saint - demografi · title: saint author: ��s�ren fiig jarner keywords: powerpoint...
TRANSCRIPT
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Levetidsmodellering: SAINT-modellen
Dansk Demografisk Forening
27. januar 2010
Søren Fiig Jarner
Esben Masotti Kryger
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Levetidsmodellering: SAINT
Indhold
Hvad er SAINT?
Gængse levetidsmodeller
Dødeligheden i små populationer
SAINT-modellen
- trend
- spread
Resultater
Implementering i ATP
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Levetidsmodellering: SAINT
ATP’s mortality model
SAINT = Spread Adjusted InterNational Trend
- describes small population mortality as temporary deviations from underlying trend
- developed in-house in 2007
Stochastic mortality model
- produce a range of possible, future evolutions of mortality intensities, μ(t,x)
- the mean forecast is used to calculate the tariff and set the reserve for POM
- calibrated annually
- no systematic, future increases in reserves due to mortality (if the model is right!)
Discrete version of SAINT for ATP implemented in P&H
Continuous version applied to DK developed in academic paper
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Levetidsmodellering: SAINT
Mortality models
Main methodologies
1. Expert judgment (data free)
2. Deterministic improvements
3. Lee-Carter family
4. Parametric time-series modelling
Deterministic
Stochastic
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Levetidsmodellering: SAINT
Mortality modelling
Lee-Carter (1992)
- log μ(t,x) = a(x) + b(x)k(t) + noise
- assumes age-specific, constant, relative rates of improvement
- conceptually simple; improvements driven by single index
- projections overly confident when based only on index variability
- no structural limitations to the shape of mortality rates; problematic when applied to small population mortality data
- ”future improvements = historic improvements”;the mortality of very old will never improve
- not very robust; in particular so in small populations
- various extensions suggested
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Levetidsmodellering: SAINT
Mortality modelling
Parametric time-series modelling
- assume functional form of (population) mortality, i.e. μ(t,x) = F(θt,x),
e.g. Makeham or logistic period life tables
- time-series model for (low-dimensional) parameter vector (θt)
- easy to fit and typically provides good description of data
- provides no insight into what causes the drift in (θt)
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Levetidsmodellering: SAINTD
ea
th r
ate
(m
)
1950 2000 2050
1%
2%
3%
4%
Forecasting principle
”In the absence of additional information the best one
can do is to extrapolate past trends”
- sounds sensible, but what does it actually mean?
Model
log m(t) = a + b t + c t2 + εt
log m(t) = a + b t + εt
m(t)1/2 = a + b t + εt
m(t) = a + b t + εt
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Levetidsmodellering: SAINT
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Simple projections lack structure and robustness
1990
40
5060
70
80
90
Age
100
30
20
Reasonable short-term projections
Implausible long-term projections lacking (biological) structure
Danish female mortality
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Levetidsmodellering: SAINT
Small population mortality
Modelling challenge: Produce plausible, long-term forecasts reflecting
both the general pattern and the ”wildness” seen in data
- General pattern
- mortality increases with age
- age-specific death rates decline over time
- rates of improvement decrease with age
- rates of improvement for old age groups increase over time
- Deviations
- substantial deviations from the general pattern
- even periods with increasing mortality for some age groups
The SAINT model structure
mortality = international trend + spread
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Levetidsmodellering: SAINT
Data and terminology
Human Mortality Database (www.mortality.org)
Danish and international female mortality from 1933 to 2005
- 19 countries in the international dataset: USA, Japan, West Germany, UK,
France, Italy, Spain, Australia, Canada, Holland, Portugal, Austria, Belgium,
Switzerland, Sweden, Norway, Finland, Iceland & Denmark.
Death counts and exposures for each year and each age group
D(t,x) = number of deaths
E(t,x) = exposure (”years lived”)
Death rate, D(t,x)/E(t,x), is an estimate of (the
average of) underlying intensity, μ(t,x)
Death probability, q(t,x) = 1-e-∫μ(t,x) ≈ ∫μ(t,x)t t+1 time
x
x+1
age
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Levetidsmodellering: SAINT
Year
De
ath
ra
te
1940 1950 1960 1970 1980 1990 2000
0.1
%1
%1
0%
10
0%
Danish fluctuations around stable international trend
Danish and international female mortality
40
50
60
70
80
90
Age
100
3020
Danish life expectancy
among the highest in
the world
Denmark falling behind
the international trend
Is this the beginning
of a catch up period?
