Signature
dRMST = RMST_1 - RMST_0; dLY_life = LY_1 - LY_0; phi = dRMST / dLY_life; dLY_beyond = dLY_life - dRMST
| Inputs | Definition | Unit |
|---|---|---|
RMST_1 | Restricted mean survival time of the intervention arm at the follow-up time tau | years |
RMST_0 | Restricted mean survival time of the comparator arm at the same follow-up time tau | years |
LY_1 | Mean survival of the intervention arm over the model horizon, the area under its extrapolated curve | years |
LY_0 | Mean survival of the comparator arm over the same model horizon | years |
dRMST | Life years gained within follow-up, the difference in restricted mean survival time at tau | years per person |
|---|---|---|
dLY_life | Life years gained over the model horizon | years per person |
phi | Share of the life years gained over the model horizon that lies within follow-up | proportion |
dLY_beyond | Life years gained after the end of follow-up | years per person |
Function
Life years gained as the area between two survival curves and the share observed within a trial
Maps the survival curves of an intervention and its comparator, a trial follow-up time tau and a model horizon T to the life years gained within follow-up, the life years gained over the horizon, the share of that gain lying within follow-up and the gain lying after it. The gain over a horizon is the area between the two curves, the integral of S_1(t) minus S_0(t), which Royston and Parmar give as the difference in restricted mean survival time between two arms. Existing records are cited and not restated. The area under one curve is the restricted mean survival time of HE-FM-ADMC-003 for a Kaplan-Meier step function, the interval area A_j of HE-FM-BTH-001 under a constant hazard and the trapezoid life years of HE-CF-PSM-001, so the undiscounted gain is the difference of two such areas. The discounted gain is the present value of a stream, HE-FM-DR-002, applied to the yearly gains; the article's step 3, with year 1 undiscounted, places the gain of year k at time k minus 1 (t starting at 0 in HE-FM-DR-002, the start-of-year timing of HE-FM-DISC-002), whereas HE-CF-PSM-001 discounts each interval at its midpoint. The share of a gain that discounting removes, 1 minus the discounted total over the undiscounted total, equals rise divided by (1 plus rise) for the level stream of HE-CF-DISC-001 (derived here), so it is not drafted either. The cost per life year gained is HE-FM-ICER-001 with life years as the effect, and the QALY gain when both arms live at one constant utility is that utility times the life years gained, the constant-utility case of HE-FM-QALY-001 and of HE-FM-PSM-003 with equal state utilities. Mean survival with a cure fraction is HE-FN-CRM-001, the expected cycles alive of a constant-risk cohort follow from HE-FM-ABS-001, and the share of one arm's mean survival that comes from an exponential tail is P_tail of HE-FM-ADMC-004. Discounted quality-adjusted life expectancy belongs to the QALY Shortfall concept. Notation follows the Life Years Gained article.
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Implementations
Excel
Excel cells for the within-trial share of life years gained
With the restricted means in RMSTNew and RMSTStd and the means over the model horizon in MeanSurvNew and MeanSurvStd, the four formulas return the within-trial gain in RMSTGain, the gain over the horizon in LifetimeLYGain, the share in WithinTrialShare and the gain after follow-up in GainBeyondTrial.
=RMSTNew-RMSTStd; =MeanSurvNew-MeanSurvStd; =RMSTGain/LifetimeLYGain; =LifetimeLYGain-RMSTGain
Assumptions
Restricted means of both arms taken at one follow-up time
Both restricted means run from time 0 to the same tau, no later than the last follow-up observed in either arm, the restriction time of HE-FM-ADMC-003, and both means over the model horizon start at time 0 and run to the same horizon.
Model horizon means from extrapolations that follow the trial data
The means over the model horizon come from curves that agree with the observed data up to tau, so that each is its restricted mean plus the area after tau. TSD 14 section 2 states that the base case should use extrapolation of the fitted distribution and that results based only on the observed data can show the importance of the extrapolated period in the mean; phi expresses that importance as a share of the gain, a reading derived here that TSD 14 does not state.
Within-trial share of life years gained read as a share only for a positive gain
phi is at least 0 and at most 1 when the gain over the horizon is positive and neither the within-trial gain nor the gain after follow-up is negative. If the curves cross after follow-up the within-trial gain can exceed the gain over the horizon, phi rises above 1 and dLY_beyond is negative (HE-EX-LYG-007), and phi is not defined when dLY_life is 0.
