Misplaced Pages

Gompertz–Makeham law of mortality: Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editContent deleted Content addedVisualWikitext
Revision as of 08:31, 6 April 2023 editLumos3 (talk | contribs)Autopatrolled, Extended confirmed users29,319 edits Future of human longevityTag: Visual edit← Previous edit Latest revision as of 17:48, 26 November 2024 edit undoOzzie10aaaa (talk | contribs)Autopatrolled, Extended confirmed users, New page reviewers213,670 edits Description 
(7 intermediate revisions by 6 users not shown)
Line 6: Line 6:
| cdf_image = | cdf_image =
| notation = | notation =
| parameters = <math>\alpha \in \mathbb{R}^+</math><br/><math>\beta \in \mathbb{R}^+</math><br/> <math>\lambda \in \mathbb{R}^+</math> | parameters = <math>\alpha \in \mathbb{R}^+</math><br /><math>\beta \in \mathbb{R}^+</math><br /> <math>\lambda \in \mathbb{R}^+</math>
| support = <math>x \in \mathbb{R}^+</math> | support = <math>x \in \mathbb{R}^+</math>
| pdf = <math>\left( \alpha e^{\beta x} + \lambda \right) \cdot \exp \left</math> | pdf = <math>\left( \alpha e^{\beta x} + \lambda \right) \cdot \exp \left</math>
| cdf = <math>1-\exp \left</math> | cdf = <math>1-\exp \left</math>
| mean = | mean =
Line 23: Line 23:
}} }}


The '''Gompertz–Makeham law''' states that the human death rate is the sum of an age-dependent component (the ], named after ]),<ref name="Gompertz1825">{{cite journal |last=Gompertz |first=B. |year=1825 |title=On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies |journal=] |volume=115 |pages=513–585 |url=http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-55920 |doi=10.1098/rstl.1825.0026|jstor=107756|s2cid=145157003 }}</ref> which ] with age<ref name="Leonid">{{citation |last1=Gavrilov|first1=Leonid A.|last2=Gavrilova|first2=Natalia S.|year=1991 |title=The Biology of Life Span: A Quantitative Approach. |publisher=Harwood Academic Publisher |location=New York|isbn=3-7186-4983-7}}</ref> and an age-independent component (the Makeham term, named after ]).<ref name="Makeham1860">{{cite journal|last=Makeham|first=W. M.|year=1860|title=On the Law of Mortality and the Construction of Annuity Tables|url=https://archive.org/details/jstor-41134925|journal=J. Inst. Actuaries and Assur. Mag.|volume=8|issue=6|pages=301–310|doi=10.1017/S204616580000126X|jstor=41134925}}</ref> 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''' states that the human death rate is the sum of an age-dependent component (the ], named after ]),<ref name="Gompertz1825">{{cite journal |last=Gompertz |first=B. |year=1825 |title=On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies |journal=] |volume=115 |pages=513–585 |url=http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-55920 |doi=10.1098/rstl.1825.0026|jstor=107756|s2cid=145157003 |doi-access=free }}</ref> which ] with age<ref name="Leonid">{{citation |last1=Gavrilov|first1=Leonid A.|last2=Gavrilova|first2=Natalia S.|year=1991 |title=The Biology of Life Span: A Quantitative Approach. |publisher=Harwood Academic Publisher |location=New York|isbn=3-7186-4983-7}}</ref> and an age-independent component (the Makeham term, named after ]).<ref name="Makeham1860">{{cite journal|last=Makeham|first=W. M.|year=1860|title=On the Law of Mortality and the Construction of Annuity Tables|url=https://archive.org/details/jstor-41134925|journal=J. Inst. Actuaries and Assur. Mag.|volume=8|issue=6|pages=301–310|doi=10.1017/S204616580000126X|jstor=41134925}}</ref> 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.
__TOC__

==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 ]<ref name="Leonid" /> – but more recent studies disagree.<ref>{{cite journal|last1=Gavrilov|first1=Leonid A.|last2=Gavrilova|first2=Natalia S.|title=Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File|journal=North American Actuarial Journal|date=2011|volume=15|issue=3|pages=432–447|url=http://longevity-science.org/pdf/Mortality-NAAJ-2011.pdf|doi=10.1080/10920277.2011.10597629|pmid=22308064|pmc=3269912}}</ref> 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 ]<ref name="Leonid" /> – but more recent studies disagree.<ref>{{cite journal|last1=Gavrilov|first1=Leonid A.|last2=Gavrilova|first2=Natalia S.|title=Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File|journal=North American Actuarial Journal|date=2011|volume=15|issue=3|pages=432–447|url=http://longevity-science.org/pdf/Mortality-NAAJ-2011.pdf|doi=10.1080/10920277.2011.10597629|pmid=22308064|pmc=3269912}}</ref>


. Mortality rates increase exponentially with age after age 30.]] . Mortality rates increase exponentially with age after age 30.]]


