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

อ่านฉบับภาษาไทย (Thai version)

Abstract

The restricted mean survival time (RMST) is the average event-free time up to a fixed horizon. Give each patient a 24-month window and count the months spent free of the event inside it. In a hand example with five patients per arm, the treatment group averages 20.2 months and the control group 18.4 months, so the RMST difference through 24 months is 1.8 months in favour of treatment. The word restricted means capped: a patient still event-free at month 60 counts only 24. The same number is the area under the survival curve from 0 to 24 months, because the curve gives the share still event-free at each moment. Two extremes show why the area is a time: no events gives 24 months, and every event at month 12 gives 12. This article decodes the 1.8 months as a population average and shows why it is not a gain for every patient, a longer median, a lower hazard or a higher survival percentage.

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One line in a trial report

A clinician reads the main result of a randomised trial of a new treatment. The report sums it up in one line: the RMST difference through 24 months is 1.8 months in favour of treatment. The trial and its numbers are a hand example.

The line is short, and it is easy to hear something else in it: that every treated patient gained 1.8 months, that the median moved by 1.8 months, that the hazard fell 1.8 times, or that survival rose by 1.8 percentage points. None of those is what the number counts. This article builds the 1.8 months from five patients per arm before any survival curve appears, and then shows why each of the other readings is a different quantity.

Start without the survival curve

Leave the survival curve aside for now. Follow every patient for at most 24 months and ask one plain question:

Within these 24 months, for how many months did each patient stay free of the event?

The event is whatever the trial counts, so event-free time is the time a patient lives without it. A patient who has the event at month 15 scores 15 months. A patient still event-free at month 24 scores 24, the whole window.

Hand example. The treatment group has five patients.

Treatment group

Average the event-free months: treatment group

1. Mean event-free time, treatment group

   \[ \frac{24+24+20+18+15}{5}=20.2 \text{ months} \]

   Add the five event-free times and divide by the five patients.

Result: The treatment group averages 20.2 event-free months within the first 24 months.

Control group

Average the control group, then take the difference

1. Mean event-free time, control group

   \[ \frac{24+21+18+15+14}{5}=18.4 \text{ months} \]

   The same average for the five control patients.

2. Difference, treatment minus control

   \[ 20.2-18.4=1.8 \text{ months} \]

   This is the meaning of an RMST difference of 1.8 months.

Result: Within the first 24 months, the treatment group has on average 1.8 more event-free months than the control group.

Where the word restricted comes from

The horizon is set in advance: $\tau$ (tau) = 24 months. Only months 0 to 24 count.

Suppose one patient stays event-free until month 60. For the RMST through 24 months, that patient is not credited with 60 months. The most anyone can be credited with is 24, because the question is:

Within the first 24 months, how many months was this patient event-free?

So each patient contributes:

• event at month 10: 10 months
• event at month 17: 17 months
• event at month 23: 23 months
• event after month 24: 24 months

That is why patients 1 and 2 in the treatment group, and patient 1 in the control group, score exactly 24: whatever happens to them after month 24 lies outside the window.

Where the survival curve comes in

Only now bring in the survival curve. Write $S(t)$ for the probability of still being event-free at time $t$: the survival function, the height of the survival curve at month $t$. For a treatment group in a larger trial, the curve might read:

• month 1: 99% still event-free
• month 6: 95%
• month 12: 85%
• month 18: 78%
• month 24: 70%

Take the area under the survival curve from month 0 to month 24. That area is the RMST [1, 2]:

$$\mathrm{RMST}(24)=\int_0^{24} S(t)\,dt$$

There is no need to work the integral to understand it. Think of it this way:

The survival curve says, at each moment, what share of patients is still event-free.

Add that share up over all 24 months and the area under the curve becomes a time. In real trials the curve is estimated from patients whose follow-up ends at different times; the series part on Kaplan-Meier and Cox shows how that curve is drawn and read.

Why an area becomes an average time

Hand example. Two very simple curves over the window from month 0 to month 24.

1. Everyone stays event-free for the whole window

   \[ S(t)=1 \;\Rightarrow\; 1\times24=24 \]

   Every patient is 100% event-free from month 0 to month 24, so the area is 24 and RMST = 24 months. That makes sense: everyone is event-free for all 24 months.

2. Everyone has the event at month 12

   \[ S(t)=1 \text{ for months 0 to 12}, \quad S(t)=0 \text{ for months 12 to 24} \]

   The curve stays at 1 until month 12, when every patient has the event, and then sits at 0.

3. Area under that curve

   \[ 1\times12+0\times12=12 \]

   So RMST = 12 months: every patient was event-free for 12 of the 24 months.

Result: In both cases the area under the curve is the average event-free time.

Back to the result: reading 1.8 months word by word

Now return to the line in the trial report:

RMST difference through 24 months = 1.8 months in favour of treatment

Take it one part at a time.

The most common confusion: did everyone gain 1.8 months?

No. Suppose the treatment helps some patients a great deal and others not at all:

• patient A gains 5 months
• patient B gains 0 months
• patient C gains 3 months
• patient D gains almost nothing
• patient E may even do worse

Across the whole population, the average difference is 1.8 months. So the statement that every treated patient remains event-free for 1.8 extra months is badly wrong.

