Concept Architecture
Concept
Theoretically, the Gompertz Distribution is a continuous probability distribution commonly used to model time-to-event data in which the hazard changes exponentially over time. It was originally developed to describe human mortality and has become a standard distribution in survival analysis because it captures age-dependent increases or decreases in risk. In health economics, the Gompertz distribution is frequently used to model survival, disease progression and long-term mortality for cost-effectiveness analyses and health technology assessments.
Mathematically, the Gompertz distribution is defined by scale and shape parameters that determine the exponential change in the hazard function over time. Unlike the exponential distribution, which assumes a constant hazard, the Gompertz distribution permits hazards that increase or decrease exponentially. Closed-form expressions exist for the hazard, cumulative hazard and survival functions, facilitating likelihood-based estimation and long-term extrapolation.
In practice, the Gompertz distribution is fitted to survival data using maximum likelihood estimation. It is routinely compared with Weibull, exponential, log-normal, log-logistic and generalised gamma distributions when selecting an appropriate survival model. The fitted distribution is subsequently used to estimate survival probabilities, life expectancy, quality-adjusted life-years and healthcare costs within health economic models.
Purpose
Used to model survival data with exponentially changing hazards, estimate long-term mortality and extrapolate survival outcomes for health economic evaluation.
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))
Mean model parameters are estimated using:
?? = arg max L(?)
Related Mathematical Methods
- Maximum likelihood estimation
- Parametric survival modelling
- Gompertz model
- Weibull distribution
- Exponential distribution
- Survival analysis
- Akaike Information Criterion (AIC)
- Bayesian Information Criterion (BIC)
Example
An economic evaluation of a cardiovascular intervention requires lifetime survival estimates beyond a six-year clinical trial. A Gompertz distribution is fitted to the observed survival data, reflecting the increasing mortality risk associated with ageing. The projected survival curve is then used to estimate lifetime quality-adjusted life-years and healthcare costs.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B1*A2) | Calculate the exponential hazard at time t. |
| EXP | =EXP(-(Lambda/Gamma)*(EXP(Gamma*A2)-1)) | Calculate the Gompertz survival probability. |
| LN | =LN(A2) | Calculate log survival times during exploratory analysis. |
| Solver | Minimise negative log-likelihood | Estimate Gompertz distribution parameters by maximum likelihood. |
VBA (Optional)
Automate maximum likelihood estimation of Gompertz distribution parameters and generate projected survival curves for health economic models.
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 14: Survival analysis for economic evaluations alongside clinical trials – extrapolation with patient-level data — Nicholas R. Latimer, TSD 14 ed., 2013 (NICE Decision Support Unit (University of Sheffield))
The reference guidance on survival analysis for economic evaluation: fitting standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal) to censored trial data and extrapolating to estimate lifetime survival benefit, with a process guide for model selection and justification.
Frequently Asked Questions (6)
What is the Gompertz distribution?
A probability distribution characterised by a hazard rate rising exponentially with time, originally developed to describe human mortality increasing with age.
Source: Gompertz 1825
Why was the Gompertz distribution originally created?
The distribution is named after Benjamin Gompertz, an actuary who in the early nineteenth century observed that adult human mortality rises at a roughly constant proportional rate with age, so that the risk of death multiplies over equal intervals. He captured this with a hazard that increases exponentially over time, which is the distribution's defining feature. Its origin in describing ageing mortality is why it fits age-related death well. Gompertz (1825) set out the original law.
Source: Gompertz 1825
What is the key feature of the Gompertz distribution?
The key feature of the Gompertz distribution is its exponentially increasing hazard: the instantaneous risk of the event rises by a constant proportion per unit time, so the hazard accelerates over time. This produces survival that falls slowly at first and then rapidly as the hazard grows. The exponential rise in hazard distinguishes the Gompertz from distributions with constant or more slowly changing hazards, and it is what makes the Gompertz well suited to age-related mortality, which increases in a similar exponential manner.
Source: Gompertz 1825
Why does the Gompertz distribution suit human mortality?
The Gompertz distribution suits human mortality because, over much of adult life, the risk of death rises approximately exponentially with age, doubling over roughly regular intervals, which is exactly the pattern the Gompertz hazard describes. Gompertz observed this regularity and formulated the distribution to represent it. Consequently, the Gompertz law captures the acceleration of mortality with age well, making the distribution a standard model for adult human mortality and for actuarial and demographic work on ageing populations.
Source: Collett 2015
How is the Gompertz distribution used in survival analysis?
In survival analysis, the Gompertz distribution is used to model time-to-event data where the hazard increases exponentially with time, such as mortality that accelerates with age. It can be fitted to survival data and used to describe and extrapolate survival, and it is a candidate distribution in survival modelling. Its exponentially rising hazard is appropriate for age-related mortality, but because this can imply very steep hazards over long horizons, its extrapolated behaviour is examined, and it is compared with other distributions.
Source: Latimer 2013
What are the limitations of the Gompertz distribution?
The Gompertz distribution's exponentially increasing hazard can rise very steeply over long horizons, which may overstate mortality when extrapolated far beyond the data, and its hazard is monotonic, so it cannot represent hazards that fall or turn. It fits accelerating mortality well but is inappropriate where the hazard is constant, decreasing, or non-monotonic. As with any parametric model, its extrapolation depends on the assumed form. These limitations mean the Gompertz is used where an exponentially rising hazard is plausible and compared with alternatives.
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
- Term code
- HE-EM-SM-031
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