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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, [email protected]
Joana Malta, Statistics Portugal, [email protected]
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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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