The Gompertz–Makeham law states that the human death rate is the sum of an age-dependent component (the
Gompertz function
The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. Th ...
, named after
Benjamin Gompertz
Benjamin Gompertz (5 March 1779 – 14 July 1865) was a British self-educated mathematician and actuary, who became a Fellow of the Royal Society. Gompertz is now best known for his Gompertz law of mortality, a demographic model published in 1 ...
),
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.
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 population, scaled to the size of that population, per unit of time. Mortality rate is typically expressed in units of de ...
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
Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering.
The failure rate of a ...
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, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equ ...
can be expressed in a
closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
using the
Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential function ...
:
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