Restricted mean survival time (RMST) vs the hazard ratio

Restricted mean survival time (RMST) is the average time a person survives, or stays event-free, between time zero and a chosen horizon $τ$ . The difference in RMST between two groups is the difference in expected event-free time within that window, which in a randomised trial is the time the treatment gains or loses, and it keeps that meaning whether or not hazards are proportional. A hazard ratio compares event rates instead, so the two answer different questions, and a report can give both.

What RMST measures

The survival function $S (t)$ is the probability of being event-free at time $t$ , and RMST is the area under it up to $τ$ , $R M S T (τ) = \int_{0}^{τ} S (t) d t$ . If $τ$ is five years and RMST is 4.2 years, a person in that group can expect to spend, on average, 4.2 of the first five years after time zero event-free. The area above the curve, $τ - R M S T (τ)$ , is the restricted mean time lost (RMTL). With competing risks, time lost can be split by cause, as the area under each cause’s cumulative incidence curve up to $τ$ (Andersen, 2013).

The mean survival time is the area under the whole survival curve, but when the last people followed have not had the event, the Kaplan–Meier curve does not reach zero and the data cannot say how the tail continues, so the mean cannot be estimated without assuming it. The documentation of R’s survival package calls the mean undefined in that case and offers, as the first remedy, an upper limit. Restricting the area to a horizon $τ$ fixed in advance gives a quantity the data can estimate: the average survival time within 0 to $τ$ (Royston and Parmar, 2013).

Why use it

Estimated from Kaplan–Meier curves, RMST needs no proportional hazards assumption, whereas a hazard ratio when hazards are not proportional is roughly a weighted average of a hazard ratio that changes over follow-up, with weights that depend on the study’s follow-up and censoring. Like any Kaplan–Meier estimate, it does assume that censoring is independent of the event. The RMST difference is also collapsible. In a randomised trial it is a weighted average of the differences within strata of any baseline covariate (see Ni, Lin and Lu, 2021), which a hazard ratio is not (see what is non-collapsibility?). And it is in units of time, which patients and committees can weigh against harms and costs. It has costs of its own: the answer depends on $τ$ , a difference that comes after $τ$ does not count, and under proportional hazards a log-rank test can be slightly more powerful (see below).

How it is estimated

From Kaplan–Meier curves. Take the area under each group’s curve up to $τ$ , with no model. Uno et al. (2014) recommend model-free summaries like this when hazards may not be proportional. The survRM2 package in R and the community-contributed strmst2 in Stata (Cronin, Tian and Uno, 2016) compare groups this way; both also offer a covariate-adjusted comparison, which is a regression on RMST and so depends on a model.
From a model. Integrate the survival curve predicted by a fitted model, such as a flexible parametric model (Royston and Parmar, 2013). The model’s assumptions then apply. Our merlin package for Stata predicts RMST, and RMST differences and ratios, after fitting.
By regression on pseudo-observations, which gives covariate effects directly on the RMST scale (Andersen, Hansen and Klein, 2004).

A worked calculation

Take the ten hypothetical patients of the worked Kaplan–Meier calculation, with events at 2, 4 and 7 months and follow-up to 12 months. The Kaplan–Meier curve is 1 until month 2, 0.90 from month 2, 0.7875 from month 4 and 0.65625 from month 7, and it stays there to month 12. It never reaches zero, so the mean survival time cannot be estimated, but RMST up to $τ = 12$ months is the area under the steps, one rectangle at a time:

From 0 to 2 months the curve is 1: $1 \times 2 = 2$ months.
From 2 to 4 months it is 0.90: $0.90 \times 2 = 1.8$ .
From 4 to 7 months it is 0.7875: $0.7875 \times 3 = 2.3625$ .
From 7 to 12 months it is 0.65625: $0.65625 \times 5 = 3.28125$ .

The areas add to 9.44375, or 9.44 months. A patient in this group is event-free for 9.44 of the first 12 months on average, so the restricted mean time lost is $12 - 9.44 = 2.56$ months. With $τ = 6$ months the third rectangle is cut at month 6, $0.7875 \times 2 = 1.575$ , and RMST is $2 + 1.8 + 1.575 = 5.375$ , or 5.38 months. The same data give a different RMST for each $τ$ , which is why $τ$ is chosen in advance. With ten patients the confidence interval, which software computes from the variance of the Kaplan–Meier curve, would be wide.

