Signature
h_t = lambda_1 * gamma_1 * t^(gamma_1 - 1) + lambda_2 * gamma_2 * t^(gamma_2 - 1); S_t = exp(-(lambda_1 * t^gamma_1 + lambda_2 * t^gamma_2))
| Inputs | Definition | Unit |
|---|---|---|
lambda_1 | Scale parameter of the early component | years to the power minus gamma_1 |
gamma_1 | Shape parameter of the early component; below 1 for a hazard that falls over time | none |
t | Time since the procedure, above zero | years |
lambda_2 | Scale parameter of the late component | years to the power minus gamma_2 |
gamma_2 | Shape parameter of the late component; above 1 for a hazard that rises over time | none |
h_t | Sum of the two Weibull component hazards at time t | events per person-year |
|---|---|---|
S_t | Proportion of the cohort alive at time t under the two-component model | proportion of the starting cohort |
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.
Try this function
Implementations
Excel
Poly-Weibull hazard and survival in two cells
With named cells Lam1, Gam1, Lam2, Gam2 and Time, the two formulas return the total hazard and survival at Time.
=Lam1*Gam1*Time^(Gam1-1)+Lam2*Gam2*Time^(Gam2-1); =EXP(-(Lam1*Time^Gam1+Lam2*Time^Gam2))
Assumptions
Independent additive components of a bathtub hazard
The two components act as independent competing risks whose hazards add, for example an early risk tied to the procedure and a late risk tied to ageing and chronic complications. Each component has its own Weibull form, with its own parameters.
Distinct shapes for the two Weibull components
The bathtub shape needs gamma_1 below 1 and gamma_2 above 1. Demiris and colleagues note that identifiability problems can arise when the shape parameters are equal.
Worked examples
Poly-Weibull hazard three months after the procedure
Illustrative parameters, not taken from a fitted model: lambda_1 of 0.10 with gamma_1 of 0.5, and lambda_2 of 0.0001 with gamma_2 of 3. At three months the falling component dominates and the hazard is about 0.100019 per year, with survival of 0.951228.
lambda_1 = 0.10; gamma_1 = 0.5; lambda_2 = 0.0001; gamma_2 = 3; t = 0.25; h_t = 0.100019; S_t = 0.951228
Poly-Weibull hazard near the floor of the bathtub at five years
With the same illustrative parameters, the hazard at five years has fallen to about 0.029861 per year, near the lowest point of the curve, and survival is 0.789696.
lambda_1 = 0.10; gamma_1 = 0.5; lambda_2 = 0.0001; gamma_2 = 3; t = 5; h_t = 0.029861; S_t = 0.789696
Poly-Weibull hazard in the rising arm at twenty years
At twenty years the rising component dominates and the hazard is about 0.131180 per year, above its value at three months, with survival of 0.287304.
lambda_1 = 0.10; gamma_1 = 0.5; lambda_2 = 0.0001; gamma_2 = 3; t = 20; h_t = 0.131180; S_t = 0.287304
Common errors
Mixing two Weibull distributions to obtain a bathtub hazard
Averaging two Weibull survival curves with mixing weights is a mixture model, not a sum of hazards. Demiris and colleagues cite reliability work showing that a mixture of two Weibull distributions cannot give a bathtub-shaped hazard, whereas a sum of Weibull hazards can.
Sources
Poly-Weibull hazard and survivor function for a bathtub hazard
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, equations 2.1 and 2.2 for the hazard as a sum of Weibull hazards and the survivor function, the identifiability remark on equal shape parameters, and the Introduction on mixtures of two Weibull distributions.
Poly-hazard models 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 3.2.5, equations 5 and 6, which define a poly-hazard model as a sum of hazards and report that Demiris proposed Weibull components to capture a bathtub hazard function.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0