Restricted Mean Survival Time

Restricted Mean Survival Time in Trials and Economic Models

Restricted mean survival time (RMST) is the average length of time patients stay free of a defined event, usually death or disease progression, counted only within a window that opens at randomisation and closes at a chosen horizon $\tau$. An RMST of 2.6 years at a five-year horizon means that patients lived, on average, 2.6 of their first five years. The difference in RMST between two arms is an absolute gain in time and does not require hazards to be proportional. Health economists meet the measure in two roles, as a trial effect summary beside the hazard ratio, and as the observed part of the mean survival that a cost-effectiveness model has to estimate over a lifetime.

The restricted mean as the area under a survival curve

Survival time is capped at the horizon, so a patient who dies at 1.5 years contributes 1.5 years and a patient still alive at $\tau$ contributes exactly $\tau$. The mean of this capped time equals the area under the survival function up to $\tau$.

$$\mathrm{RMST}(\tau)=E\left[\min(T,\tau)\right]=\int_0^{\tau} S(t)\,dt$$

where $T$ is the time from randomisation to the event, $\tau$ is the restriction time or horizon, $S(t)$ is the probability of remaining event-free beyond time $t$, and $E[\cdot]$ is the expectation over patients. Royston and Parmar note that when $T$ is measured in years to death, the quantity can be read as the $\tau$-year life expectancy.

Two arms are compared through the area between their curves.

$$\Delta(\tau)=\int_0^{\tau}\left[S_1(t)-S_0(t)\right]dt=\mathrm{RMST}_1(\tau)-\mathrm{RMST}_0(\tau)$$

where $S_1(t)$ and $S_0(t)$ are the survival functions in the research and control arms and $\Delta(\tau)$ is the difference in RMST. A positive $\Delta(\tau)$ is the average event-free time gained in the research arm within the window, and this difference, unlike a ratio of RMSTs, is in units an economic model can use.

Estimating RMST from censored trial data

In a trial with right censoring, the survival curve is estimated first and then integrated. The non-parametric route uses the Kaplan-Meier estimator, and because that curve is a step function its area is a sum of rectangles.

$$\widehat{\mathrm{RMST}}(\tau)=\sum_{j=0}^{m}\widehat{S}(t_j)\,\left(t_{j+1}-t_j\right)$$

where $t_1<\dots<t_m$ are the distinct event times up to $\tau$, $t_0=0$ with $\widehat{S}(t_0)=1$, $t_{m+1}=\tau$, and $\widehat{S}(t_j)$ is the Kaplan-Meier estimate just after $t_j$. A trapezoid rule applied to the Kaplan-Meier points understates this area, because each sloping segment runs below the flat step.

Royston and Parmar list three estimation methods, namely direct integration of Kaplan-Meier curves, a jackknife method and flexible parametric regression. They describe the jackknife as slow and direct integration of Kaplan-Meier curves as possibly unreliable, and prefer flexible parametric models fitted to each arm separately, which do not assume proportional hazards. In a randomised trial they require the estimation method to be predefined, so that no modelling decision depends on the trial data. The difference is then tested by dividing $\widehat{\Delta}$ by its standard error, $\sqrt{SE(\widehat{\mu}_0)^2+SE(\widehat{\mu}_1)^2}$, where $\widehat{\mu}_0$ and $\widehat{\mu}_1$ are the estimated RMSTs in each arm, and comparing the result with a standard normal distribution.

Parametric curves often give RMST in closed form. For a piecewise exponential model, with a constant hazard inside each interval, the area is a sum of exponential pieces.

$$\mathrm{RMST}(\tau)=\sum_{j=0}^{k} e^{-H_j}\,\frac{1-e^{-h_{j+1}\delta_{j+1}}}{h_{j+1}}$$

where $h_{j+1}$ is the constant hazard in interval $j+1$, $\delta_{j+1}$ is the length of that interval lying inside the window, $H_j$ is the cumulative hazard at the start of the interval with $H_0=0$, and $k$ is the number of knots, the interval boundaries between 0 and $\tau$. With a single interval and hazard $h$, the expression reduces to $(1-e^{-h\tau})/h$.

