Joint EUROSTAT/UNECE Work Session on Demographic Projections Lisbon, 29 th April 2010

1. « Estimating life expectancy in small population areas « Jorge Miguel Bravo, University of Évora / CEFAGE-UE, jbravo@uevora.pt Joana Malta, Statistics Portugal, joana.malta@ine.pt Joint EUROSTAT/UNECE Work Session on Demographic Projections Lisbon, 29th April 2010 «

2. Presentation « • Introduction: implications of estimating life expectancy in small population areas • Overview of mortality graduation methods • Graduation of sub-national mortality data in Portugal • The CMIB methodology • Assessing model fit • Projecting probabilities of death at older ages • Applications to mortality data « «

3. Estimating life expectancy in small population areas « • Increasing demand of indicators of mortality for smaller (sub-national, sub-regional) areas. • Due to the particularities of small population areas’ data, calculating life expectancy is often not possible or requires more complex methods • There are several methods to deal with the challenges posed to the analyst in these situations. • Statistics Portugal currently uses solutions that combine traditional complete life table construction techniques with smoothing or graduation methods. «

4. Overview of mortality graduation methods « • Graduation is the set of principles and methods by which the observed (or crude) probabilities are fitted to provide a smooth basis for making practical inferences and calculations of premiums and reserves. • One of the principal applications of graduation is the construction of a survival model, normally presented in the form of a life table. «

5. Overview of mortality graduation methods « • The need for graduation is an outcome of • Small population • Absence of deaths in some ages • Variability of probabilities of death between consecutive ages • Graduation methods • Non-parametric • Parametric «

6. Overview of mortality graduation methods « • Beginning with a crude estimation of , , we wish to produce smoother estimates, , of the true but unknown mortality probabilities from the set of crude mortality rates, , for each age x. • The crude rate at age x is usually based on the corresponding number of deaths recorded, , relative to initial exposed to risk, . «

7. Overview of mortality graduation methods « • Parametric approach • Probabilities of death (or mortality rates) are expressed as a mathematical function of age and a limited set of parameters on the basis of mortality statistics • Non parametric approach • Replace crude estimates by a set of smoothed probabilities «

8. Parametric graduation « • Based on the assumption that the probabilities of deaths qxcan be expressed as a function of age and a limited set of unknown parameters, i.e., • Parameters are estimated using the gross mortality probabilities obtained from the available data, using adequate statistical procedures. «

9. Graduation of sub-national mortality data in Portugal « • The method adopted by Statistics Portugal in 2007 to calculate graduated mortality rates for sub-national levels (regions NUTS II and NUTS III) is framed under the parametric graduation procedures • It is an extension of the Gompertz and Makeham models. «

10. The methodology adopted by Statistics Portugal « • Consider a group of consecutive ages x and the series of independent deaths and corresponding exposure to risk • The graduation procedures uses a family of parametric functions know as Gompertz-Makeham of the type . They are functions with parameters of the form (1) «

11. The methodology adopted by Statistics Portugal « • In some applications it is useful to establish the following Logit Gompertz-Makeham functions of the type , defined as (2) • The methodology developed by CMIB states that the expression in (3) results in an adequate adjustment (3) «

12. General Linear Models (GLM) « • Given the non linear nature of the parametric functions, estimations using classic linear models is not possible. • General Linear Models (GLM) are an extension of linear models for non normal distributions and non linear transformations of the response variables, giving them special interest in this context. «

13. General Linear Models (GLM) « • As an alternative to classic linear regression models, GLM allow, through a link function, estimation of a function for the mean of the response variable, defined in terms of a linear combinations of all independent variables. «

14. GLM and graduation of probabilities of death « • Considering that we intend to apply a logit transformation with a linear predictor of the type Gompertz-Makeham to the probabilities of death, and assuming that , the suggested link function is given by (4) And its inverted function is given by (5) «

15. Data used « • Life-tables corresponding to three-year periodt, t+1 e t+2 • Deaths by age, sex and year of birth • Live-births by sex • Population estimates by age and sex «

16. Estimation, evaluation and construction of life tables « • The graduation procedure begins by determining the order (r,s) for the Gompertz-Makeham function that best fits the data. • In each population different combinations are tested, varying s and r between and , respectively. • The choice for the optimal model is based on the evaluation of several measures and tests for model fit. «

17. Estimation, evaluation and construction of life tables « • The graduated life table preserves the gross probability of death at age 0. • In ages where the number of registered deaths is very small or null it can be advisable to aggregate the number of deaths until they add up to 5 or more occurrences. The age to consider for this group of aggregated observations is the mid point of all ages considered in the interval. «