The decline in the human ] 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.<ref name="Leonid" /><ref>{{cite journal |last1=Gavrilov |first1=L. A. |last2=Gavrilova |first2=N. S. |last3=Nosov |first3=V. N. |year=1983 |title=Human life span stopped increasing: Why? |journal=] |volume=29 |issue=3 |pages=176–180 |doi=10.1159/000213111 |pmid=6852544 }}</ref> 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.<ref name="Gavrilov1985">{{cite journal |last=Gavrilov |first=L. A. |author2=Nosov, V. N. |year=1985 |title=A new trend in human mortality decline: derectangularization of the survival curve |journal=Age |volume=8 |issue=3 |pages=93|doi=10.1007/BF02432075|s2cid=41318801 }}</ref><ref>{{cite journal |last1=Gavrilova |first1=N. S. |last2=Gavrilov |first2=L. A. |year=2011 |trans-title=Ageing and Longevity: Mortality Laws and Mortality Forecasts for Ageing Populations |language=cs |title=Stárnutí a dlouhovekost: Zákony a prognózy úmrtnosti pro stárnoucí populace |journal=Demografie |volume=53 |issue=2 |pages=109–128 }}</ref> The decline in the human ] 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.<ref name="Leonid" /><ref>{{cite journal |last1=Gavrilov |first1=L. A. |last2=Gavrilova |first2=N. S. |last3=Nosov |first3=V. N. |year=1983 |title=Human life span stopped increasing: Why? |journal=] |volume=29 |issue=3 |pages=176–180 |doi=10.1159/000213111 |pmid=6852544 }}</ref> 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.<ref name="Gavrilov1985">{{cite journal |last=Gavrilov |first=L. A. |author2=Nosov, V. N. |year=1985 |title=A new trend in human mortality decline: derectangularization of the survival curve |journal=Age |volume=8 |issue=3 |pages=93|doi=10.1007/BF02432075|s2cid=41318801 }}</ref><ref>{{cite journal |last1=Gavrilova |first1=N. S. |last2=Gavrilov |first2=L. A. |year=2011 |trans-title=Ageing and Longevity: Mortality Laws and Mortality Forecasts for Ageing Populations |language=cs |title=Stárnutí a dlouhovekost: Zákony a prognózy úmrtnosti pro stárnoucí populace |journal=Demografie |volume=53 |issue=2 |pages=109–128 |pmid=25242821 |pmc=4167024 }}</ref>


The ] for the Gompertz-Makeham distribution is most often characterised as <math>h(x)=\alpha e^{\beta x} + \lambda </math>. The empirical magnitude of the beta-parameter is about .085, implying a doubling of mortality every .69/.085 = 8 years (Denmark, 2006). The ] for the Gompertz-Makeham distribution is most often characterised as <math>h(x)=\alpha e^{\beta x} + \lambda </math>. The empirical magnitude of the beta-parameter is about .085, implying a doubling of mortality every .69/.085 = 8 years (Denmark, 2006).


The ] can be expressed in a ] using the ]:<ref name="Jodra2009">{{cite journal |last=Jodrá |first=P. |year=2009 |title=A closed-form expression for the quantile function of the Gompertz–Makeham distribution |journal=Mathematics and Computers in Simulation |volume=79 |issue= 10|pages=3069–3075 |doi=10.1016/j.matcom.2009.02.002}}</ref> The ] can be expressed in a ] using the ]:<ref name="Jodra2009">{{cite journal |last=Jodrá |first=P. |year=2009 |title=A closed-form expression for the quantile function of the Gompertz–Makeham distribution |journal=Mathematics and Computers in Simulation |volume=79 |issue= 10|pages=3069–3075 |doi=10.1016/j.matcom.2009.02.002}}</ref>
Line 38: Line 39:


The Gompertz law is the same as a ] for the negative of age, restricted to negative values for the ] (positive values for age). The Gompertz law is the same as a ] for the negative of age, restricted to negative values for the ] (positive values for age).

