm{HR}_0$: Non-inferiority margin on the hazard-ratio scale
$\alpha$: 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 against the margin

D = \left\lceil \frac{(z_{1-\alpha} + z_{1-\beta})^{2}}{p_1\,p_2\,(\ln \mathrm{HR} - \ln \mathrm{HR}_0)^{2}} \right\rceil

Schoenfeld’s formula with the distance from the margin in place of the distance from 1, tested one-sided [1, 2]. A true hazard ratio of 1 is the usual planning assumption.

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 [3].

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 [5]. 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

True hazard ratio expected $\mathrm{HR}$

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.

Non-inferiority margin $\mathrm{HR}_0$

Values: Above 0, not 1; above the expected hazard ratio when a higher hazard is worse

The largest loss of effect that would still be acceptable, justified clinically and from the control treatment’s known benefit. Regulators expect it to be set in advance.

Significance level $\alpha$

Values: One-sided; usually 0.025

One-sided, since only one direction is being ruled out. 0.025 matches a two-sided 0.05 test.

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 [4].
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 margin of 1.3

True HR 1, margin 1.3, one-sided α = 0.025, 80% power: D = 7.849 / (0.25 × (ln 1.3)²) = 456.1, so 457 events; with half the participants having the event, N = 914.

Events: 457 · Group 1: 457 · Group 2: 457 · Total: 914

Protocol statement
To show non-inferiority within a hazard-ratio margin of 1.3 with 80% power at a one-sided significance level of 0.025, using a log-rank test, and assuming a true hazard ratio of 1 and 50% of the control group event-free at the end of the study, 457 events are required; with an expected event probability of 50%, the required sample size is 457 per group (914 in total). Calculated with StatWisely (statwisely.com) using: Events = [z_(1-α)+z_(1-β)]²/[p₁p₂(log(HR)-log(HR₀))²]. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

A tighter margin

Margin 1.2: 945 events and N = 1890. Halving the log-margin roughly quadruples the events.

Events: 945 · Group 1: 945 · Group 2: 945 · Total: 1890

Open this example in the calculator

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.

How to cite

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

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.Chow SC, Shao J, Wang H. Sample Size Calculations in Clinical Research. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2008.
3.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.
4.Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: CRC Press; 2015.
5.Pintilie M. Dealing with competing risks: testing covariates and calculating sample size. Statistics in Medicine. 2002;21(22):3317–3324.