Restricted Mean Survival Time (RMST)

Suppose cancer patients received one of two treatments and were followed for 24 months. Patients on treatment A did well early on, but their survival probability dropped quickly over time. Patients on treatment B, by contrast, had a relatively lower survival probability at first, but it declined less afterward.

Which of the two treatments should we say had the better outcome?

Comparing Kaplan–Meier survival curves shows how survival probability changes over time. A log-rank test can also tell us whether the two groups’ survival functions differ statistically. But these results alone make it hard to say how much the actual survival time differs.

The same goes for the hazard ratio (HR) used in Cox regression. A hazard ratio describes the relative size of the instantaneous risk of the event, but it does not tell you directly how many more months patients survived on average.

The method used to answer this kind of question is restricted mean survival time (RMST) analysis.

RMST calculates the expected mean survival time within a period the researcher chooses. For example, if the analysis is based on 24 months, you can estimate how many of those 24 months patients in each treatment group survive on average.

In this article we’ll cover how RMST is calculated, a worked comparison of two treatments, choosing the restriction time, the RMST difference and ratio, confidence intervals, how RMST differs from Kaplan–Meier and Cox regression, and how to interpret results and what to watch out for.

The ordinary mean survival time is the expected value of the full time until the event occurs. In real survival data, however, it is hard to observe the time of death for every patient.

For example, if patients were followed for 24 months and many were still alive when the study ended, we cannot know how much longer they will live. That can make the overall mean survival time hard to estimate. RMST handles this problem by limiting the calculation to a time point fixed in advance.

If T is the time to the event and τ (tau) is the restriction time, it is written as follows.

This means that if the event happens before the restriction time, the actual time to the event counts, and if no event has happened by the restriction time, only the time up to τ is counted.

In real data with censoring, we don’t simply average the observed survival times. Instead, we estimate the survival function with a method such as Kaplan–Meier and calculate RMST from the area under that survival function.

For example, a 24-month RMST of 20.5 months means that patients in that group are estimated to survive about 20.5 months on average within the restricted 24-month period.

Why is RMST needed?

Survival analysis often uses Kaplan–Meier estimation, the log-rank test and Cox regression.

But each of these answers a different question. Kaplan–Meier analysis shows the probability of surviving to a given time. The log-rank test assesses whether the survival functions differ statistically between groups. Cox regression expresses the relationship between several variables and the instantaneous risk of the event as hazard ratios.

RMST, on the other hand, shows how much the two groups’ mean survival time differs over a fixed observation period. For example, suppose we got the following results.

Here the difference in survival probability at 24 months is 10 percentage points. Comparing RMST, though, we can say that over the 24 months the mean survival time in the treatment A group was 1.8 months longer than in the treatment B group.

The two measures mean different things, however. The 24-month survival probability is the probability of surviving to that time, while RMST is the mean survival time accumulated over the 24 months. So two groups with the same survival probability can still have different RMSTs if their survival curves differed before that point.

The formula is as follows.

Here S(t) is the probability that the event has not occurred by time t, and τ is the restriction time chosen by the analyst. For example, suppose the survival probability of 100 patients changed as follows.

Between event times, the Kaplan–Meier survival probability is assumed to stay constant. The 12-month RMST can then be calculated as follows.

So the 12-month RMST is 10.2 months. This means that patients in this group are estimated to survive about 10.2 months on average during the 12 months after starting treatment. The important point is that RMST is not simply the 12-month survival probability of 60% multiplied by 12 months.

RMST reflects how the survival probability changes over the entire observation period.

Patients were assigned 120 to each treatment group and followed for up to 24 months after starting treatment. In this example we assume a randomized study and define the event of interest as death from any cause.

The research question is as follows.

To answer this question, assume we estimated each group’s Kaplan–Meier survival function and calculated the RMST up to 24 months. We can then evaluate the treatment effect using the RMST difference and ratio between the two groups and their confidence intervals.

What do you look for in the Kaplan–Meier survival curves?