== Future of human longevity ==
The birth year cohorts of those born after 1950 should be the first to see a considerable delay in the historical course of death, according to 2023 research paper that draws this conclusion solely from maths. The study predicts a future in which longevity records will frequently be broken after 2073, with some prediction graphs reaching into the 140s. <ref>{{Cite web |last=Jackson |first=Justin |last2=Xpress |first2=Medical |title=Lifespans into the 140s predicted by centuries-old Gompertz law |url=https://medicalxpress.com/news/2023-04-lifespans-140s-centuries-old-gompertz-law.html |access-date=2023-04-06 |website=medicalxpress.com |language=en}}</ref>


==See also== ==See also==
* ] * ]
*] * ]
* ] * ]
* ] * ]
Line 53: Line 51:


==References== ==References==
<references/> <references />
{{ProbDistributions|continuous-semi-infinite}} {{ProbDistributions|continuous-semi-infinite}}
{{DEFAULTSORT:Gompertz-Makeham Law Of Mortality}} {{DEFAULTSORT:Gompertz-Makeham Law Of Mortality}}

Latest revision as of 17:48, 26 November 2024

Mathematical equation related to human death rate
Gompertz–Makeham
Parameters α R + {\displaystyle \alpha \in \mathbb {R} ^{+}}
β R + {\displaystyle \beta \in \mathbb {R} ^{+}}
λ R + {\displaystyle \lambda \in \mathbb {R} ^{+}}
Support x R + {\displaystyle x\in \mathbb {R} ^{+}}
PDF ( α e β x + λ ) exp [ λ x α β ( e β x 1 ) ] {\displaystyle \left(\alpha e^{\beta x}+\lambda \right)\cdot \exp \left}
CDF 1 exp [ λ x α β ( e β x 1 ) ] {\displaystyle 1-\exp \left}

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.

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.

Estimated probability of a person dying at each age, for the U.S. in 2003 . Mortality rates increase exponentially with age after age 30.

The decline in the human mortality rate 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 for the Gompertz-Makeham distribution is most often characterised as h ( x ) = α e β x + λ {\displaystyle h(x)=\alpha e^{\beta x}+\lambda } . 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 can be expressed in a closed-form expression using the Lambert W function:

Q ( u ) = α β λ 1 λ ln ( 1 u ) 1 β W 0 [ α e α / λ ( 1 u ) ( β / λ ) λ ] {\displaystyle Q(u)={\frac {\alpha }{\beta \lambda }}-{\frac {1}{\lambda }}\ln(1-u)-{\frac {1}{\beta }}W_{0}\left}

The Gompertz law is the same as a Fisher–Tippett distribution for the negative of age, restricted to negative values for the random variable (positive values for age).

See also

References

  1. Gompertz, B. (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society. 115: 513–585. doi:10.1098/rstl.1825.0026. JSTOR 107756. S2CID 145157003.
  2. ^ Gavrilov, Leonid A.; Gavrilova, Natalia S. (1991), The Biology of Life Span: A Quantitative Approach., New York: Harwood Academic Publisher, ISBN 3-7186-4983-7
  3. Makeham, W. M. (1860). "On the Law of Mortality and the Construction of Annuity Tables". J. Inst. Actuaries and Assur. Mag. 8 (6): 301–310. doi:10.1017/S204616580000126X. JSTOR 41134925.
  4. Gavrilov, Leonid A.; Gavrilova, Natalia S. (2011). "Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File" (PDF). North American Actuarial Journal. 15 (3): 432–447. doi:10.1080/10920277.2011.10597629. PMC 3269912. PMID 22308064.
  5. Gavrilov, L. A.; Gavrilova, N. S.; Nosov, V. N. (1983). "Human life span stopped increasing: Why?". Gerontology. 29 (3): 176–180. doi:10.1159/000213111. PMID 6852544.
  6. Gavrilov, L. A.; Nosov, V. N. (1985). "A new trend in human mortality decline: derectangularization of the survival curve ". Age. 8 (3): 93. doi:10.1007/BF02432075. S2CID 41318801.
  7. Gavrilova, N. S.; Gavrilov, L. A. (2011). "Stárnutí a dlouhovekost: Zákony a prognózy úmrtnosti pro stárnoucí populace" [Ageing and Longevity: Mortality Laws and Mortality Forecasts for Ageing Populations]. Demografie (in Czech). 53 (2): 109–128. PMC 4167024. PMID 25242821.
  8. Jodrá, P. (2009). "A closed-form expression for the quantile function of the Gompertz–Makeham distribution". Mathematics and Computers in Simulation. 79 (10): 3069–3075. doi:10.1016/j.matcom.2009.02.002.
Probability distributions (list)
Discrete
univariate
with finite
support
with infinite
support
Continuous
univariate
supported on a
bounded interval
supported on a
semi-infinite
interval
supported
on the whole
real line
with support
whose type varies
Mixed
univariate
continuous-
discrete
Multivariate
(joint)
Directional
Univariate (circular) directional
Circular uniform
Univariate von Mises
Wrapped normal
Wrapped Cauchy
Wrapped exponential
Wrapped asymmetric Laplace
Wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
Bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
Categories: