Force of mortality

In actuarial science and demography, force of mortality, also known as death intensity,¹² is a function, usually written ${\displaystyle \mu (x)}$, that gives the instantaneous rate at which deaths occur at age x, conditional on survival to age x.³ In survival analysis it corresponds to the hazard function, and in reliability theory it corresponds to the failure rate.⁴⁵ It has units of inverse time, and integrating it over an interval gives the survival probability over that interval.⁴

Let ${\displaystyle X}$ be a non-negative random variable representing an individual's age at death (or lifetime). Write ${\displaystyle F(x)=\Pr(X\leq x)}$ for its cumulative distribution function and ${\displaystyle S(x)=\Pr(X>x)}$ for its survival function.⁴

The force of mortality at age ${\displaystyle x}$, written ${\displaystyle \mu (x)}$, is defined as the instantaneous conditional rate of death at age ${\displaystyle x}$. Formally, it is the limit of the conditional probability of dying in a short interval after ${\displaystyle x}$, divided by the interval length³: $${\displaystyle \mu (x)=\lim _{\Delta x\to 0^{+}}{\frac {\Pr(x<X\leq x+\Delta x\mid X>x)}{\Delta x}}.}$$

When ${\displaystyle X}$ is continuous with probability density function ${\displaystyle f(x)}$, the force of mortality can be written in terms of ${\displaystyle f}$ and ${\displaystyle S}$ as⁴ $${\displaystyle \mu (x)={\frac {f(x)}{S(x)}}={\frac {f(x)}{1-F(x)}}.}$$

Equivalently, where ${\displaystyle S}$ is differentiable, it is the negative derivative of the log-survival function⁴: $${\displaystyle \mu (x)=-{\frac {\mathrm {d} }{\mathrm {d} x}}\ln S(x).}$$

The force of mortality ${\displaystyle \mu (x)}$ is an instantaneous rate rather than a probability. For a short interval ${\displaystyle \Delta x}$, the conditional probability of dying shortly after age ${\displaystyle x}$ is approximately ${\displaystyle \mu (x)\,\Delta x}$, provided ${\displaystyle \Delta x}$ is small enough that the rate does not change much over the interval.³

In survival analysis, ${\displaystyle \mu (x)}$ is the hazard function.⁴ In reliability theory, the same mathematical object is commonly called the failure rate.⁵

The cumulative force of mortality (also called the cumulative hazard) is the integral of the force over age. Writing $${\displaystyle H(x)=\int _{0}^{x}\mu (u)\,\mathrm {d} u}$$ then the survival function can be expressed as⁴ $${\displaystyle S(x)=\exp \!{\bigl (}-H(x){\bigr )}.}$$

These identities imply the differential relationship $${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}S(x)=-\mu (x)\,S(x)}$$ and, for a continuous lifetime distribution, the density can be written as⁴ $${\displaystyle f(x)=\mu (x)\,S(x).}$$

In actuarial notation, the probability that a life aged ${\displaystyle x}$ survives for a further ${\displaystyle t}$ years is written ${\displaystyle {}_{t}p_{x}}$. In terms of the lifetime random variable ${\displaystyle X}$, it is³ $${\displaystyle {}_{t}p_{x}=\Pr(X>x+t\mid X>x)={\frac {S(x+t)}{S(x)}}.}$$

Using the force of mortality, this conditional survival probability can be expressed as an exponential of the integrated force³⁴: $${\displaystyle {}_{t}p_{x}=\exp \!\left(-\int _{x}^{x+t}\mu (u)\,\mathrm {d} u\right).}$$

Life tables often tabulate survival and death probabilities at integer ages. In that setting, the one-year survival probability is ${\displaystyle p_{x}={}_{1}p_{x}}$ and the one-year death probability is ${\displaystyle q_{x}=1-p_{x}}$.³ The force of mortality provides a continuous-age description that can be used to relate probabilities over different intervals through the integral relationship above.³

Several parametric models are used to describe how the force of mortality varies with age. A constant force of mortality, ${\displaystyle \mu (x)=\lambda }$ for ${\displaystyle \lambda >0}$, corresponds to an exponential distribution for ${\displaystyle X}$ and gives a memoryless survival pattern.⁴

In actuarial work, the Gompertz–Makeham law of mortality is often written as the sum of an age-independent component and an exponentially increasing component, for example $${\displaystyle \mu (x)=A+B\,c^{x}}$$ with ${\displaystyle A\geq 0}$, ${\displaystyle B>0}$, and ${\displaystyle c>1}$.³ The Gompertz model is the special case ${\displaystyle A=0}$, giving:³ $${\displaystyle \mu (x)=B\,c^{x}}$$

A common model in survival analysis and reliability uses a Weibull hazard, which has the form $${\displaystyle \mu (x)={\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}}$$ for shape ${\displaystyle k>0}$ and scale ${\displaystyle \lambda >0}$. This family includes decreasing, constant, and increasing forces of mortality depending on the value of ${\displaystyle k}$.⁴⁵