Choosing the restriction time

The horizon changes the answer, so it is fixed and stated in advance. Royston and Parmar write that the primary RMST estimate is tied to a chosen horizon that must be made explicit, though the effect can also be explored over a range of alternative values. They warn of a temptation to pick the horizon that gives the most significant result, and treat any such search as secondary to the prespecified analysis. For the final analysis they suggest limiting the horizon to the largest uncensored event time, so that estimates from a flexible parametric model are not extrapolated.

Tian and colleagues showed that, under a mild condition on the censoring distribution, inference about RMST remains valid with a horizon as late as the largest follow-up time, observed or censored. They still regard prespecification at the design stage, on clinical and feasibility grounds, as the ideal. A late horizon uses more data, but the end of the curve rests on few patients at risk.

The shape of the effect also drives trial size. In Royston and Parmar's designs based on the SORCE kidney cancer trial, the RMST approach needed about 8% more patients than the log-rank test when hazards were proportional, while with a treatment effect that faded over time the log-rank design needed 27% to 42% more patients than the RMST design.

How RMST compares with the hazard ratio and the median

A hazard ratio is a relative measure of instantaneous risk and says nothing about time gained until it is applied to a baseline curve. When the proportional hazards assumption fails, a single ratio also depends on the length of follow-up. Part of Royston and Parmar's comparison of effect measures, each taken as a difference between arms, is shown below.

The median is a snapshot at one point on the curve. RMST uses the whole curve up to $\tau$, and the authors argue that its need for a time point is a merit, because it puts the time dimension of the trial into the result. Their example compares two MRC cancer trials, ASTEC and OE02, with almost identical absolute log hazard ratios. At a five-year horizon the absolute RMST difference was 0.29 years in OE02 against 0.09 years in ASTEC, more than three times larger, because five-year survival was much higher in ASTEC. Uno and colleagues argue that the clinical meaning of a hazard ratio is difficult, if not impossible, to interpret when proportional hazards fail, and encourage a prespecified, clinically meaningful, model-free measure.

Worked example with an early harm and a later benefit

All figures in this example are illustrative. A control treatment has a constant death hazard of 0.30 per year. A new treatment carries early risk, with a hazard of 0.40 per year in the first year, and then lowers the hazard to 0.15 per year. The hazard ratio is $\mathrm{HR}=0.40/0.30=1.33$ in year one and $\mathrm{HR}=0.15/0.30=0.50$ afterwards, so no single ratio describes the effect, and the survival curves cross at about 1.67 years.

1. Control arm at five years. With one constant hazard, $\mathrm{RMST}_0(5)=(1-e^{-1.5})/0.30=(1-0.2231)/0.30=2.5896$ years.

2. Treatment arm, first year. The first interval contributes $(1-e^{-0.40})/0.40=(1-0.6703)/0.40=0.8242$ years, and survival at one year is $S_1(1)=e^{-0.40}=0.6703$.

3. Treatment arm, years one to five. The second interval has length 4 and hazard 0.15, so it contributes $S_1(1)\times(1-e^{-0.60})/0.15=0.6703\times3.0079=2.0163$ years.

4. Treatment arm total. $\mathrm{RMST}_1(5)=0.8242+2.0163=2.8405$ years.

5. Difference at five years. $\Delta(5)=2.8405-2.5896=0.2509$ years, which is $\Delta(5)\times12=3.01$ months.

6. Difference at two years. The control arm gives $\mathrm{RMST}_0(2)=(1-e^{-0.60})/0.30=0.4512/0.30=1.5040$. The treatment arm gives $\mathrm{RMST}_1(2)=0.8242+0.6703\times(1-e^{-0.15})/0.15=0.8242+0.6703\times0.9286=0.8242+0.6225=1.4467$. The difference is $\Delta(2)=1.4467-1.5040=-0.0573$ years, about 21 days lost.

7. Beyond the trial. At five years survival is $S_0(5)=e^{-1.5}=0.2231$ and $S_1(5)=e^{-1.0}=0.3679$. If the hazards were assumed to stay at 0.30 and 0.15 for life, the mean time lived after year five would be $S_0(5)/0.30=0.7438$ and $S_1(5)/0.15=2.4525$ years. Lifetime means would be 3.3333 and 5.2930 years, a gain of about 1.96 years.

