Force of mortality

In actuarial science and demography, force of mortality is a function, usually written $\mu (x)$ , that gives the instantaneous rate at which deaths occur at age x, conditional on survival to age x.^[1] In survival analysis it corresponds to the hazard function, and in reliability theory it corresponds to the failure rate.^[2]^[3] It has units of inverse time, and integrating it over an interval gives the survival probability over that interval.^[2]

Definition

Let $X$ be a non-negative random variable representing an individual's age at death (or lifetime). Write $F(x)=\Pr(X\leq x)$ for its cumulative distribution function and $S(x)=\Pr(X>x)$ for its survival function.^[2]

The force of mortality at age $x$ , written $\mu (x)$ , is defined as the instantaneous conditional rate of death at age $x$ . Formally, it is the limit of the conditional probability of dying in a short interval after $x$ , divided by the interval length^[1]: $\mu (x)=\lim _{\Delta x\to 0^{+}}{\frac {\Pr(x<X\leq x+\Delta x\mid X>x)}{\Delta x}}.$

When $X$ is continuous with probability density function $f(x)$ , the force of mortality can be written in terms of $f$ and $S$ as^[2] $\mu (x)={\frac {f(x)}{S(x)}}={\frac {f(x)}{1-F(x)}}.$

Equivalently, where $S$ is differentiable, it is the negative derivative of the log-survival function^[2]: $\mu (x)=-{\frac {\mathrm {d} }{\mathrm {d} x}}\ln S(x).$

Interpretation and related quantities

The force of mortality $\mu (x)$ is an instantaneous rate rather than a probability. For a short interval $\Delta x$ , the conditional probability of dying shortly after age $x$ is approximately $\mu (x)\,\Delta x$ , provided $\Delta x$ is small enough that the rate does not change much over the interval.^[1]

In survival analysis, $\mu (x)$ is the hazard function.^[2] In reliability theory, the same mathematical object is commonly called the failure rate.^[3]

The cumulative force of mortality (also called the cumulative hazard) is the integral of the force over age. Writing $H(x)=\int _{0}^{x}\mu (u)\,\mathrm {d} u$ then the survival function can be expressed as^[2] $S(x)=\exp \!{\bigl (}-H(x){\bigr )}.$

These identities imply the differential relationship ${\frac {\mathrm {d} }{\mathrm {d} x}}S(x)=-\mu (x)\,S(x)$ and, for a continuous lifetime distribution, the density can be written as^[2] $f(x)=\mu (x)\,S(x).$

Survival probabilities and life tables

In actuarial notation, the probability that a life aged $x$ survives for a further $t$ years is written ${}_{t}p_{x}$ . In terms of the lifetime random variable $X$ , it is^[1] ${}_{t}p_{x}=\Pr(X>x+t\mid X>x)={\frac {S(x+t)}{S(x)}}.$

Using the force of mortality, this conditional survival probability can be expressed as an exponential of the integrated force^[1]^[2]: ${}_{t}p_{x}=\exp \!\left(-\int _{x}^{x+t}\mu (u)\,\mathrm {d} u\right).$

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

Examples of mortality models

Several parametric models are used to describe how the force of mortality varies with age. A constant force of mortality, $\mu (x)=\lambda$ for $\lambda >0$ , corresponds to an exponential distribution for $X$ and gives a memoryless survival pattern.^[2]

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 $\mu (x)=A+B\,c^{x}$ with $A\geq 0$ , $B>0$ , and $c>1$ .^[1] The Gompertz model is the special case $A=0$ , giving:^[1] $\mu (x)=B\,c^{x}$

A common model in survival analysis and reliability uses a Weibull hazard, which has the form $\mu (x)={\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}$ for shape $k>0$ and scale $\lambda >0$ . This family includes decreasing, constant, and increasing forces of mortality depending on the value of $k$ .^[2]^[3]

References

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Dickson, David C. M.; Hardy, Mary R.; Waters, Howard R. (2013). Actuarial Mathematics for Life Contingent Risks (2nd ed.). Cambridge University Press. ISBN 9781107044074.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k Klein, John P.; Moeschberger, Melvin L. (2003). Survival Analysis: Techniques for Censored and Truncated Data (2nd ed.). Springer. ISBN 9780387953991.
^ ^a ^b ^c Rausand, Marvin; Høyland, Arnljot (2004). System Reliability Theory: Models, Statistical Methods, and Applications (2nd ed.). Wiley-Interscience. ISBN 9780471471332.

[rdp-we-cite_note-DicksonHardyWaters-LCR-2013-1] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Dickson, David C. M.; Hardy, Mary R.; Waters, Howard R. (2013). Actuarial Mathematics for Life Contingent Risks (2nd ed.). Cambridge University Press. ISBN 9781107044074.

[rdp-we-cite_note-KleinMoeschberger-Survival-2003-2] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k Klein, John P.; Moeschberger, Melvin L. (2003). Survival Analysis: Techniques for Censored and Truncated Data (2nd ed.). Springer. ISBN 9780387953991.

[rdp-we-cite_note-RausandHoyland-Reliability-2004-3] Rausand, Marvin; Høyland, Arnljot (2004). System Reliability Theory: Models, Statistical Methods, and Applications (2nd ed.). Wiley-Interscience. ISBN 9780471471332.

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