Concept Architecture
Concept
Theoretically, the Weibull Model is a parametric survival model that assumes survival times follow a Weibull distribution. It is one of the most widely used survival models because it can represent increasing, decreasing and constant hazard rates through a single shape parameter. In health economics, the Weibull model is extensively used for survival extrapolation, treatment-effect estimation and long-term economic evaluation.
Mathematically, the Weibull model specifies the survival, hazard and cumulative hazard functions using the Weibull distribution. Covariates may be incorporated through accelerated failure time or proportional hazards parameterisations, with model parameters estimated using maximum likelihood estimation. The flexibility of the Weibull model arises from the shape parameter, which determines whether hazards increase, decrease or remain constant over time.
In practice, the Weibull model is fitted to censored patient-level survival data from clinical trials or observational studies. The estimated model is used to predict survival probabilities, estimate median and mean survival, extrapolate outcomes beyond observed follow-up and generate transition probabilities for decision-analytic models used in health technology assessment.
Purpose
Used to model time-to-event data, estimate survival and hazard functions, extrapolate long-term survival and provide parametric survival estimates for health economic decision models.
Mathematical Formulae
Primary Formula
Survival function:
S(t) = exp[?(t/?)?]
Where:
t = time
? = scale parameter
k = shape parameter
Supporting Formulae
Hazard function:
h(t) = (k/?)(t/?)???
Cumulative hazard:
H(t) = (t/?)?
Probability density function:
f(t) = (k/?)(t/?)??? exp[?(t/?)?]
Accelerated failure time model:
ln(T) = X? + �W
Where:
X = covariate matrix
? = regression coefficients
� = scale parameter
W = extreme value error term
Related Mathematical Methods
- Maximum likelihood estimation
- Weibull distribution
- Parametric survival modelling
- Accelerated failure time modelling
- Cox proportional hazards regression
- Survival extrapolation
Example
A health technology assessment evaluates a new oncology therapy using patient-level trial data. A Weibull model estimates a shape parameter of 1.35 and a scale parameter of 28 months, indicating that mortality increases over time. The fitted model is extrapolated beyond the trial period to estimate lifetime survival and calculate quality-adjusted life-years for cost-effectiveness analysis.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| WEIBULL.DIST | =WEIBULL.DIST(A2,Shape,Scale,FALSE) | Calculates Weibull probability density values. |
| WEIBULL.DIST | =WEIBULL.DIST(A2,Shape,Scale,TRUE) | Calculates cumulative Weibull probabilities. |
| EXP | =EXP(-(A2/Scale)^Shape) | Calculates Weibull survival probabilities. |
| GAMMA | =Scale*GAMMA(1+1/Shape) | Calculates expected survival time. |
VBA (Optional)
VBA can automate Weibull model fitting, parameter estimation, survival prediction and long-term extrapolation for health economic analyses.
Sources
Weibull W. A statistical distribution function of wide applicability. Journal of Applied Mechanics. 1951;18:293?297.
Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Wiley; 2002.
Collett D. Modelling Survival Data in Medical Research. 3rd ed. Chapman & Hall/CRC; 2015.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
NICE. Health Technology Evaluations: The Manual. National Institute for Health and Care Excellence; 2022.
Related Concepts (2)
Library
Tools & Resources
1
survHE — Survival Analysis for Health Economic Evaluation (R package) — Gianluca Baio, R package ed., 2023 (CRAN)
An R package for fitting and comparing parametric survival models for health economic evaluation, including Bayesian estimation, and for extrapolating time-to-event data to inform cost-effectiveness models.
Software (R package)View source →
Frequently Asked Questions (7)
What is a weibull model?
A survival model based on the Weibull distribution, representing an increasing or decreasing hazard over time, though restricted to a monotonic pattern.
Source: Weibull 1951
Why is the Weibull model a common first choice for survival extrapolation?
The Weibull model is often tried first because it adds just enough flexibility to the exponential to be useful while staying simple and well behaved. Its single shape parameter lets the hazard rise or fall steadily, which fits many diseases better than a constant risk, and its parameters are usually estimated reliably even from modest data. It also extrapolates in a stable, interpretable way. Its limitation is that the hazard can only move in one direction, so it cannot follow risk that turns. Latimer (2013) notes its common use.
Source: Latimer 2013
What is a Weibull model?
A Weibull model is a survival model based on the Weibull distribution, representing a hazard that increases or decreases over time, though restricted to a monotonic pattern, according to its shape parameter. It generalises the exponential model, which has a constant hazard, by allowing the hazard to rise or fall. Fitted to survival data, the Weibull model provides a smooth survival curve for description and extrapolation, and it is one of the most common candidate parametric survival models.
Source: Weibull 1951
What hazard pattern does the Weibull model capture?
The Weibull model captures a monotonic hazard: increasing over time when the shape parameter exceeds one, decreasing when it is below one, or constant when equal to one, reducing to the exponential model. It cannot represent hazards that turn, such as unimodal or bathtub shapes. This lets it fit data where risk rises or falls steadily, but not where the hazard peaks or reverses, so its use depends on the hazard being monotonic over the modelled period.
Source: Weibull 1951
When is a Weibull model used?
A Weibull model is used when survival data show a monotonically increasing or decreasing hazard, and where it fits better than the constant-hazard exponential or than distributions with turning hazards. It is a common candidate in survival modelling and extrapolation, chosen where a monotonic hazard is plausible for both the observed and projected periods. Its fit within the data and the plausibility of its extrapolated monotonic hazard are compared with other distributions before it is selected.
Source: Latimer 2013
How does the Weibull model behave in extrapolation?
In extrapolation, the Weibull model projects its monotonic hazard indefinitely: an increasing hazard keeps rising, giving accelerating mortality and declining survival, while a decreasing hazard keeps falling, giving long survival. This behaviour suits data with a genuinely monotonic hazard but may be implausible if projected far beyond the observed period, where the hazard could change. Because the extrapolated tail affects mean survival, the Weibull model's projected hazard is examined for plausibility and compared with alternatives before its extrapolation is relied upon.
Source: Latimer 2013
What are the limitations of the Weibull model?
The Weibull model's hazard is monotonic, so it cannot capture hazards that turn, such as unimodal or bathtub patterns, limiting its fit where the hazard is not monotonic, and its projected hazard is extended indefinitely in extrapolation, which may be implausible far beyond the data. Its extrapolation depends on the assumed monotonic form and is uncertain. As one of several candidate distributions, it may fit worse than alternatives for particular data. These limitations mean it is used where a monotonic hazard is appropriate and compared with other models.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Oct 2025
Content version: 1.0.0
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- Persistent URI
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- Term code
- HE-EM-SM-091
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