The within-trial result changes sign with the horizon, a small loss at two years becoming a gain of three months at five. Under the constant-hazard tail assumed in step 7, only about 13% of the lifetime gain falls inside the five observed years. A cost-effectiveness model built on these data would depend mainly on the extrapolation, and the five-year RMST would serve as a check on the observed period, not as the base-case benefit.

RMST in cost-effectiveness models and NICE appraisals

The NICE manual (PMG36, section 4.2.22) asks for a time horizon long enough to reflect all important differences in costs or outcomes, and section 4.2.23 states that a lifetime horizon is needed when technologies lead to differences in survival. A trial-horizon RMST is therefore rarely enough for a base case when many patients are still alive. Latimer, in NICE Decision Support Unit Technical Support Document 14, recommends that mean time to event be estimated rather than medians, and that parametric models be used rather than restricted means approaches unless the data are almost entirely complete. His review found restricted means analyses in 17 of 45 appraisals (38%), in the base case or a sensitivity analysis, usually as the area under the Kaplan-Meier curve up to the final observation.

Matching the observed RMST is a weak test of a survival model. In the simulation study in Technical Support Document 21, standard parametric models produced low bias for RMST up to the end of trial follow-up, apart from the exponential and in some scenarios the Gompertz, while bias could become substantial when extrapolating to a lifetime horizon. PMG36 (section 4.6.25) also asks for the clinical plausibility of extrapolated hazard functions to be assessed routinely.

RMST also sits inside the arithmetic of a partitioned survival model. Technical Support Document 19 describes how the progression-free survival curve gives the proportion of patients progression-free over time, and how membership of the progressed state is the difference between the overall and progression-free survival curves. Integrated to the model horizon, the expected time progression-free is the RMST of the progression-free curve, and the time in the progressed state is the RMST of overall survival minus that amount. Undiscounted life years gained are the RMST difference for overall survival at the model horizon, and QALYs and costs follow by weighting the same areas with utilities, unit costs and discount factors.

Limits and common misreadings

Most errors with RMST come from forgetting the window it refers to. Five are common.

1. Reading RMST as lifetime mean survival. It covers only the window, and a lifetime mean needs an assumption about what happens after $\tau$.
2. Comparing values across studies with different horizons. RMSTs at five and ten years describe different quantities, and baseline risk changes the difference too.
3. Choosing the horizon after seeing the results. The test statistic depends on $\tau$, so a horizon picked for significance undermines the comparison.
4. Ignoring the tail. Late portions of a Kaplan-Meier curve rest on few patients, and the estimate assumes censoring is unrelated to prognosis. In trials with treatment switching, an intention-to-treat RMST also reflects the switch.
5. Dismissing a small difference. Royston and Parmar note that significant RMST differences from trials may look small yet be more realistic and clinically meaningful than relative effects on the hazard that look more impressive.

Sources

• Latimer NR. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; June 2011 (last updated March 2013). https://www.sheffield.ac.uk/nice-dsu/tsds/survival-analysis
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 4.2.22, 4.2.23 and 4.6.25. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• 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. https://doi.org/10.1186/1471-2288-13-152
• Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020. Section 5. https://www.sheffield.ac.uk/nice-dsu/tsds/flexible-methods-survival-analysis
• Tian L, Jin H, Uno H, Lu Y, Huang B, Anderson KM, Wei LJ. On the empirical choice of the time window for restricted mean survival time. Biometrics. 2020;76(4):1157-1166. https://doi.org/10.1111/biom.13237
• Uno H, Claggett B, Tian L, et al. Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis. Journal of Clinical Oncology. 2014;32(22):2380-2385. https://doi.org/10.1200/JCO.2014.55.2208
• Woods B, Sideris E, Palmer S, Latimer N, Soares M. NICE DSU Technical Support Document 19: Partitioned survival analysis for decision modelling in health care, a critical review. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2 June 2017. https://www.sheffield.ac.uk/nice-dsu/tsds/partitioned-survival-analysis