Social conditions and the Gompertz rate of ageing

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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

Jon Anson

YishaiFriedlander

Deparment of Social Work

Ben-GurionUniversity of the Negev

84105 BeerSheva, Israel

TakingGompertzSeriously

Complexity in social systems: from data to models,

Cergy-Pontoise, France, June 2013

Funding: ISF 677/11

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
• 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
• 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
• 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 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
• 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

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?

Question: what explains the residual variation in b?

= delayed or premature adult mortality

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 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

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

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
• 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
• 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
Conclusion
• 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

anson@bgu.ac.il