Before calculating RMST, it’s a good idea to look at the Kaplan–Meier survival curve for each treatment group.

For example, suppose the following survival probabilities were observed.

Survival probability by treatment

In this example, treatment A had a higher survival probability than treatment B at every time point. In particular, survival probability at 24 months was 70% for treatment A and 60% for treatment B, a difference of 10 percentage points.

But the survival probability at 24 months alone cannot fully describe the two groups’ overall survival experience, because when the deaths occurred during those 24 months also matters. For example, if deaths in one group were concentrated right after treatment started, its RMST could be lower even if its final survival probability was similar to the other group’s.

So RMST has the advantage of showing the difference in survival time over the whole observation period, not just the survival probability at a single time point.

The results for the two groups are as follows.

24-month RMST results

These are not results from analyzing real individual patient data.

The RMST for treatment A was 21.0 months. This means patients in the treatment A group are estimated to survive about 21.0 months on average within the 24-month observation window. The RMST for treatment B was 19.2 months.

So the difference in mean survival time between the two groups is 1.8 months. The restricted mean time lost (RMTL) can also be calculated by subtracting RMST from the restriction time. For treatment A it is 24 - 21.0 = 3.0 months, and for treatment B it is 24 - 19.2 = 4.8 months.

This is the average survival time lost to the event within the restricted period up to 24 months. Note, though, that RMST and RMTL adding up to the restriction time is a relationship that holds by definition; they are not two independent analysis results.

The formula is as follows.

In the earlier example: 21.0 - 19.2 = 1.8 months This means that within the restricted 24-month observation period, the mean survival time of the treatment A group was 1.8 months longer than that of the treatment B group.

Because the RMST difference is expressed in units of time, it is intuitive to interpret. For example, if the RMST difference is 3 months, you can say the two groups’ restricted mean survival times differ by 3 months.

However, the RMST difference depends on the restriction time chosen. The RMST difference up to 12 months and the RMST difference up to 24 months can be different values. So when you report results, always give the restriction time along with the RMST difference.

What is the RMST ratio?

The RMST ratio is a measure calculated by dividing one group’s restricted mean survival time by the other’s.

The formula is as follows.

In the earlier example: 21.0 ÷ 19.2 ≈ 1.094 In other words, the 24-month RMST for treatment A was about 9.4% higher than for treatment B.

This should not be read as meaning that actual survival time under treatment A increased by 9.4% for every patient, though. The RMST ratio is a relative measure comparing mean survival time up to a fixed restriction time.

Comparing the RMST difference and ratio

The RMST difference shows the difference in actual time, and the RMST ratio shows the relative size. Reporting both gives a fuller picture of the difference in survival time between groups.

So when evaluating the RMST difference between two groups, it’s a good idea to check the confidence interval and p-value as well. Suppose the earlier example gave the following test results.

Comparison of RMST between groups

These results were not calculated from real data or from a particular estimation procedure.

The 95% confidence interval for the RMST difference runs from 0.5 months to 3.1 months. The null value for a difference is 0 months. So if the 95% confidence interval does not include 0, as in this result, that is consistent with a statistically significant difference at the 0.05 significance level.

For the RMST ratio, the null value is 1. If the confidence interval for the ratio does not include 1, that is evidence against the hypothesis that the two groups have the same RMST. But a statistically significant result does not necessarily mean a clinically important effect.

How much a 1.8-month difference in mean survival time actually matters has to be judged by also considering the nature of the disease, the side effects of treatment, patients’ quality of life and so on.

The restriction time determines how far out the mean survival time is calculated. For example, the 12-month RMST and the 24-month RMST summarize survival experience over different periods. Depending on how you set the restriction time, the RMST and the difference between groups can change.

Criteria for choosing the restriction time

For example, in a study where one-year survival after starting treatment is what matters, you might choose 12 months as the restriction time. If long-term survival effects matter, on the other hand, you might consider 24 or 36 months.

