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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

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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