Signature
h_all = h_bg + h_exc
| Inputs | Definition | Unit |
|---|---|---|
h_bg | Hazard of death in the general population of the same age, sex and calendar year | events per person-year |
h_exc | Hazard of death above the background hazard, attributed to the disease or procedure | events per person-year |
h_all | All-cause hazard of death at a given time and attained age | events per person-year |
|---|
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
All-cause hazard from background and excess cells
With the life-table hazard for the attained age in BackgroundHazard and the modelled excess hazard in ExcessHazard, the cell returns the all-cause hazard. A lookup on attained age usually fills BackgroundHazard.
=BackgroundHazard+ExcessHazard
Assumptions
Background hazard fixed from population rates
The background hazard is taken from national mortality rates by age, sex and calendar year and treated as fixed, with no uncertainty. It must be evaluated at the attained age of the cohort at each time.
Excess hazard of zero or above in a bathtub model
The excess hazard is not negative, so the all-cause hazard never falls below population mortality at the same age, which TSD 21 regards as expected in nearly all cases.
Worked examples
Excess hazard dominating the middle of the bathtub
Illustrative figures: a few years after the procedure the background hazard is 0.010 per year and the excess hazard 0.040, giving an all-cause hazard of 0.050.
h_bg = 0.010; h_exc = 0.040; h_all = 0.050
Background hazard driving the late arm of the bathtub
Illustrative figures: decades later the excess hazard has fallen to 0.020 per year while the background hazard has risen with age to 0.060, so the all-cause hazard rises to 0.080.
h_bg = 0.060; h_exc = 0.020; h_all = 0.080
Common errors
General population rates for a cohort with more comorbidity
Taking the background hazard from national life tables when patients carry more comorbidity than the general population understates expected mortality, and the late arm of the bathtub then rises too slowly.
Excess hazard extrapolated below zero
A fitted excess hazard that keeps falling can turn negative in the extrapolated period, pulling the all-cause hazard below population mortality and inflating survival in the tail.
Sources
Excess mortality and relative survival equations in DSU TSD 21
Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020 (updated March 2022). Section 3.7.2, equations 13 and 14, which split all-cause mortality into background and excess mortality and define relative survival as all-cause over expected survival; section 6.2 on disease mortality not falling below the general population; and section 7.1, point III, on national rates by age, sex and calendar year understating mortality when patients have more comorbidity.
Canonical Identity
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