Sample size for a Cox regression

When to use it

An observational survival analysis of an exposure adjusted for confounders, or an adjusted trial analysis.

Not for

An unadjusted comparison, where the log-rank test is the same calculation.

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
$R^{2}_{X|Z}$: Share of the predictor’s variance explained by the other covariates
$\alpha$: 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, inflated for correlated covariates

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

Schoenfeld’s events, multiplied by the variance inflation from covariates that predict the exposure [1, 2].

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

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.

R² of the predictor on the other covariates $R^{2}_{X|Z}$

Values: At least 0 and below 1

How well the other covariates predict the exposure, from previous data. With 0 the result equals the log-rank test.

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 [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

An exposure related to the covariates

HR = 0.7, with the other covariates explaining 0.2 of the exposure: 246.8 / (1 − 0.2) = 308.5, so 309 events and N = 699.

Events: 309 · Group 1: 350 · Group 2: 349 · Total: 699

Protocol statement
To detect a hazard ratio of 0.7 with 80% power at a two-sided significance level of 0.05, using a Cox regression, and assuming 50% of the control group event-free at the end of the study and other covariates explaining 0.2 of the predictor’s variance, 309 events are required; with an expected event probability of 44.22%, the required sample size is 350 in the first group and 349 in the second group (699 in total). Calculated with StatWisely (statwisely.com) using: Events = [z_(1-α/2)+z_(1-β)]²/[p₁p₂(log(HR))²] × 1/(1-R²_X|Z). These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

An exposure unrelated to them

With R² = 0, the same as the log-rank test: 247 events, N = 559.

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

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 Cox regression. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/survival-cox/

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.Hsieh FY, Lavori PW. Sample-size calculations for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials. 2000;21(6):552–560.
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.