The Gompertz–Makeham law states that the human death rate is the sum of an age-dependent component (the
Gompertz function, named after
Benjamin Gompertz),
which
increases exponentially with age,
and an age-independent component (the Makeham term, named after
William Makeham).
In a protected environment where external causes of death are rare (laboratory conditions, low mortality countries, etc.), the age-independent mortality component is often negligible. In this case the formula simplifies to a Gompertz law of mortality. In 1825, Benjamin Gompertz proposed an exponential increase in death rates with age.
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Description
The Gompertz–Makeham law of mortality describes the age dynamics of human mortality rather accurately in the age window from about 30 to 80 years of age. At more advanced ages, some studies have found that death rates increase more slowly – a phenomenon known as the
late-life mortality deceleration – but more recent studies disagree.
The decline in the human
mortality rate
Mortality rate, or death rate, is a measure of the number of deaths (in general, or due to a specific cause) in a particular Statistical population, population, scaled to the size of that population, per unit of time. Mortality rate is typically ...
before the 1950s was mostly due to a decrease in the age-independent (Makeham) mortality component, while the age-dependent (Gompertz) mortality component was surprisingly stable.
Since the 1950s, a new mortality trend has started in the form of an unexpected decline in mortality rates at advanced ages and "rectangularization" of the survival curve.
The
hazard function
A hazard is a potential source of harm. Substances, events, or circumstances can constitute hazards when their nature would potentially allow them to cause damage to health, life, property, or any other interest of value. The probability of that ...
for the Gompertz-Makeham distribution is most often characterised as
. The empirical magnitude of the beta-parameter is about .085, implying a doubling of mortality every .69/.085 = 8 years (Denmark, 2006).
The
quantile function
In probability and statistics, the quantile function is a function Q: ,1\mapsto \mathbb which maps some probability x \in ,1/math> of a random variable v to the value of the variable y such that P(v\leq y) = x according to its probability distr ...
can be expressed in a
closed-form expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...
using the
Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...
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