Choosing τ

Choose $τ$ in advance, as the horizon the decision is about, within the follow-up the study can deliver (Uno et al., 2014). The Kaplan–Meier estimate stops where follow-up stops, so $τ$ should be no later than the shorter of the two groups’ longest follow-up times, up to which inference is valid under a mild condition on censoring (Tian et al., 2020). Near that point few remain at risk and the estimate is imprecise.

A $τ$ taken from the data this way is itself random, but the usual confidence intervals remain valid for it (Tian et al., 2020). Choosing it where the curves look furthest apart is not valid in this way, because it picks the most favourable comparison. So fix $τ$ in advance where the design allows it, use one taken from the data only when none was set, and say which it was. A $τ$ beyond follow-up needs a model that extrapolates, as in health economic models, and the answer then rests on the extrapolation.

An example

Uno et al. (2014) reanalysed the ECOG E4A03 trial of lenalidomide with low- or high-dose dexamethasone in newly diagnosed multiple myeloma. The hazard ratio for death, low dose against high, was 0.87 (95% CI 0.60 to 1.27), and the estimated hazard ratio rose over time, crossing one partway through follow-up. Up to 40 months, about the length of follow-up, RMST was 35.4 months on the low dose and 33.3 on the high dose; the paper reports the difference as 2.2 months (95% CI 0.1 to 4.2), presumably from unrounded estimates. Time lost was 4.6 against 6.7 months, a reported RMTL ratio of 0.68, about a third less. Survival at 40 months was slightly lower on the low dose (0.70 against 0.74), so the time gained came earlier in follow-up. The hazard ratio’s interval included one, while the RMST difference favoured the low dose by about two months in 40, with a lower confidence limit close to zero.

How RMST relates to a hazard ratio

Under proportional hazards, survival in the treated group is $S_{0} (t)^{H R}$ , where $S_{0} (t)$ is survival in the control group, so the RMST in each group follows from the hazard ratio and the control curve. The same hazard ratio can be worth very different amounts of time. For illustration, take a hazard ratio of 0.7 and exponential survival in the control group with a median of 12 months. Up to 24 months, RMST is 13.0 months in the control group and 15.4 in the treated group, a gain of 2.4 months; up to 60 months the gain is 5.8 months.

When hazards are not proportional, the two can disagree, as in the example. Royston and Parmar (2013) found the log-rank test slightly more powerful than a test of the RMST difference when hazards were proportional. In their non-proportional scenario, a benefit that was largest early and then waned, the log-rank test, run at the sample size set for 90% power of the RMST test, had 81 to 84% power, while the RMST test stayed close to 90%. For designs with severe non-proportional hazards the sample sizes the two tests need can differ markedly. Our hazard ratio tool shows both for delayed, waning and crossing effects.

RMST or the median?

Median survival is the time at which the estimated survival probability first falls to 0.5 or below, so it reads each curve at a single height. It cannot be estimated when the Kaplan–Meier curve is still above one half at the end of follow-up, which happens when event rates are low or follow-up is short (Uno et al., 2014). RMST uses the whole curve up to $τ$ .

What to report

$τ$ , why it was chosen, and whether it was fixed before the data were seen.
RMST in each group and the difference, with confidence intervals, and the RMST or RMTL ratio if useful.
Whether RMST came from Kaplan–Meier curves or a model, and any covariate adjustment.
The Kaplan–Meier curves with numbers at risk, and, as a sensitivity analysis around the $τ$ chosen, how the conclusion changes with other values.

In software

In R, the survRM2 package compares two groups’ RMST from their Kaplan–Meier curves; in the survival package, printing a survfit object with rmean set to a number gives the RMST up to that time, so keep the number within each group’s follow-up, because beyond it survfit holds the last estimate flat; and flexsurv reports it from a fitted parametric model with summary(fit, type = "rmst", t = 12), where t gives the horizon. In Stata, strmst2 (Cronin, Tian and Uno, 2016) compares groups from Kaplan–Meier curves, and merlin predicts RMST after fitting a model.

Choosing the effect measure for a submission?

The hazard ratio, the difference in restricted mean survival time and the survival curves answer different questions, and a reviewer may ask for more than one. We can help you choose what to report and justify it: see our HTA survival modelling service, or talk to us about your data.

Want to learn more?

MethodSurvival analysis

RMST or a hazard ratio?