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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 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