Restricted Mean Survival Time: How Much Event-Free Time Did Treatment Buy?

อ่านฉบับภาษาไทย (Thai version)

Abstract

When survival curves separate early, then draw closer again, one hazard ratio (a ratio of event rates) averages a strong early benefit with no benefit later. It cannot say how much time treatment buys. The restricted mean survival time, RMST(tau), is the area under the survival curve up to a chosen horizon tau: the mean event-free time within that window. Without censoring it is the mean of event-free times capped at tau. With censoring it is estimated, under independent censoring, by the area under the Kaplan-Meier step curve, summed as rectangles, which gives 18.375 months for ten hand-worked patients followed to 24 months. In a simulated heart-failure trial whose death hazard ratio changes at 12 months, the single hazard ratio is 0.84, while the RMST difference at 24 months is 1.86 months (95% CI 1.04 to 2.67) against a true 1.74. This article concludes that RMST, with a pre-specified horizon, complements the hazard ratio and the curves with a difference in months.

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A two-year question in a heart-failure clinic

A cardiologist is going through the results of a randomised trial of a new heart-failure drug with a patient who was discharged last week. The drug and the trial are fictional, and their numbers are simulated data. The two survival curves separate in the first year and then draw closer again without crossing: the drug helps early and gives no further benefit after that. The report gives one hazard ratio for death, 0.84 (95% CI 0.74 to 0.96).

The patient asks a plain question: over the next two years, how much longer will I live if I take it? A hazard ratio cannot answer in months. It compares event rates, and here it averages a strong early benefit with no benefit later.

The restricted mean survival time (RMST) can answer it as an average over patients like those in the trial. Within 24 months of randomisation, patients assigned to the drug lived on average 1.86 months longer than patients assigned to control (95% CI 1.04 to 2.67). This article builds that number from its definition, first by hand and then in the simulated trial, and shows what it does not say.

RMST is the area under the survival curve

Write $T$ for a patient's event time in months from randomisation, and $S(t) = P(T > t)$ for the probability of still being event-free at month $t$. Choose a horizon $\tau$ (tau), the last month the question is about. The RMST up to that horizon is

The integral is the area under the survival curve from month 0 to month $\tau$. The second form says the same thing as a mean: $\min(T, \tau)$, the smaller of the event time and the horizon, is the event-free time a patient has within the window, and $\mathrm{E}[\cdot]$ averages it over patients. A patient still event-free at the horizon counts as exactly $\tau$ months, however long they go on to live; that cap is what restricted means. Because the height of the curve is a probability and its width is time, the area is in months [1, 2].

Two extremes make the link plain. If nobody has the event before $\tau$, then $S(t) = 1$ throughout, the area is $1 \times \tau$, and everyone got the whole window. If everyone had the event at the start, the area would be 0.

With censoring, measure the area instead

Averaging observed times breaks down once follow-up is incomplete. A patient lost to follow-up at month 8 was event-free for at least 8 months, but nobody knows for how much longer. Counting 8 as the event time understates the mean, and dropping the patient throws away the 8 months that were observed.

In the Kaplan-Meier curve, a censored patient stays in the risk set, the patients still event-free and under observation, until follow-up ends, and then leaves it without counting as an event. This rests on independent censoring, also called non-informative censoring: patients who leave are assumed to share the future risk of those who stay. Censoring is informative when this fails, for example when patients stop attending because they are getting worse. The simulated trial below generated censoring independently of treatment and of the risk of death, so the assumption holds there by design.

In real data it is an assumption, and the data alone cannot confirm it. Under this assumption, RMST in censored data is estimated by the area under the Kaplan-Meier curve. Because that curve is flat between events, the area is a sum of rectangles [1]. The series part on Kaplan-Meier and Cox works through the curve itself and how it is misread.

Choosing the horizon

RMST has no meaning without its horizon, so $\tau$ belongs to the question, not to the analysis. Three conditions usually guide the choice [1, 3]. The horizon is pre-specified, in the protocol or analysis plan, before anyone sees the curves. It is clinically meaningful, such as two years after a heart-failure admission, and it lies where enough patients remain at risk in both arms for the curves to be estimated well.

In the simulated trial, patients were enrolled over time and follow-up stopped on one closing date, so each patient could be followed for between 24 and 36 months. Up to month 24 the only censoring is loss to follow-up. Beyond it, administrative censoring, the planned end of follow-up rather than anything about the patient, starts removing patients from both arms, and after month 36 there are no data at all. A horizon of 24 months therefore covers the window that every patient could have reached.

Choosing $\tau$ from the data by a stated rule, for example as late as the largest follow-up time, has also been studied. With such a rule, valid inference remains possible under mild conditions on censoring [3]. A horizon moved by eye after the results are seen, to wherever the gap looks largest, loses the protection of pre-specification.

Difference, ratio and time lost

With an RMST for each arm, the most direct contrast is the RMST difference, treated minus control, in months: 1.8 months in hand example 1. The RMST ratio, treated over control, puts the same comparison on a relative scale, 1.10 in that example. Both depend on $\tau$ and are quoted with it.

The complement of the RMST is the restricted mean time lost (RMTL):

It is the area between the survival curve and 1, the average time lost to the event before $\tau$. It comes to 3.8 and 5.6 months in hand example 1, and 5.625 months in hand example 2. A difference in RMTL is just the RMST difference with its sign reversed. The RMTL ratio is not redundant on the relative scale: when most of the window is spent event-free, a modest RMST ratio can sit beside a large relative cut in time lost.

The simulated trial: a hazard ratio that changes at 12 months

The dataset example is a simulated randomised trial of 1,500 adults enrolled after a heart-failure admission, 750 per arm, with a fictional drug. Here the event is death from any cause, so event-free time is time alive. During follow-up, 444 patients died in the control arm and 417 in the drug arm.

The data were generated with a known truth: the hazard ratio for death is 0.60 during the first 12 months and 1.10 afterwards. This is non-proportional hazards, a hazard ratio that changes over follow-up, so no single hazard ratio describes the effect. The two arms' cumulative hazards (the running totals of the hazard over time) would become equal only at month 60, far beyond the 36-month maximum follow-up.

So the survival curves do not cross within follow-up. They separate early and then draw closer again as the late hazard ratio of 1.10 takes effect. True survival is 0.681 without the drug and 0.794 with it at 12 months, and 0.464 and 0.521 at 24 months, so the gap roughly halves. The series hub describes the other questions this trial can answer, about repeated admissions and causes of death.

What RMST says about the same trial

At $\tau$ = 24 months, the RMST is 16.88 months in the control arm and 18.74 months in the drug arm. The difference is 1.86 months (95% CI 1.04 to 2.67), close to the true 1.74 months built into the simulation. Over the first two years after randomisation, patients assigned to the drug lived on average 1.86 months longer.

The ratio scales tell the same story differently. The RMST ratio is 1.11 (95% CI 1.06 to 1.16), so the drug arm spent 1.11 times as much of the window alive. The control arm lost on average 7.12 of the 24 months to death and the drug arm 5.26, an RMTL ratio of 0.74 (95% CI 0.65 to 0.84). The RMTL ratio moves much further from 1 because most of the window is spent alive in both arms.

A stronger-looking hazard ratio, less time gained

The same simulation also produced a version of the trial with the same control arm and a constant effect. Proportional hazards therefore hold by design (the check gives P = 0.92). Its hazard ratio is further from 1, 0.78 (95% CI 0.68 to 0.90). Yet its RMST difference at 24 months is smaller, 1.33 months (95% CI 0.49 to 2.18), against a true 1.22.

In the trial with the early benefit, the true hazard ratio in the first 12 months is 0.60, stronger than the constant 0.79 of the second version. Each death averted early adds event-free time over most of the 24-month window. The single hazard ratio of that trial, 0.84, is also pulled toward 1 by the later period, when the drug no longer helps. That period includes months 24 to 36, which RMST(24) does not use.

The two estimated RMST differences have overlapping intervals, 0.49 to 2.18 and 1.04 to 2.67. But the true values show the same ordering as the estimates: 1.22 months with the constant effect against 1.74 months with the early benefit. The hazard ratios rank the two versions the other way: 0.79 for the constant effect against 0.84 for the early benefit, in a very large simulated sample. The two summaries answer different questions, which is why a report reads best with both and the curves.

The code: strmst2 in Stata and rmst2 in R

In Stata, the user-written command strmst2 [4] computes each arm's RMST up to $\tau$ from the Kaplan-Meier curve, with standard errors, the difference and the ratio; its rmtl option adds the time lost. In R, the function rmst2 from the survRM2 package does the same. Each pane below shows only the script lines that run this analysis, with the run's output beside them. The simulated trial file is not published; the lines shown are the RMST analysis to adapt to your own data.

In Stata, import delimited reads the simulated trial and stset fu_months, failure(death == 1) id(id) declares each patient's follow-up time and event. Then strmst2 arm, tau(24) rmtl produces the numbers in the RMST table above. In R, the data frame wn (the simulated trial) is read earlier in the script, and print(nrm) later prints the output shown.