18. Assessing model fit « • Measures and tests for assessing model fit: • Absolute and relative deviations; • Deviance, Chi-Square; • Signs Test / Runs Test; • Kolmogorov-Smirnov Test; • Auto-correlation Tests; • Graphical representation of adjustment of estimated mortality curve. «

19. Projecting probabilities of death at older ages « • Why? • less reliability of the available data • Irregularities observed in the gross mortality rates at older ages • Applied method (Denuit and Goderniaux, 2005): • Compatible with the tendencies observed in mortality at older ages • Imposes restrictions to life tables closing and an age limit (115 years) • Adjustable to the observed conditions in every moment • Smoothing of the mortality curve around the cutting age «

20. Application to mortality data: Lisbon, 2006-2008, sexes combined « • NUTS II: Lisbon, 2006-2008, sexes combined • Population estimate at 31/12/2006: 2794226 • Risk exposure: 5627699 • Registered deaths: 50169 • Aprox. 91.3% of deaths after the age of 50 «

21. LL and (unscaled) deviance, Lisbon 2006-2008, MF « «

22. LGM(r,s) - Goodness-of-fit measures, Lisbon, 2006-2008, MF « (…) (…) (…) (…) (…) (…) (…) (…) (…) (…) (…) «

23. Coefficients of model LGM(3,6), Lisbon, 2006-2008, MF « «

24. Adjusted mortality curve, and CI, Lisbon, 2006-2008, MF « «

25. Residuals from LGM(3,6) model, Lisbon, 2006-2008, MF « «

26. Gross Grad Grad+DG Comparison between crude and fitted death probabilities « «

27. Application to mortality data: Madeira, 2001-2003, M « • NUTS II: Madeira, M, 2001-2003 • Population estimate at 31/12/2001: 113140 • Registered deaths: 2755 • Ages with 0 registered deaths «

28. 0 -2 -4 log(qx) -6 -8 Age Idade 0 20 40 60 80 100 Gross mortality curve « «

29. 0 -2 -4 log(qx) -6 -8 Idade 0 20 40 60 80 100 Gross prob vs. Graduated prob. – LGM (0,7) « Age «

30. « 70 75 80 85 90 95 100 105 110 115 1 0 -1 -1 -2 ln(qx) -2 -3 -3 -4 age brutos graduados grad+DG Gross Grad Grad+DG Comparison between crude and fitted death probabilities «

31. Application to mortality data: Beira Interior Sul, 2004-2006, sexes combined « • NUTS III: Beira Interior Sul, sexes combined, 2004-2006 • Population estimate at 31/12/2004: 75925 • Registered deaths: 2516 • Ages with 0 registered deaths • Grouping of contiguous ages as to aggregate at least 5 deaths • Attribute aggregated deaths to the middle age point «

32. 1,0 0,0 -1,0 -2,0 -3,0 -4,0 -5,0 -6,0 -7,0 -8,0 -9,0 1 11 21 31 41 51 61 71 81 91 101 111 -10,0 Gross Grad Grad+DG brutos graduados grad+DG Beira Interior Sul: LGM (2,4)g « «

33. 70 75 80 85 90 95 100 105 110 115 1 1 0 -1 -1 -2 ln(qx) -2 -3 -3 -4 -4 -5 age Gross Grad+DG Grad brutos graduados grad+DG Comparison between crude and fitted death probabilities « «

34. Selected bibliography « • Benjamin, B. and Pollard, J. (1993). The Analysis of Mortality and other Actuarial Statistics. Third Edition. The Institute of Actuaries and the Faculty of Actuaries, U.K. • Bravo, J. M. (2007). Tábuas de Mortalidade Contemporâneas e Prospectivas: Modelos Estocásticos, Aplicações Actuariais e Cobertura do Risco de Longevidade. Tese de Doutoramento, Universidade de Évora. • Chiang, C. (1979). Life table and mortality analysis. World Health Organization, Geneva. • Denuit, M. and Goderniaux, A. (2005). Closing and projecting life tables using log-linear models. Bulletin of the Swiss Association of Actuaries, 29-49. • Forfar, D., McCutcheon, J. and Wilkie, D. (1988). On Graduation by Mathematical Formula. Journal of the Institute of Actuaries 115, 1-135. • Gompertz, B. (1825). On the nature of the function of the law of human mortality and on a new mode of determining the value of life contingencies. Philosophical Transactions of The Royal Society, 115, 513-585. «

35. « THANK YOU « «