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Social conditions and the Gompertz rate of ageing

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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

- 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

- Probability of
- Surviving x years

2. Average years

Lived between t and x

3. Density distribution

4. Modal age at death

- 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

- 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

- 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

- 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?

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

Question: what explains the residual variation in b?

= delayed or premature adult mortality

- 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 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)

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

- 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?)

- 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

- 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

anson@bgu.ac.il