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Social conditions and the Gompertz rate of ageing. Jon Anson Yishai Friedlander Deparment of Social Work Ben- Gurion University of the Negev 84105 Beer Sheva , Israel. Taking Gompertz Seriously. Complexity in social systems: from data to models,

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Social conditions and the gompertz rate of ageing

Social conditions and the Gompertz rate of ageing

Jon Anson


Deparment of Social Work

Ben-GurionUniversity of the Negev

84105 BeerSheva, Israel


Complexity in social systems: from data to models,

Cergy-Pontoise, France, June 2013

Funding: ISF 677/11

The segmented mortality curve france total population 1913
The Segmented Mortality CurveFrance, total population, 1913

The gompertz model
The Gompertz Model

  • Samuel Gompertz (1825): Adult mortality increases exponentially with age

    (x) = atbx

    with t the mortality risk at age t and x the number of years past t

  • Gompertz argued for t = 25. In practice, initial checks suggest we use t = 50

Corollaries life table functions
Corollaries: Life table functions

  • Probability of

  • Surviving x years

2. Average years

Lived between t and x

3. Density distribution

4. Modal age at death

Criteria for goodness of fit
Criteria for goodness of fit

  • Probability of surviving from age 50 to age 95

  • Partial life expectancy over 45 years, between age 50 and 95

  • Modal age at death in density distribution

Data i a historical sample
Data I: A historical sample

  • Sampled 108 male and female life tables from the Human Mortality Database (3,774 pairs)

  • No two tables from the same year

  • Same country at least 25 years apart

  • Countries with historical long series over represented

Fitting m x ages 50 to 95
Fitting mx: ages 50 to 95

  • 3-stage fitting process

    • x = x – 50 (modelling years past age 50

    • Fit log(mx) = a1 + x•log(b1)

    • Use a1 and b1 as starting points, fit

      • mx = a2b2x (non-linear model)

    • Use a2 and b2 as starting points, fit

      • xp50 =

    • Use a3 and b3 for further analysis

Conclusions stage i
Conclusions Stage I

  • At ages 50 to 95 (mature adult mortality) the Gompertz model:

    • Reproduces partial life expectancy

    • Reproduces the details of the mortality distribution (survivorship, modal age) but not perfectly

    • There is a marginal difference in the reproduction beween male and female curves. For a given observed value:

      • p(surviving): Male > Female

      • Mode: Female > Male

  • Question: which is more reliable, the data or the model?

Dependence of b on a
Dependence of b on a

Sample mortality slopes for

Sample of values of a

  • Large relative variation in

  • mortality rate at age 50

  • Little variation at age 95

  • Implies: the lower is a, the

  • the steeper the increase

A and b one parameter or two
a and b : One parameter or two?

Question: what explains the residual variation in b?

= delayed or premature adult mortality

Data ii who contemporary
Data II: WHO contemporary

  • Slope (b) not determined uniquely by prior mortality (a). Look at social conditions

  • 193 pairs of contemporary life tables for 2009, source: WHO.

    • Note: quality mixed, some data based; some data + model; some model based.

  • Social data from UN Human Development Index; Economist Intelligence Unit, etc.

The social meaning of b
The social meaning of b

  • The human life span is effectively limited to about 110 years, by which age all societies reach a similar level of mortality

  • If mortality at mid adulthood (50) is low, mortality rates will increase more rapidly to attain this maximum – hence the strong negative relation between a and b

  • All else being equal, advantageous social conditions will hold back the increase in the mortality rate (i. e. reduce b)

Predicting b from social data
Predicting b from social data

Multi-level model with sex|Country variation, variables centred at median

Interpreting social effects
Interpreting social effects

  • The major determinant of the slope is the level of mortality at younger ages (a)

  • The rate of increase for females is less steep than for males

  • There is a considerable amount of missing data, particularly concerning income and income distributions, mostly for poorer countries

  • At lower levels of average income the mortality slope is steeper than at higher levels

  • The more democratic a country, the less steep the mortality slope

  • The greater the inequality, the less steep the mortality slope!!! (Survival effect?)

Summary i
Summary I

  • The humanmortalitycurvecanbebroken down into a number of log-linear segments, each of whichcanbefitted by a Gompertz model

    mx = abx

  • The Gompertz model aboveage 50 adequatelyreproduces the generallevel of mortalityattheseages (partial life expectancy), but differsin detailfrom the published life table

  • Wecannot tell if thesedifferences are due to the inadequacies of the model, or shortcomings in the data on which the life tables are based

Summary ii
Summary II

  • The rate of increase in mortality (slope) above age 50 is heavily dependent on the level of mortality at age 50: the lower the mortality, the steeper the slope

  • Given the starting level (a)

    • Female slopes are less steep than male slopes

    • High national income reduces the slope

    • Democratic government reduces the slope

    • Inequality reduces the slope!!!

    • The effects of wealth and democracy are greater for females than for males


  • Even allowing for mortality at younger ages, there are important variations in mortality levels and rates of increase in mature adulthood

  • These differences are related to the level of wealth and forms of social, economic and political organisation

  • The Gompertz model provides a useful shorthand for summarising and investigating these differences

Jon Anson

[email protected]