An RMST difference is a population-average effect: an average over the group, not a prediction that each person gains 1.8 months.

Why it is not a median 1.8 months longer

The median is a different concept altogether. Median survival asks:

When does the survival curve come down to 50%?

For example, if the median were 19 months with treatment and 17.2 months with control, the medians would differ by 1.8 months. But the result reports an RMST difference; it says nothing about the medians.

RMST uses the whole area under the curve from month 0 to month 24. The median looks only at the single point where $S(t)=0.5$.

RMST = the whole area. Median = one point on the curve.

Why it is not a hazard 1.8 times lower

The units alone rule this out. An RMST difference is measured in months, and the result says 1.8 months.

A hazard ratio, the ratio of two groups' event rates among patients still event-free, is a ratio with no unit: HR = 0.70, or HR = 1.2. There is no such thing as HR = 1.8 months.

So reading the result as a hazard 1.8 times lower fails on dimensional logic, before any statistics come in.

Why it is not survival 1.8% higher

A difference in survival probability is measured in percent. For example, at 24 months survival might be 70% with treatment and 60% with control, a survival difference of 10%.

An RMST difference is the area accumulated over the whole window from month 0 to month 24, not a probability at one moment.

Survival probability at 24 months asks about one point, month 24. RMST from month 0 to month 24 adds up the experience of all 24 months.

Four readings of 1.8 months, and their fixes

• "Every treated patient gained 1.8 months."

  The RMST difference is an average over the population; individual gains differ, from several months to none, or even harm.

  Fix: Say that, on average, within the first 24 months, the treatment group had 1.8 more event-free months.

• "Median survival is 1.8 months longer."

  The median is one point on the curve, where it comes down to 50%; RMST is the whole area from month 0 to month 24.

  Fix: Take a median difference only from the two medians, and read the RMST difference as average event-free months within the window.

• "The hazard is 1.8 times lower."

  An RMST difference is in months; a hazard ratio is a ratio with no unit.

  Fix: Read 1.8 months as time, and take a hazard comparison only from a hazard ratio.

• "Survival is 1.8% higher."

  A survival difference is a probability at one time point, in percent; RMST is time accumulated over the whole window, in months.

  Fix: Read the survival difference at 24 months from the two curves at that point, and the RMST difference as months gained across months 0 to 24.

The picture to keep

Picture a time bucket of 24 months for every patient. Each patient fills it with event-free months until the event, or to the brim if the event does not come within 24 months.

On average, the treatment group fills 20.2 months of its bucket and the control group 18.4 months. Treatment therefore adds, on average across the population, 1.8 event-free months during the first 24 months.

Glossary

event-free time (เวลาปลอดเหตุการณ์)

The time a patient lives without the event the study counts; within a horizon, it is capped at that horizon.

restricted mean survival time (ระยะเวลารอดชีพเฉลี่ยแบบจำกัดช่วง (RMST))

RMST(tau), the average event-free time up to a fixed horizon tau, equal to the area under the survival curve from 0 to tau.

tau (the horizon) (ขอบเขตเวลา (time horizon))

The fixed time, chosen in advance, up to which a restricted mean is computed.

survival function (ฟังก์ชันการรอดชีพ)

S(t), the probability of still being event-free at time t; the height of the survival curve.

median survival (มัธยฐานการรอดชีพ)

The time at which the survival curve comes down to one half: a single point on the curve.

hazard ratio (อัตราส่วนฮาซาร์ด (hazard ratio, HR))

The ratio of two groups' event rates among patients still event-free; a ratio with no unit, not a ratio of risks.

population-average effect (ผลระดับประชากร)

An effect that compares average outcomes across a group, not the gain of any one patient.

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 Med Res Methodol. 2013;13:152. doi:10.1186/1471-2288-13-152 https://doi.org/10.1186/1471-2288-13-152
2. Uno H, Claggett B, Tian L, et al. Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis. J Clin Oncol. 2014;32(22):2380-2385. doi:10.1200/JCO.2014.55.2208 https://doi.org/10.1200/JCO.2014.55.2208
3. Kim DH, Uno H, Wei LJ. Restricted mean survival time as a measure to interpret clinical trial results. JAMA Cardiol. 2017;2(11):1179-1180. doi:10.1001/jamacardio.2017.2922 https://doi.org/10.1001/jamacardio.2017.2922

Key takeaways

• RMST is the average event-free time up to a fixed horizon: give each patient a 24-month window and average the months spent free of the event, 20.2 with treatment against 18.4 with control.
• Restricted means capped: a patient still event-free at month 60 counts 24, because only months 0 to 24 are asked about.
• The same number is the area under the survival curve from month 0 to month 24; a curve that stays at 1 gives 24 months, and events for everyone at month 12 give 12.
• An RMST difference of 1.8 months through 24 months means treated patients had, on average, 1.8 more event-free months in the first 24 months: a population average, not a gain for every patient.
• It is time over the whole window, so it is not a median difference (one point), not a hazard ratio (no unit) and not a survival percentage (one time point).

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