t

1

Estimating life expectancy in small population
areas

Jorge Miguel Bravo, University of Évora /
CEFAGE-UE, jbravo_at_uevora.pt Joana Malta,
Statistics Portugal, joana.malta_at_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 qx 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.

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 period t,
  t1 e t2
• 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
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
Gross mortality curve


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

Age

30
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
Beira Interior Sul LGM (2,4)g


33
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