Signature
lambda_hat = D / PY; S_t = exp(-lambda_hat * t); E_T = 1 / lambda_hat
| Inputs | Definition | Unit |
|---|---|---|
D | Deaths during follow-up per person entering, equal to 1 minus survival at the end of follow-up when no one is censored earlier | deaths per person entering |
PY | Person-years lived during follow-up per person entering | years |
t | Time since the procedure at which survival under the fitted constant hazard is read | years |
lambda_hat | Constant hazard of the exponential model fitted to the follow-up period | events per person-year |
|---|---|---|
S_t | Proportion alive at time t under the fitted exponential model | proportion of the starting cohort |
E_T | Mean survival under the fitted exponential model, extrapolated for life | 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
Exponential fit to bathtub follow-up in three cells
With named cells Deaths, PersonYears and Time, the first formula returns the fitted hazard in a cell named FittedHazard, and the other two return survival at Time and the implied mean survival.
=Deaths/PersonYears; =EXP(-FittedHazard*Time); =1/FittedHazard
Assumptions
Complete follow-up to the end of the bathtub trial period
Every patient is followed to the end of the follow-up period or to death, so D and PY are complete. With censoring, D is the number of observed deaths and PY the total observed time at risk.
Exponential model applied to a non-constant hazard
The formula is the correct fit of a constant hazard, but the bathtub cohort does not have one. The fitted hazard is the person-time weighted average of the hazards seen during follow-up and says nothing about the hazard after it.
Worked examples
Five-year exponential fit and five-year survival
In the article's example, 0.286448 of the cohort die within five years over 4.063613 person-years, giving a fitted hazard of about 0.070491 per year. Five-year survival under the fit is 0.7030 against 0.7136 under the bathtub hazard, and the implied mean survival is 14.1862 years, the article's 14.19.
D = 0.286448; PY = 4.063613; t = 5; lambda_hat = 0.070491; S_t = 0.7030; E_T = 14.1862
Three-month survival under the five-year exponential fit
The same fit predicts three-month survival of 0.9825, against 0.9048 under the bathtub hazard, so it places far too few deaths in the post-procedure period.
D = 0.286448; PY = 4.063613; t = 0.25; lambda_hat = 0.070491; S_t = 0.9825; E_T = 14.1862
Common errors
Accepting a constant hazard because five-year survival matches
A fit that matches the Kaplan-Meier curve at the end of follow-up can still misplace deaths and the tail. In the article's example the exponential gives five-year survival of 0.7030 against 0.7136, yet overstates mean survival by 3.61 life-years (14.19 against 10.58), most of it after year fifteen.
Sources
Exponential model with a constant hazard 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 2.1, which gives the exponential distribution a constant hazard lambda and survivor function exp(-lambda t), and section 2, which equates mean survival with the area under the survival curve.
Event rate as events divided by time at risk for a bathtub fit
Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. Section on converting to the cycle length, which defines a rate as the number of events divided by the total time at risk.
Canonical Identity
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