1. Social conditions and the Gompertz rate of ageing Taking Gompertz Seriously Jon Anson Yishai Friedlander Deparment of Social Work Ben-Gurion University of the Negev 84105 Beer Sheva, Israel Complexity in social systems: from data to models, Cergy-Pontoise, France, June 2013 Funding: ISF 677/11

2. 2 The Segmented Mortality Curve France, total population, 1913

3. 3 The Gompertz Model Samuel Gompertz (1825): Adult mortality increases exponentially with age  (x) = a t b x 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

4. 4 Corollaries: Life table functions 2. Average years Lived between t and x 1.Probability of Surviving x years 3. Density distribution 4. Modal age at death

5. 5 Criteria for goodness of fit 1.Probability of surviving from age 50 to age 95 2.Partial life expectancy over 45 years, between age 50 and 95 3.Modal age at death in density distribution

6. 6 Example: Two populations, at high and low mortality

7. 7 Gompertz lines at ages 50 to 95

8. 8 Fitted survivorship curves: l' 50 = 1

9. 9 Density curves and modal ages at death

10. 10 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

11. 11 Fitting m x : ages 50 to 95 3-stage fitting process –x = x – 50 (modelling years past age 50 –Fit log(m x ) = a 1 + xlog(b 1 ) –Use a 1 and b 1 as starting points, fit m x = a 2 b 2 x (non-linear model) –Use a 2 and b 2 as starting points, fit x p 50 = –Use a 3 and b 3 for further analysis

12. 12 Reproducing partial life expectancy, ages 50 to 95

13. 13 Reproducing p(surviving) from age 50 to 95

14. 14 Reproducing the modal age at death

15. 15 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?

16. Dependence of b on a 16 Large relative variation in mortality rate at age 50 Little variation at age 95 Implies: the lower is a, the the steeper the increase Sample mortality slopes for Sample of values of a

17. 17 a and b : One parameter or two? Question: what explains the residual variation in b? = delayed or premature adult mortality

18. 18 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.

19. 19 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)

20. 20 Predicting b from social data VariableMales Females MalesFemales Intercept1.0943 (0.00101) 1.0913 (0.00133) 1.0948 (0.00121) 1.0921 (0.00158) Log(a)-0.0282 (0.00190) -0.0284 (0.00100) -0.0291 (0.000723) -0.0289 (0.00105) Log(GDP)-0.00219 (0.000671) -0.00427 (0.000851) -0.00303 (0.000741) -0.00483 (0.000968) Democracy-0.00129 (0.000277) -0.00190 (0.000322) -0.00166 (0.000322) -0.00236 (0.000388) Log(Gini)-0.00708 (0.00236) -0.00722 (0.00283) N Countries 132 104 R² 0.9939 0.9945 Multi-level model with sex|Country variation, variables centred at median

21. 21 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?)

22. 22 Summary I The human mortality curve can be broken down into a number of log-linear segments, each of which can be fitted by a Gompertz model m x = ab x The Gompertz model above age 50 adequately reproduces the general level of mortality at these ages (partial life expectancy), but differs in detail from the published life table We cannot tell if these differences are due to the inadequacies of the model, or shortcomings in the data on which the life tables are based

23. 23 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

24. 24 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

25. 25 Jon Anson anson@bgu.ac.il