Signature
E_T = A_closed + S_c / h_K
| Inputs | Definition | Unit |
|---|---|---|
A_closed | Sum of the life-years A_j over all intervals before the last cut point | years |
S_c | Proportion of the starting cohort alive at the start of the open final interval | proportion of the starting cohort |
h_K | Constant hazard applied from the last cut point onwards, above zero | events per person-year |
E_T | Expected survival per person entering the cohort, from time zero onwards | years |
|---|
Function
Survival and mean survival from a bathtub-shaped hazard
Maps a hazard that is high soon after a procedure, falls to a stable level and later rises with age to the survival curve and the mean survival that a health economic model needs. Survival at time t is the exponential of minus the cumulative hazard up to t, and mean survival is the area under the survival curve. The formulae below apply this to a piecewise-constant hazard schedule, to a constant hazard fitted to early follow-up, to a sum of two Weibull hazards and to an all-cause hazard built from background and excess components. Turning an interval hazard into a cycle transition probability uses the conversion already given on the Transition Probability page (HE-FM-TP-001), which is not repeated here.
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Implementations
Excel
Mean survival from closed bathtub intervals and an open tail
With the summed closed-interval life-years in ClosedLifeYears, survival at the last cut point in SurvCut and the final hazard in FinalHazard, the cell returns mean survival.
=ClosedLifeYears+SurvCut/FinalHazard
Assumptions
Final bathtub hazard held constant without limit
The hazard in the final interval stays constant for the rest of the cohort's life. This interval is the extrapolated tail, so its hazard usually comes from outside the trial, for example from background mortality, and deserves sensitivity analysis.
Undiscounted mean survival from a bathtub schedule
E_T is undiscounted. Discounted life-years need the survival curve to be integrated year by year with a discount factor, so they cannot be obtained by discounting E_T itself.
Worked examples
Mean survival of 10.58 years under the article's bathtub hazard
The three closed intervals of the article's example contribute 8.975268 life-years and leave 0.320620 alive at fifteen years. A hazard of 0.20 per year thereafter adds 1.6031 life-years, giving mean survival of about 10.5784, the article's 10.58 years.
A_closed = 8.975268; S_c = 0.320620; h_K = 0.20; E_T = 10.5784
Post-peak hazard of 0.05 carried forward for life
If the stable hazard of 0.05 per year is carried forward from three months with no late rise, the open interval starts at survival of 0.904837 after 0.237906 life-years. Mean survival becomes about 18.3346, the article's 18.33 years.
A_closed = 0.237906; S_c = 0.904837; h_K = 0.05; E_T = 18.3346
Common errors
Bathtub mean survival stopped at the last cut point
Summing only the closed intervals gives a restricted mean. In the article's example it returns 8.9753 years instead of 10.5784, leaving out the 1.6031 life-years lived after year fifteen.
Stable post-peak hazard extrapolated without the late rise
Carrying the middle hazard of a bathtub forward for life omits the age-related rise. In the article's example mean survival becomes 18.33 years instead of 10.58, and every arm of a cost-effectiveness model is credited with too many life-years and QALYs.
Sources
Mean survival as the area under the survival curve in DSU TSD 14
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 only a restricted mean can be estimated directly while patients remain alive at the end of follow-up, and section 2.7 on the limits of piecewise models for the extrapolated portion.
Mean and discounted mean survival in the poly-Weibull paper
Demiris N, Lunn D, Sharples LD. Survival extrapolation using the poly-Weibull model. Statistical Methods in Medical Research. 2015;24(2):287-301. Section 2.1, equation 2.6 for mean survival as the integral of the survivor function and the following equation for discounted mean survival.
Canonical Identity
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