Stata: strmst2 with the rmtl option

* ---------------------------------------------------------------- non-proportional-hazards file: RMST and RMTL by arm
* the simulated trial file (not published)
import delimited using ../../datasets/W3/W3_nonph.csv, clear asdouble
stset fu_months, failure(death == 1) id(id)
strmst2 arm, tau(24) rmtl

. strmst2 arm, tau(24) rmtl

Number of observations for analysis = 1500

The truncation time: tau = 24 was specified.

Restricted Mean Survival Time (RMST) by arm
-----------------------------------------------------------
   Group |  Estimate    Std. Err.      [95% Conf. Interval]
---------+-------------------------------------------------
   arm 1 |    18.742       0.276       18.201       19.284
   arm 0 |    16.884       0.311       16.275       17.494
-----------------------------------------------------------

Restricted Mean Time Lost (RMTL) by arm
-----------------------------------------------------------
   Group |  Estimate    Std. Err.      [95% Conf. Interval]
---------+-------------------------------------------------
   arm 1 |     5.258       0.276        4.716        5.799
   arm 0 |     7.116       0.311        6.506        7.725
-----------------------------------------------------------

Between-group contrast (arm 1 versus arm 0)
------------------------------------------------------------------------
           Contrast  |  Estimate       [95% Conf. Interval]     P>|z|
---------------------+--------------------------------------------------
RMST (arm 1 - arm 0) |     1.858        1.042        2.673      0.000
RMST (arm 1 / arm 0) |     1.110        1.060        1.163      0.000
RMTL (arm 1 / arm 0) |     0.739        0.646        0.845      0.000
------------------------------------------------------------------------

R: rmst2 from the survRM2 package

# RMST still has a plain meaning when hazards are not proportional
nrm <- rmst2(wn$fu_months, wn$death, wn$arm, tau = 24)

> print(nrm)

The truncation time: tau = 24  was specified.

Restricted Mean Survival Time (RMST) by arm
               Est.    se lower .95 upper .95
RMST (arm=1) 18.742 0.276    18.201    19.283
RMST (arm=0) 16.884 0.311    16.275    17.493


Restricted Mean Time Lost (RMTL) by arm
              Est.    se lower .95 upper .95
RMTL (arm=1) 5.258 0.276     4.717     5.799
RMTL (arm=0) 7.116 0.311     6.507     7.725


Between-group contrast
                      Est. lower .95 upper .95 p
RMST (arm=1)-(arm=0) 1.858     1.043     2.672 0
RMST (arm=1)/(arm=0) 1.110     1.060     1.163 0
RMTL (arm=1)/(arm=0) 0.739     0.646     0.845 0

What RMST is not

RMST is not the median survival, the time at which the survival curve falls to one half, which is a single point on the curve. In the simulated trial the true medians are 21.66 months without the drug and 25.15 months with it, a gap that answers a different question. Nor is RMST the unrestricted mean survival time, the whole area under the curve, which needs follow-up to the end of the curve. Here about half of each arm is still alive at month 24, and nobody is followed beyond month 36.

RMST is not free of $\tau$: the RMST difference generally changes with the horizon, so a difference quoted without its horizon cannot be read. Nor does RMST replace the curves. Different pairs of curves can share one RMST difference, so the Kaplan-Meier curves, with the numbers at risk, still belong in the report [5].

What to do in your own analysis

• Choose $\tau$ from the clinical question and the planned minimum follow-up, write it into the protocol or analysis plan, and check that enough patients remain at risk in both arms at that time.
• Each arm's RMST, the difference with its confidence interval and the horizon are usually reported together in one sentence.
• Consider adding the RMST ratio or the RMTL ratio; the RMTL ratio is often the more telling relative measure when most patients stay event-free through the window.
• The Kaplan-Meier curves, with the numbers at risk, are usually shown beside any RMST or hazard ratio.
• Consider pre-specifying RMST beside the hazard ratio rather than adding it only after a proportional-hazards check fails. When hazards look non-proportional, period-specific hazard ratios may also show where the effect sits; a later-period ratio is best read as descriptive, because it compares only the patients who survived the earlier period.
• To adjust for baseline covariates, RMST can be modelled directly. One route computes a pseudo-observation for each patient from the whole-sample RMST and the RMST recomputed without that patient. A generalised linear model is then fitted to these values with robust (sandwich) standard errors, which allow for the pseudo-observations not being independent of one another [6]. Another route is a regression model for the restricted mean [7].

Related in the wiki: [[survival-analysis-kaplan-meier-cox]] [[time-to-event-survival-analysis]]