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

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Estimating life expectancy in small population areas

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Jorge Miguel Bravo, University of Évora / CEFAGE-UE, jbravo@uevora.pt

Joana Malta, Statistics Portugal, joana.malta@ine.pt

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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

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Presentation

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«Estimating life expectancy in small population areas

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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.

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«Overview of mortality graduation methods

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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.

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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

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«Overview of mortality graduation methods

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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, .

xq minˆ : ,...,xQ q x

ˆxqxq

xd xE

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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

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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.

Parametric graduation

( , )f x

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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.

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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

( , )r s

r s1 1

,

0 0

( ) expr s

r s i ji j

i j

GM x x x

(1)

xd xE

The methodology adopted by Statistics Portugal

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In some applications it is useful to establish the following

Logit Gompertz-Makeham functions of the type ,

defined as

( , )r s

(2),

,,

( )( )

1 ( )

r sr s

r s

GM xLGM x

GM x

The methodology developed by CMIB states that the

expression in (3) results in an adequate adjustment

, ( )r sxq LGM x (3)

The methodology adopted by Statistics Portugal

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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.

, ( )r sGM x

General Linear Models (GLM)

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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)

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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

log1

xx

x

q

q

And its inverted function is given by

exp( )

1 exp( )x

xx

q

(4)

(5)

( , )x x xD Bin E q

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Data used

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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

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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.

0,4 2,7

Estimation, evaluation and construction of life tables

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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

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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

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«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

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«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

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r s = 2 s = 3 s = 4 s = 5 s = 6 s = 70 217947.2 216716.8 216523.6 216521.1 216512.1 216489.11 217283.4 216689.8 216523.1 216505.9 216504.6 216488.42 216767.0 216500.2 216522.0 216505.1 216481.2 216449.73 216505.6 216500.1 216498.9 216489.4 216451.4 216441.54 216502.2 216498.8 216473.3 216461.0 216450.6 216442.4

r s = 2 s = 3 s = 4 s = 5 s = 6 s = 70 3464.65 1003.89 617.41 612.47 594.35 548.331 2136.99 949.78 616.46 582.05 579.41 547.082 1104.23 570.63 614.16 508.38 432.61 469.583 581.46 570.44 568.02 548.92 473.01 453.204 574.62 567.75 516.78 492.13 471.48 455.06

Lisboa 2006-2008, HM

Log-Likelihood

Deviance

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

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

(…) (…) (…) (…) (…) (…) (…) (…) (…) (…) (…)

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

Coef. se t -ratio p -value

α0 0,003332 0,00009 35,743 < 0.0001α1 0,009357 0,00031 30,308 < 0.0001α2 0,006380 0,00027 23,997 < 0.0001β0 -7,895357 0,15654 -50,436 < 0.0001β1 5,667404 0,27884 20,325 < 0.0001β2 10,428748 0,71621 14,561 < 0.0001β3 -8,786856 0,94184 -9,329 < 0.0001β4 -9,507530 0,87421 -10,876 < 0.0001β5 10,509087 0,98526 10,666 < 0.0001

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

0 20 40 60 80 100

-8-6

-4-2

0Prob. Brutas vs Prob Graduadas

Idade

log

(qx)

0 20 40 60 80 100

-8-6

-4-2

0

Gross vs. Graduated probabilities of death

Age

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

Quantiles of Standard Normal

rel.d

ev

-2 -1 0 1 2

-50

510

Scaled Deviations

age

scal

ed.r

elde

v

0 20 40 60 80 100-2

02

4

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«Comparison between crude and fitted death probabilities «

-12,0

-10,0

-8,0

-6,0

-4,0

-2,0

0,0

2,0

1 11 21 31 41 51 61 71 81 91 101 111

Gross Grad Grad+DGGross Grad Grad+DG

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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

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Gross mortality curve «

0 20 40 60 80 100

-8-6

-4-2

0

Idade

log(

qx)

Age

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

0 20 40 60 80 100

-8-6

-4-2

0

Idade

log(

qx)

Age

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-4

-3

-3

-2

-2

-1

-1

0

1

70 75 80 85 90 95 100 105 110 115

age

ln(qx)

brutos graduados grad+DGGross Grad Grad+DG

Comparison between crude and fitted death probabilities

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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

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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

1 11 21 31 41 51 61 71 81 91 101 111

brutos graduados grad+DGGross Grad Grad+DG

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-5

-4

-4

-3

-3

-2

-2

-1

-1

0

1

1

70 75 80 85 90 95 100 105 110 115

age

ln(qx)

brutos graduados grad+DGGross Grad Grad+DG

Comparison between crude and fitted death probabilities

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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.

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«THANK YOU