Sample size for a difference in restricted mean survival time

When to use it

A trial reporting the RMST difference, months of event-free time gained, as its primary or a supportive analysis when hazards may not be proportional.

Not for

A τ beyond the minimum follow-up, where the Kaplan–Meier curve is too poorly estimated.

Notation

$N$: Number of participants
$\lambda$: Control hazard, from the control survival at the end of the study
$d$: Delay after entry before the treatment effect begins
$\mathrm{HR}$: Hazard ratio after the delay, treatment against control
$h_g,\ S_g$: Hazard and survival of group g; the treated hazard is λ before d and HR·λ after it
$G$: Chance of still being followed at time t, from entry, losses and competing events
$\pi_0,\ \pi_1$: Shares of participants in control and treatment
$a,\ f$: Accrual period and additional follow-up after it
$\eta$: Loss-to-follow-up hazard
$c,\ \mu$: Chance of a competing event by the end of the study, and its hazard
$\tau$: Restriction time; the minimum follow-up unless given
$\Delta$: Difference in RMST, ∫₀^τ (S₁ − S₀) dt
$\sigma_g^{2}$: Per-participant variance of group g’s Kaplan–Meier RMST
$\alpha$: 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

The RMST difference and its variance

N = \left\lceil \frac{(z_{1-\alpha/2} + z_{1-\beta})^{2}\,(\sigma_0^{2}/\pi_0 + \sigma_1^{2}/\pi_1)}{\Delta^{2}} \right\rceil, \qquad \sigma_g^{2} = \int_0^{\tau} \frac{\left(\int_t^{\tau} S_g\right)^{2} h_g}{S_g\, G}\,dt

Each group’s Kaplan–Meier RMST up to τ has the Greenwood-type variance σ²/n [1]. A delay shrinks Δ more than it shrinks the log-rank drift, so RMST usually needs more participants under a delayed effect, while reading as time gained.

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 [2]. 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 after the delay $\mathrm{HR}$

Values: Above 0 and not 1

The effect once it has begun; 0.7 is a 30% lower hazard from the delay on.

Delay before the effect $d$

Values: 0 or more, before the end of the study

From earlier trials of the same kind of treatment, often where the published Kaplan–Meier curves separate.

Control survival at the end of the study $S_0$

Values: Between 0 and 1

Sets the control hazard, taken as constant.

Allocation ratio $k$ Optional

Values: Above 0; 1 for equal groups

Treatment participants per control participant.

Accrual period $a$ Optional

Values: 0 or more, with the follow-up

Entry is taken as uniform over it; without it, one study period of follow-up for everyone, in which the delay is a fraction.

Follow-up after accrual $f$ Optional

Values: 0 or more, with the accrual

Longer follow-up gives a delayed effect more time to show.

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

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

Participants lost before the delay ends add nothing to the comparison.

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.

Restriction time $\tau$ Optional

Values: After the delay and no later than the minimum follow-up

A clinically meaningful horizon, such as two years; later τ gains signal but loses precision.

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

Hazards are equal until the delay and in a constant ratio after it, with a constant control hazard; entry is uniform and censoring unrelated to the outcome.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

An effect from a quarter of the way in

HR = 0.7 from 0.25 on, τ = 1: RMSTs of 0.721 and 0.758, a difference of 0.037, so N = 2729.

Events: 1250 · Group 1: 1365 · Group 2: 1364 · Total: 2729

Protocol statement
To detect the difference in restricted mean survival that a hazard ratio of 0.7 beginning 0.25 units of time after entry produces with 80% power at a two-sided significance level of 0.05, using a comparison of restricted mean survival time up to 1 units of time, and assuming 50% of the control group event-free at the end of the study and equal hazards in both groups until the delay, 1250 events are required; with an expected event probability of 45.78%, the required sample size is 1365 in the first group and 1364 in the second group (2729 in total). Calculated with StatWisely (statwisely.com) using: N = (z_(1-α/2)+z_(1-β))²(σ₀²/π₀ + σ₁²/π₁)/ΔRMST², Kaplan-Meier RMST up to τ. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

The same effect from the start

With no delay the difference doubles to 0.071: N = 684.

Events: 303 · Group 1: 342 · Group 2: 342 · Total: 684

Open this example in the calculator

Two-and-a-half-year RMST

HR = 0.6 after 0.5 years, recruitment over 1 year, τ = 2.5 years: a gain of 0.129 years, N = 1294.

Events: 577 · Group 1: 647 · Group 2: 647 · Total: 1294

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.
Simulated trials with piecewise-exponential survival, analysed with Kaplan–Meier RMST differences and Greenwood-type variances, gave power of 0.81 to 0.83 against a target of 0.80.

How to cite

StatWisely. Sample size for a difference in restricted mean survival time. Method document, engine version 1.0.0, reviewed 2026-10-10. https://statwisely.com/methods/survival-rmst/

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

References

1.Royston P, Parmar MKB. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. BMC Medical Research Methodology. 2013;13:152. doi:10.1186/1471-2288-13-152
2.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

The RMST difference and its variance

Competing risks

Dropout

Parameters

Hazard ratio after the delayHR\mathrm{HR}HR

Delay before the effectddd

Control survival at the end of the studyS0S_0S0​

Allocation ratiokkkOptional

Accrual periodaaaOptional

Follow-up after accrualfffOptional

Loss-to-follow-up rateη\etaηOptional

Chance of a competing eventcccOptional

Restriction timeτ\tauτOptional

Significance levelα\alphaα

Power1−β1-\beta1−β

Expected dropoutqqqOptional

Assumptions

Worked examples

An effect from a quarter of the way in

The same effect from the start

Two-and-a-half-year RMST

Validation

How to cite

References

Hazard ratio after the delay $\mathrm{HR}$

Delay before the effect $d$

Control survival at the end of the study $S_0$

Allocation ratio $k$ Optional

Accrual period $a$ Optional

Follow-up after accrual $f$ Optional

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

Chance of a competing event $c$ Optional

Restriction time $\tau$ Optional

Significance level $\alpha$

Power $1-\beta$

Expected dropout $q$ Optional