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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Joint EUROSTAT/UNECE Work Session on Demographic Projections
Lisbon, 29th April 2010
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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
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 Applications to mortality data
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Estimating life expectancy in
small population areas
 Increasing demand of indicators of mortality for smaller (subnational, 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
 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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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
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Overview of mortality
graduation methods
 Beginning with a crude estimation of qx , Qˆ  qx : xmin ,..., , we
wish to produce smoother estimates,
qˆ x
, of the true but unknown
mortality probabilities from the set of crude mortality rates,
qx , for
each age x.
 The crude rate at age x is usually based on the corresponding
d x , relative to initial exposed to risk, Ex .
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number of deaths recorded,
6
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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
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
7
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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., f ( x,  )
 Parameters are estimated using the gross mortality probabilities
obtained from the available data, using adequate statistical
procedures.
8
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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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The methodology adopted by
Statistics Portugal
 Consider a group of consecutive ages x and the series
of independent deaths d x  and corresponding exposure
to risk Ex 
 The graduation procedures uses a family of parametric
functions know as Gompertz-Makeham of the type ( r, s ) .
They are functions with r  s parameters of the form
s 1


r ,s
i
j
GM ( x )  i x  exp    j x 
i 0
 j 0

r 1
(1)
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The methodology adopted by
Statistics Portugal
 In some applications it is useful to establish the following
Logit Gompertz-Makeham functions of the type ( r, s ) ,
defined as
r ,s
GM
r ,s
 ( x)
LGM  ( x) 
r ,s
1  GM  ( x)
(2)
 The methodology developed by CMIB states that the
expression in (3) results in an adequate adjustment
(3)
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qx  LGM r , s ( x)
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General Linear Models (GLM)
 Given the non linear nature of the GM r , s ( x) 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.
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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.
13
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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
Dx
Bin( Ex , qx ) ,
the suggested link function is given by
 qx 
 x  log 

 1  qx 
(4)
And its inverted function is given by
(5)
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exp( x )
qx 
1  exp( x )
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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
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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
2,7 and 0, 4 , respectively.
 The choice for the optimal model is based on the evaluation of
several measures and tests for model fit.
16
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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.
17
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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.
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 Why?
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Projecting probabilities of death at
older ages

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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
19
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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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LL and (unscaled) deviance, Lisbon
2006-2008, MF
Lisboa 2006-2008, HM
r
0
1
2
3
4
s=2
217947.2
217283.4
216767.0
216505.6
216502.2
s=2
3464.65
2136.99
1104.23
581.46
574.62
s=3
216716.8
216689.8
216500.2
216500.1
216498.8
s=3
1003.89
949.78
570.63
570.44
567.75
Deviance
s=4
617.41
616.46
614.16
568.02
516.78
s=5
612.47
582.05
508.38
548.92
492.13
s=6
216512.1
216504.6
216481.2
216451.4
216450.6
s=7
216489.1
216488.4
216449.7
216441.5
216442.4
s=6
594.35
579.41
432.61
473.01
471.48
s=7
548.33
547.08
469.58
453.20
455.06
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r
0
1
2
3
4
Log-Likelihood
s=4
s=5
216523.6 216521.1
216523.1 216505.9
216522.0 216505.1
216498.9 216489.4
216473.3 216461.0
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LGM(r,s) - Goodness-of-fit measures,
Lisbon, 2006-2008, MF
(…)
(…)
(…)
(…)
(…)
(…)
(…)
(…)
(…)
(…)
«
(…)
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se
0,00009
0,00031
0,00027
0,15654
0,27884
0,71621
0,94184
0,87421
0,98526
t -ratio
35,743
30,308
23,997
-50,436
20,325
14,561
-9,329
-10,876
10,666
p -value
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
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α0
α1
α2
β0
β1
β2
β3
β4
β5
Coef.
0,003332
0,009357
0,006380
-7,895357
5,667404
10,428748
-8,786856
-9,507530
10,509087
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Coefficients of model LGM(3,6),
Lisbon, 2006-2008, MF
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Adjusted mortality curve, and CI,
Lisbon, 2006-2008, MF
-8
-6
log(qx)
-4
-2
0
Brutas vsprobabilities
Prob Graduadas
GrossProb.
vs. Graduated
of death
20
40
60
Idade
Age
80
100
«
0
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Residuals from LGM(3,6) model,
Lisbon, 2006-2008, MF
-2
0
scaled.reldev
0
-5
-2
-1
0
Quantiles of Standard Normal
1
2
0
20
40
60
80
100
age
«
rel.dev
2
5
4
10
Scaled Deviations
25
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Comparison between crude and fitted
death probabilities
2,0
0,0
1
11
21
31
41
51
61
71
81
91
101
111
-2,0
-4,0
-6,0
-8,0
-10,0
Gross
Gross
Grad
Grad
Grad+DG
Grad+DG
«
-12,0
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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
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«
-4
-6
-8
0
20
40
60
80
100
Age
Idade
«
log(qx)
-2
0
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Gross mortality curve
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-4
-6
-8
0
20
40
60
80
100
Age
Idade
«
log(qx)
-2
0
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Gross prob vs. Graduated prob. –
LGM (0,7)
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Comparison between crude and fitted
death probabilities
70
75
80
85
90
95
100
105
110
115
1
0
-1
-2
-2
-3
-3
-4
age
Gross
brutos
Grad
graduados
Grad+DG
grad+DG
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ln(qx)
-1
30
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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
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«
Beira Interior Sul: LGM (2,4)g
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
Gross
brutos
41
51
61
Grad
graduados
71
81
91
101
111
Grad+DG
grad+DG
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-10,0
32
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Comparison between crude and fitted
death probabilities
70
75
80
85
90
95
100
105
110
115
1
1
0
-1
-2
-2
-3
-3
-4
-4
-5
age
Gross
brutos
Grad
graduados
Grad+DG
grad+DG
«
ln(qx)
-1
33

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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
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