So whenever possible, the restriction time should be set in advance, taking the research aim and the observation period into account.

Do the results change if the restriction time changes?

Because RMST is the area under the survival curve up to the restriction time, changing τ changes the result.

For example, suppose we calculated the RMST for the two treatment groups at several time points.

In this example, the RMST difference up to 6 months was small at 0.1 months, but the difference up to 24 months grew to 1.8 months. This suggests that the difference in survival experience between the two treatments may have built up over time. But a later restriction time does not necessarily make the RMST difference larger.

If the two groups’ survival curves cross, the RMST difference can shrink or even change direction. So when comparing RMST, it’s a good idea to look at the overall shape of the Kaplan–Meier survival curves, not just the difference at a particular restriction time.

For example, suppose treatment A has a high survival probability early on that drops quickly over time, while treatment B starts with a lower survival probability that then stays relatively stable. In a case like this, the hazard ratio between the two groups can change over time.

The Cox proportional hazards model relies on the proportional hazards assumption, which says the hazard ratio is constant over time. So when survival curves cross substantially, a single hazard ratio may not describe the relationship over the whole observation period well.

Because RMST uses the area under the survival curve, it can compare mean survival time over a restricted period without the proportional hazards assumption. When survival curves cross, though, the RMST difference may turn out to be small.

For example, if treatment A is better early and treatment B is better later, the differences in survival time over the different periods can cancel out. So RMST does not always solve the problem of crossing survival curves.

RMST answers a clear question, the difference in mean survival time over a fixed period, and studies that want to separate early effects from late effects may need additional analyses.

Kaplan–Meier estimation calculates the survival probability over time. RMST accumulates that survival curve up to a particular time point and summarizes it as a mean survival time.

For example, a 24-month survival probability of 70% for treatment A means the estimated probability that a patient survives to 24 months after starting treatment is 70%. A 24-month RMST of 21 months, on the other hand, means that within the same period you can expect about 21 months of survival time on average.

So the two methods do not replace each other; they describe survival outcomes from different angles.

How do RMST and the log-rank test differ?

The log-rank test is a method for testing statistically whether two groups’ survival functions are the same.

An RMST analysis, by contrast, estimates the mean survival time over a particular period and calculates the difference between groups directly.

For example, a log-rank p-value of 0.01 gives statistical evidence against the hypothesis that the two groups’ survival functions are the same. But that result alone does not tell you how many months patients’ mean survival time differs. An RMST comparison gives that time difference directly.

So depending on the research aim, you may report both methods together.

Cox regression expresses the relationship between covariates and the instantaneous risk of the event as a hazard ratio. RMST, on the other hand, calculates mean survival time over a restricted observation period.

For example, if Cox regression gives a hazard ratio of 0.75 for treatment A, then, provided the proportional hazards assumption and other model assumptions hold, you can interpret treatment A’s instantaneous risk of the event as 25% lower than the comparison group’s.

An RMST difference of 1.8 months, by contrast, is interpreted as a 1.8-month difference in mean survival time within a fixed period. Because the two measures answer different questions, a hazard ratio cannot be converted directly into an RMST difference.

Also, RMST comparisons are possible even when the proportional hazards assumption does not hold, but RMST still needs appropriate censoring assumptions and an observation range over which it can be estimated.

For example, if patients’ age or disease severity is distributed differently between the two treatment groups, you can estimate an RMST that statistically adjusts for these variables. Survival regression models, standardization, weighting-based methods and similar approaches can be used for this.

For example, you can estimate each patient’s survival function taking age and disease severity into account, and then calculate the average survival function based on the covariate distribution of a common target population. Integrating that survival function up to the restriction time then gives an estimate of the adjusted RMST.

However, the assumptions required and how the result is interpreted depend on which adjustment method you use. In observational studies in particular, adjusting for several variables does not remove all confounding. So an adjusted RMST difference should not be read straight away as the causal effect of the treatment.

