Signature
RMST_tau = sum_(k=1)^K [S_k * L_k]
| Inputs | Definition | Unit |
|---|---|---|
S_k | Survival during interval k, equal to the Kaplan-Meier value at its start | probability |
L_k | Width of interval k; the widths add to tau | months |
RMST_tau | Mean event-free time within the first tau units of follow-up, the area under the survival curve from 0 to tau | months |
|---|
KNumber of intervals between 0, the distinct event times up to tau, and tau (count)
Function
Observed follow-up, Kaplan-Meier survival and restricted mean under administrative censoring
Maps each patient's entry date, the shared data cut-off and the event times seen by the cut-off to the observed follow-up and event indicator, then to the Kaplan-Meier survival curve, the restricted mean survival time up to the last observed follow-up and the share of mean survival that has to come from extrapolation beyond it. Staggered entry makes the censoring time differ between patients, so early recruits are observed for longer than late ones. The constant hazard estimated as events divided by person-time is the rate formula of the Adverse Event Rate page (HE-FM-AER-002), and mean survival with an exponential tail has the form of HE-FM-BTH-002 on the Bathtub Hazard page; neither is repeated here.
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Implementations
Excel
Restricted mean from survival and interval widths
With the survival in each interval in SurvRange and the matching widths in WidthRange, SUMPRODUCT returns the area under the step curve.
=SUMPRODUCT(SurvRange,WidthRange)
Assumptions
Restriction time within the observed follow-up
tau is no later than the last observed follow-up, because the Kaplan-Meier curve is undefined beyond it. When patients remain alive at that point only a restricted mean can be estimated directly, and the lifetime mean needs an assumption about the tail.
Undiscounted area under a step function
Survival is constant between event times, so each interval contributes a rectangle. The result is undiscounted and in the time unit of the follow-up.
Worked examples
Restricted mean of 21.7 months to month 34
The article's curve is 1 from 0 to 6 months, 0.875 to month 10, 0.75 to month 14, 0.6 to month 20 and 0.4 to month 34, so the restricted mean is 1 × 6 + 0.875 × 4 + 0.75 × 4 + 0.6 × 6 + 0.4 × 14 = 21.7 months.
K = 5; S_k = [1,0.875,0.75,0.6,0.4]; L_k = [6,4,4,6,14]; RMST_tau = 21.7
Restricted mean of 17.7 months to month 24
Restricting the same curve at month 24, when the curve still rests on two patients, shortens the last interval to 4 months and gives 17.7 months. The figure is computed here for illustration and is not printed in the article.
K = 5; S_k = [1,0.875,0.75,0.6,0.4]; L_k = [6,4,4,6,4]; RMST_tau = 17.7
Common errors
Survival at the end of each interval used as its height
Multiplying each width by the survival reached at the end of the interval places every drop too early. For the article's curve it gives 18.65 months instead of 21.7. The 18.65 is computed here for illustration.
Restricted mean read as lifetime mean survival
The 21.7 months covers only the first 34 months. With 40% still alive at month 34, the lifetime mean also includes the area of the tail, between about a third and a half of the total under the article's tail assumptions, so using the restricted mean in a lifetime model understates survival in both arms.
Sources
Restricted mean survival time as the area under the survival curve
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. Methods, definition of RMST, equation 1: the restricted mean is the mean of the survival time limited to a horizon t* and equals the area under the survival curve from 0 to t*.
Only a restricted mean when patients remain alive at the end of follow-up
Latimer N. 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; 2011, last updated March 2013. Section 2, which states that mean survival equals the area under the survival curve and that, when a proportion of patients remain alive at the end of follow-up, only a mean restricted to that time point can be estimated directly.
Canonical Identity
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