Worked examples
Article's two-year follow-up against the life years gained over the horizon
In the article's step 6 follow-up ends at year 2, where the restricted means are 1.55 and 1.25 life years against means of 2.15 and 1.50 over the horizon, all from illustrative curves. The within-trial gain of 0.30 is 46.2% of the gain of 0.65, so 0.35 life years, more than half, lie after follow-up, as the article states.
RMST_1 = 1.55; RMST_0 = 1.25; LY_1 = 2.15; LY_0 = 1.5; dRMST = 0.3; dLY_life = 0.65; phi = 0.4615; dLY_beyond = 0.35
Late survival gain largely unseen at two years of follow-up
Illustrative curves with the same gain of 0.65 life years arising later, standard care at 0.90, 0.70, 0.40, 0.10 and 0 and the new treatment at 0.90, 0.75, 0.65, 0.45 and 0 at the ends of years 1 to 5, have trapezoid restricted means of 1.775 and 1.75 at two years and means of 3.25 and 2.60 over the horizon, so a two-year trial sees 0.025 of the 0.65 life years gained, 3.8%, and 0.625 lie after follow-up (computed here for illustration).
RMST_1 = 1.775; RMST_0 = 1.75; LY_1 = 3.25; LY_0 = 2.6; dRMST = 0.025; dLY_life = 0.65; phi = 0.0385; dLY_beyond = 0.625
Survival curves crossing after follow-up give a within-trial share above 1
With illustrative means of 2.0 and 1.85 over the horizon, the comparator's longer tail cuts the gain from 0.30 within follow-up to 0.15 over the horizon, so phi is 2 and the gain after follow-up is negative (computed here for illustration), the case set aside in HE-AS-LYG-007.
RMST_1 = 1.55; RMST_0 = 1.25; LY_1 = 2.0; LY_0 = 1.85; dRMST = 0.3; dLY_life = 0.15; phi = 2; dLY_beyond = -0.15
Common errors
Within-trial share of life years gained read from one arm's tail share
Taking the new treatment's restricted mean as a share of its own mean, 1.55 over 2.15, suggests that 72.1% of the benefit is observed in the trial, while the share of the gain is 46.2% for the article's figures (computed here). The tail share of HE-FM-ADMC-004 describes one arm's survival; the share of a gain needs both arms.
Within-trial life years gained used as the effect in a cost per life year
Holding the article's GBP 12,000 fixed for illustration, the within-trial gain of 0.30 gives GBP 40,000 per life year against GBP 18,462 on the undiscounted gain of 0.65 over the horizon (computed here), more than double. TSD 14 warns that a survival benefit restricted to what the trial observed is likely to underestimate the true survival gain.
Restricted means of the two arms taken at different follow-up times
Setting the new treatment's restricted mean at two years, 1.55, against the comparator's at three years, 1.45, gives a within-trial gain of 0.10 and a share of 15.4% instead of 0.30 and 46.2% for the article's curves (computed here). Both restricted means are taken at the same tau.
Sources
Royston and Parmar on the difference in restricted mean survival time
Royston P, Parmar MK. 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. Background, which recalls the authors' earlier proposal to express the treatment effect as the difference in restricted mean survival time between the randomised arms at a suitable follow-up time t*, and Methods, section Definition of RMST, which reads the restricted mean of equation 1 as the t*-year life expectancy and gives the difference between arms as the area between the survival curves, in the unnumbered expression that follows equation 1.
TSD 14 on restricted means and extrapolated survival gains
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; June 2011, last updated March 2013. Executive summary (page 4), which states that without extrapolation estimates of the survival benefit are restricted to that observed in the trial and are likely to underestimate the true survival gain, and section 2 (page 12), which states that the base case should use extrapolation of the fitted probability distribution, with results based only on the observed data giving information on the importance of the extrapolated period in the mean.
Latimer and Adler on restricted and lifetime survival benefit
Latimer NR, Adler AI. Extrapolation beyond the end of trials to estimate long term survival and cost effectiveness. BMJ Medicine. 2022;1(1):e000094. Key messages, which state that extrapolation beyond the trial is usually needed to estimate long term effects and that survival models can give widely varying results, and the section on the rationale for survival modelling with figure 1, which describes the area between the curves within the trial as a restricted mean benefit and the area between the extrapolated curves as the mean lifetime survival benefit relevant for economic evaluation.
Canonical Identity
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