In a randomized study, if randomization worked well and the analysis assumptions are met, the RMST difference between groups can be used as a measure of the treatment effect.

Censoring must be handled appropriately.

If censoring is related to the risk of the event, the RMST estimate can be biased. For example, it can be a problem if patients in poorer health are more likely to drop out of follow-up.

The survival function must be estimable up to the restriction time.

RMST calculates the area under the survival curve up to a fixed restriction time, so there must be enough observed information up to that point. Choosing a common restriction time is especially important if follow-up in one group ends much earlier than in the other.

The event and the start of observation must be defined consistently.

Results differ depending on whether survival time is measured from the date treatment started or from the date of diagnosis. So the starting point of the analysis and the event of interest must be clearly defined.

The uncertainty of the group comparison must be assessed.

Reporting only the RMST estimates makes it hard to see the uncertainty due to sampling variation. So where possible, confidence intervals and appropriate test results should be reported alongside them.

Things to watch out for in RMST analysis

When interpreting RMST results, check the restriction time and the uncertainty of the estimate together.

Do not interpret RMST as the overall mean survival time.

A 24-month RMST of 21 months does not mean that patients’ overall mean lifespan is 21 months. It means the mean survival time expected within the restricted 24-month period.

Results can change with the restriction time.

The 12-month RMST and the 24-month RMST summarize different survival experience. So if the restriction time isn’t reported, the result cannot be interpreted correctly.

Do not pick only a favorable restriction time after the fact.

If you compare many restriction times and report only those that gave significant results, selection bias can arise. So whenever possible, set the restriction time before the analysis, and if you explored several time points, say so explicitly.

Check that the number at risk is large enough.

As you approach the restriction time, the risk set can shrink because of events and censoring. In that case the uncertainty of the survival function and of the RMST estimate can grow.

Distinguish statistical significance from practical importance.

Even if an RMST difference is statistically significant, whether it is large enough to matter for actual treatment choices has to be judged separately. For example, whether a one-month difference in mean survival time matters depends on patients’ health, treatment costs, side effects, quality of life and so on.

Using the two-chemotherapy comparison from above, you could write it up like this.

The estimates and test results shown are not figures calculated from a real clinical trial.

In a real study, it’s a good idea to also report both groups’ Kaplan–Meier survival curves, the follow-up period, the number censored and the rationale for the choice of restriction time. And when explaining the clinical meaning of the RMST difference, you may also consider other outcomes such as treatment side effects and patients’ quality of life.

Medicine and clinical trials

You can compare the mean survival time, or time survived without disease recurrence, of patients who received different treatments. It can be a particularly useful complementary measure in studies where the hazard ratio changes over time.

Public health and epidemiology

You can analyze whether the event-free time expected over a given period differs by particular risk factors or health conditions.

Manufacturing and reliability analysis

You can compare the mean failure-free operating time of different parts or equipment within a restricted period of use.

HR and organizational research

You can compare how long different groups of employees worked on average without leaving over a given period.

Education research

You can compare, by education program, how long students stayed enrolled on average over a fixed period.

Customer retention analysis

You can compare how long customers kept a subscription service on average over 12 or 24 months, by contract type or customer group. In this way, RMST is useful in studies that want to understand the time until an event in actual units of time rather than as a probability or hazard ratio.

If Kaplan–Meier estimation shows survival probability over time and Cox regression gives the ratio of instantaneous event risks, RMST expresses, in units of time, how long subjects survived on average within a restricted period.

For example, if the 24-month RMSTs of two treatments are 21.0 months and 19.2 months, the difference in mean survival time is 1.8 months. A result like this can help explain the difference in actual time more intuitively than a hazard ratio.

In addition, because RMST does not require the proportional hazards assumption of Cox regression, it is useful even when survival curves cross or the hazard ratio changes over time.

RMST results do depend on the restriction time chosen, however, and the effects of censoring and the observation period must be considered. So it’s best to choose a restriction time that fits the research aim in advance and to check the confidence interval and survival curves as well.