Concept Architecture
Concept
Theoretically, the Gompertz Model is a fully parametric survival model that assumes survival times follow a Gompertz distribution. It is characterised by a hazard function that changes exponentially over time, making it particularly suitable for modelling mortality processes in which risk increases with age. In health economics, the Gompertz model is widely used to estimate survival, extrapolate long-term outcomes and inform cost-effectiveness analyses and health technology assessments.
Mathematically, the Gompertz model specifies the hazard function using scale and shape parameters that determine both the initial hazard and its rate of exponential change over time. These parameters are estimated using maximum likelihood estimation, from which the cumulative hazard and survival functions are derived. The model provides closed-form survival predictions and is commonly compared with alternative parametric survival models during model selection.
In practice, the Gompertz model is fitted to individual patient survival data using specialised statistical software. Model adequacy is evaluated using goodness-of-fit statistics, visual comparison with Kaplan-Meier estimates and clinical plausibility of long-term extrapolations. The selected model is subsequently used to estimate life expectancy, quality-adjusted life-years, healthcare costs and incremental cost-effectiveness ratios.
Purpose
Used to model survival data with exponentially changing hazards, extrapolate long-term survival and estimate health and economic outcomes for decision modelling and health technology assessment.
Mathematical Formulae
Primary Formula
h(t) = ?e??
where:
- ? > 0 = scale parameter
- ? = shape parameter
Supporting Formulae
Cumulative hazard:
H(t) = (? / ?)(e?? ? 1),?? ? 0
Survival function:
S(t) = exp(?H(t))
Maximum likelihood estimation:
?? = arg max L(?)
Related Mathematical Methods
- Maximum likelihood estimation
- Parametric survival modelling
- Gompertz distribution
- Weibull model
- Exponential model
- Kaplan-Meier estimator
- Akaike Information Criterion (AIC)
- Bayesian Information Criterion (BIC)
Example
A health technology assessment evaluates a new treatment for chronic heart failure using six years of clinical trial data. Because mortality increases with age, a Gompertz model is selected to extrapolate survival beyond the trial period. The projected survival estimates are then used to calculate lifetime quality-adjusted life-years, healthcare costs and the incremental cost-effectiveness ratio.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(Gamma*A2) | Calculate the exponential change in hazard over time. |
| EXP | =EXP(-(Lambda/Gamma)*(EXP(Gamma*A2)-1)) | Calculate predicted survival probabilities from the fitted Gompertz model. |
| LN | =LN(A2) | Calculate log survival times during exploratory model assessment. |
| Solver | Minimise negative log-likelihood | Estimate Gompertz model parameters by maximum likelihood. |
VBA (Optional)
Automate fitting of Gompertz survival models, compare alternative parametric models and generate long-term survival projections for economic evaluation.
Sources
- Gompertz B. On the Nature of the Function Expressive of the Law of Human Mortality.
- Lawless JF. Statistical Models and Methods for Lifetime Data.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 15: Cost-effectiveness modelling using patient-level simulation — Davis, Stevenson, Tappenden & Wailoo, TSD 15 ed., 2014 (NICE Decision Support Unit (University of Sheffield))
Guidance on individual patient-level (microsimulation) cost-effectiveness modelling — when to use it in preference to cohort models, how to structure it, and how to handle the associated computational and uncertainty challenges.
Frequently Asked Questions (6)
What is a Gompertz model?
A survival model based on the Gompertz distribution, assuming the hazard rate rises exponentially over time, a pattern common in age-related mortality.
Source: Gompertz 1825
Why can the Gompertz model produce extreme long-term projections?
Because the Gompertz model makes the hazard rise or fall exponentially with time, small differences in its estimated shape parameter can produce very different long-term behaviour once extrapolated. A hazard estimated to decline exponentially can imply implausibly long survival, since risk shrinks toward nothing, while a rising hazard can drive survival down very fast. This sensitivity means its extrapolated tail must be checked carefully against what is clinically credible. The exponential change compounds over time. Latimer (2013) notes this behaviour.
Source: Latimer 2013
What does the Gompertz model assume about the hazard?
The Gompertz model assumes the hazard rises exponentially with time, increasing by a constant proportion per unit time, so the risk of the event accelerates. This gives survival that falls slowly at first and then rapidly as the hazard grows. The assumption of an exponentially increasing hazard defines the model and makes it suitable for age-related mortality, but it means the model cannot represent hazards that are constant, decreasing, or non-monotonic, so its use depends on the hazard genuinely rising in this way.
Source: Gompertz 1825
When is a Gompertz model used?
A Gompertz model is used when the hazard is expected to rise exponentially over time, most notably for mortality that increases with age, where its assumption matches the observed acceleration of risk. It is a candidate in survival modelling and extrapolation, chosen where its exponentially rising hazard fits the data and gives a plausible projection. Because its hazard can become very steep over long horizons, its extrapolated survival is scrutinised, and it is compared with other distributions before being selected.
Source: Latimer 2013
How does the Gompertz model behave in extrapolation?
In extrapolation, the Gompertz model projects a hazard that continues rising exponentially, so survival declines increasingly steeply and can fall rapidly at long times. This suits age-related mortality but may overstate mortality if the exponential rise is projected too far, giving shorter survival than plausible. Because the extrapolated tail strongly affects mean survival, the Gompertz model's steep projected hazard is examined for plausibility, and its long-term behaviour compared with alternatives and with background mortality, before relying on its extrapolation.
Source: Latimer 2013
What are the limitations of the Gompertz model?
The Gompertz model's exponentially rising hazard is monotonic, so it cannot represent constant, decreasing, or turning hazards, and its steep rise can overstate mortality when extrapolated far beyond the data. It is appropriate only where the hazard genuinely accelerates, such as age-related mortality, and inappropriate otherwise. As with any parametric model, its extrapolation depends on the assumed form and is uncertain. These limitations mean the Gompertz model is used where an exponentially increasing hazard is plausible and compared with other distributions.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 21 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/gompertz-model
- Term code
- HE-EM-SM-032
Stable URI · Machine-readable · Resolvable · CC BY 4.0