Gompertz-Makeham Law of Mortality
Also known as: Gompertz-Makeham Model, Makeham's Law, Gompertz Law of Mortality, Gompertz-Makeham Ölümlülük Yasası
The Gompertz-Makeham law is the foundational parametric model of adult human mortality. Benjamin Gompertz showed in 1825 that the force of mortality rises exponentially with age, and William Makeham added an age-independent background term in 1860 to account for deaths from causes unrelated to ageing. The combined law expresses the hazard of death as a constant plus an exponentially increasing component, capturing the dominant shape of adult mortality with just three parameters.
Key highlights
- Captures the dominant exponential rise of adult mortality with only two or three interpretable parameters.
- The slope parameter α gives a clean, comparable measure of the rate of biological ageing across populations and species.
- Makeham's background term substantially improves fit at younger adult ages over the pure Gompertz form.
- Yields a closed-form survival function, making it convenient for actuarial annuity and insurance calculations.
Intuition
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How it works
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When to use it
Use the Gompertz-Makeham law to model and smooth adult mortality, to estimate the rate of ageing α for cross-population comparisons, and to provide a compact, biologically interpretable hazard for actuarial valuation of annuities and life insurance. It applies best over the adult age range, roughly from the late twenties to the eighties, where the exponential pattern dominates. The law assumes mortality is the sum of a constant background and an exponentially rising senescent component; it does not represent infant and childhood mortality, the young-adult accident hump, or the deceleration of mortality at the oldest-old ages. For those features a fuller model such as Heligman-Pollard, or a logistic/Kannisto modification at extreme ages, should be used instead.
Strengths & limitations
- Captures the dominant exponential rise of adult mortality with only two or three interpretable parameters.
- The slope parameter α gives a clean, comparable measure of the rate of biological ageing across populations and species.
- Makeham's background term substantially improves fit at younger adult ages over the pure Gompertz form.
- Yields a closed-form survival function, making it convenient for actuarial annuity and insurance calculations.
- Does not describe infant, child, or young-adult mortality, so it covers only the adult portion of the age range.
- Overstates mortality at the oldest-old ages, where empirical hazards decelerate below the exponential and a logistic form fits better.
- Assumes a single, time-invariant exponential ageing process, ignoring cohort effects and heterogeneity in frailty.
- Three parameters are too few to capture the full age pattern of mortality across the whole lifespan.
Common pitfalls
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Applications
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Frequently asked
What does the Makeham term add to the original Gompertz law?
Gompertz's law alone makes the entire hazard grow exponentially with age, which underpredicts mortality at younger adult ages where there is a substantial floor of age-independent risk from accidents, infections, and violence. Makeham's constant term A adds exactly this age-independent background, so the model becomes a constant plus an exponential. The extra parameter typically improves the fit across young and middle adulthood without disturbing the senescent component at older ages.
Why does the Gompertz-Makeham law fail at the oldest ages?
Empirically, mortality rates at the oldest-old ages (roughly past 90 to 100) rise more slowly than the exponential predicts and appear to decelerate or plateau, a pattern often attributed to selection and frailty heterogeneity — the frailest individuals die first, leaving more robust survivors. The pure exponential keeps accelerating and therefore overpredicts mortality there, so logistic modifications such as the Kannisto model are used for the extreme ages.
How is the rate parameter related to the mortality doubling time?
Because the senescent hazard grows as e^(αx), mortality doubles every ln(2)/α years. For adult humans α is roughly 0.08–0.10 per year, implying a mortality doubling time of about seven to ten years. This doubling time is a widely used summary of the pace of ageing and allows direct comparison of senescence across populations and species.
Sources
- 1.Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality. Philosophical Transactions of the Royal Society of London, 115, 513–583.
- 2.Makeham, W. M. (1860). On the law of mortality and the construction of annuity tables. Journal of the Institute of Actuaries, 8(6), 301–310.
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Cite this page
ScholarGate. (2026, June 22). Gompertz-Makeham Law of Mortality. ScholarGate. https://scholargate.app/demography/gompertz-makeham-law