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Survival Analysis

Restricted Mean Survival Time (RMST) Sample Size Calculator

Estimate the sample size required for a two-arm trial whose primary treatment contrast is the difference in restricted mean survival time up to a prespecified horizon τ. The calculation uses a large-sample normal approximation with exponential survival distributions and the exact variance of min(T,τ).

Trial Design Inputs

Specify the survival assumptions and target operating characteristics.
Enter all time quantities in the same units (for example, months). The RMST is the area under each survival curve from 0 through τ.
The attrition field inflates the analyzable sample size by 1 / (1 − dropout). It is separate from the survival model and should be used only for additional loss to follow-up not already represented in the survival assumptions.

Required Sample Size

Large-sample RMST difference calculation under exponential survival.
Enter the design assumptions and click Calculate Sample Size.

Methodology

Restricted mean survival time (RMST) at a prespecified restriction time τ is the area under the survival function through τ:

RMST(τ) = ∫0τ S(t) dt

It can equivalently be viewed as the expected value of min(T,τ). The treatment effect used here is the difference Δ = RMSTT(τ) − RMSTC(τ). This is an interpretable time-scale treatment effect and does not require a proportional-hazards treatment effect.

Exponential survival model

For this calculator, each arm is assumed to have an exponential survival distribution. If the median survival is m, the corresponding hazard rate is λ = log(2) / m. Under that model:

RMST(τ) = [1 − exp(−λτ)] / λ

Because RMST equals E[min(T,τ)], the variance of the restricted survival time under the exponential model is obtained from its first two moments:

E[X] = [1 − exp(−λτ)] / λ
E[X2] = 2[1 − exp(−λτ)(1 + λτ)] / λ2
Var(X) = E[X2] − E[X]2,   X = min(T,τ)

Sample-size formula

Let VC and VT denote the variances of the restricted survival times in the control and treatment arms, respectively. If r = nT / nC is the treatment-to-control allocation ratio, the large-sample variance of the RMST difference is approximated by

Var(Δ̂) ≈ VC/nC + VT/nT = [VC + VT/r] / nC

Therefore, for a two-sided test with type-I error α and target power 1 − β, the required control-arm sample size before rounding is:

nC = (z1−α/2 + z1−β)2> [VC + VT/r] / Δ2

The treatment-arm sample size is nT = r nC. Each arm is rounded upward to a whole participant. If an additional attrition fraction q is specified, the resulting arm sizes are inflated by 1/(1−q).

Interpretation

A positive RMST difference means that, under the specified survival assumptions, the treatment arm has a larger average restricted survival time through τ. For example, an RMST difference of 2 months means 2 additional months of average survival accumulated over the prespecified 24-month horizon.

Important design assumption: this implementation uses the closed-form exponential-survival version of the RMST calculation. It does not attempt to reproduce a simulation-based design for arbitrary piecewise survival curves, nonuniform accrual, time-varying treatment effects, or complex censoring mechanisms. Those assumptions can materially change RMST power and required enrollment.

References

Royston, P. & Parmar, M. K. B. (2013). “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, 13, 152. The paper develops RMST-based trial-design methodology, including the relationship between the RMST difference, its variance, power, and required sample size. Published article.

Uno, H., Claggett, B., Tian, L., et al. (2014). “Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis.” Journal of Clinical Oncology, 32(22), 2380–2385. This paper describes RMST as an interpretable alternative summary of treatment differences in survival analysis. PubMed.

nQuery / Statsols. nQuery provides survival-design procedures and explicitly lists restricted-mean-survival-time methodology among its survival testing capabilities, including the “Non-Inferiority Test for the Difference of Two Restricted Mean Survival Times (RMST)” (STE10). The nQuery catalog also documents survival sample-size procedures and related survival design calculations. nQuery procedure catalog.

Worked validation example

A fixed numerical check for this implementation uses a two-sided α = 0.05 test, 80% power, equal allocation, a 24-month restriction time, 12-month control median survival, and 18-month treatment median survival, with no additional attrition.

Control RMST = 12.9843 months
Treatment RMST = 15.6629 months
RMST difference = 2.6786 months
Control restricted-time variance = 73.2367
Treatment restricted-time variance = 73.4887
Unrounded n per arm = 160.5028
Required n per arm = 161
Required total N = 322

The JavaScript implementation below independently reproduces this calculation and returns 322 total participants for those inputs.

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