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

2. 2 «  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 « « Presentation

3. 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. 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. 5 « «  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 Overview of mortality graduation methods

6. 6 « «  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. 7 « «  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 Overview of mortality graduation methods

8. 8 « «  Based on the assumption that the probabilities of deaths q x can 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. Parametric graduation

9. 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. 10 « «  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) The methodology adopted by Statistics Portugal

11. 11 « «  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) The methodology adopted by Statistics Portugal

12. 12 « «  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. General Linear Models (GLM)

13. 13 « «  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. General Linear Models (GLM)

14. 14 « «  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 GLM and graduation of probabilities of death And its inverted function is given by (4) (5)

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

16. 16 « «  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. Estimation, evaluation and construction of life tables

17. 17 « «  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. Estimation, evaluation and construction of life tables

18. 18 « «  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. Assessing model fit

19. 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. 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. 21 « LL and (unscaled) deviance, Lisbon 2006-2008, MF «

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

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

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

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

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

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

28. 28 « Gross mortality curve « 020406080100 -8 -6 -4 -2 0 Idade log(qx) Age

29. 29 « Gross prob vs. Graduated prob. – LGM (0,7) « Age

30. 30 « « -4 -3 -2 0 1 707580859095100105110115 age ln(qx) brutosgraduadosgrad+DG GrossGradGrad+DG Comparison between crude and fitted death probabilities

31. 31 « «  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 Application to mortality data: Beira Interior Sul, 2004-2006, sexes combined

32. 32 « « Beira Interior Sul: LGM (2,4)g -10,0 -9,0 -8,0 -7,0 -6,0 -5,0 -4,0 -3,0 -2,0 -1,0 0,0 1,0 1112131415161718191101111 brutosgraduadosgrad+DG GrossGradGrad+DG

33. 33 « « -5 -4 -3 -2 0 1 1 707580859095100105110115 age ln(qx) brutosgraduadosgrad+DG Gross Grad Grad+DG Comparison between crude and fitted death probabilities

34. 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. 35 « « « THANK YOU