Small improvements
at the highest ages
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Levetidsmodellering: SAINT
Trend modelling concepts
Population dynamics
- Ensure consistent intensity surfaces over time and ages by aggregating individual intensities to population level
- Individuals living in the same period of time are influenced by common as well as individual factors
- Factors have either a cumulative or an instant effect on mortality
Frailty (unobservable)
- People are genetically different. Only the more robust individuals will attain very high ages
- Lack of historic improvements among the very old may be due to selection effects. In the future the frailty composition at old ages will change
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Levetidsmodellering: SAINT
Homogeneous cohort – no selection
0 20 40 60 80 100
02
00
40
06
00
80
01
00
0
Age
Po
pu
latio
n s
ize
0%
5%
10
%1
5%
20
%2
5%
xex)(Gompertz-Makeham intensity:
Inte
nsity (
μ)
(x)
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Levetidsmodellering: SAINT
0 20 40 60 80 100
02
00
40
06
00
80
01
00
0
Age
Po
pu
latio
n s
ize
0%
5%
10
%1
5%
20
%2
5%
Selection effects within a cohort
)2,(x
)2
1,(x
Inte
nsi
ty (
μ)
(x)
xezzx );(Individual: xexZx )|(E)(Cohort:
)1,(x
)(x
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Levetidsmodellering: SAINT
Generalize Gompertz-Makeham intensity to allow for
time-dependent cumulative and instant factors
Underlying individual intensities
- mean 1 and variance σ2 Γ-distributed frailties, z
This yields an 8-parameter trend model for population intensity
Trend model
)()(1)(),(
1
2 tduuetextt
xt
ggu
xt
t
xt
)()),(exp()();,( tdsxtssgtzzxtt
xt
))(exp()( 021 ttt
)()(),( 03021 xxttxtg
))(exp()( 021 ttt
Previous values of κ (and g)
are ”remembered” by the
population due to selection
”treatment” level
”wear-out” rate
”accident” rate
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Levetidsmodellering: SAINT
Structure of individual intensity
t1t2t
Common and separate components of individual intensities
2x
1x
)()exp()();,(1
1
2
222 tggtzzxtt
t
t
t
)()exp()();,(1
111 tgtzzxtt
t
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Levetidsmodellering: SAINT
Rate of improvement
Individual and population intensity
Senescent component of rate of improvement
Two opposite effects in rate of improvement:
- more frail people become old (i.e. first term is negative - and vanishing)
- general mortality improvements (i.e. second term is positive)
)()(],|[),( tetxtZExt
t
xt
g
)()();,( tetzzxt
t
xt
g
t
txtxts
))(),(log(),(
t
et
t
xtZE
t
xt
g
))(log(]),|[log(
x22 tfor
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Levetidsmodellering: SAINT
Improvement rates – international trend
Female (σ=0.43, β2 lille) Male (σ=0.26, β2 stor)
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Levetidsmodellering: SAINT
Trend estimated from international data
Maximum likelihood estimation based on Poisson-model
Estimates (t0=2000, x0=60)
)),(),((Poiss~),( xtExtxtD INTINTINT
4/)1,1(),1()1,(),(),( xtxtxtxtxtINT
t t+1
x
x+1
σ α1 α2 β1 β2 β3 γ1 γ2
Female 4.29e-1 -8.78e0 -1.85e-2 9.90e-2 4.79e-6 1.31e-3 -1.18e1 -8.90e-2
Male 2.62e-1 -1.06e1 -1.78e-2 1.06e-1 8.37e-5 5.59e-5 -7.52e0 -2.50e-2
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Levetidsmodellering: SAINT
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Trend – fit and forecast
International female mortality
40
50
60
70
80
90
100
3020
Age
General, long-term rate
of improvement = 1.8%
Early, young age rate
of improvement = 9.1%
Increasing old age rate
of improvement
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Levetidsmodellering: SAINT
Spread model
Danish mortality (gender-specific)
The spread is assumed to fluctuate around zero
- that is, no mean term included in the model
The spread controls the length and magnitude of deviations
- and provides information about projection uncertainty
))()(exp(),(),( 21 xrcxrbaxtxt tttINTDK
),0(~,,,, 3 ,111 NeecbaAcba tt
t
ttt
t
ttt
40/)60()(1 xxr
1000/)3/9160120()( 22 xxxr
Mean zero, orthogonal regressors
normalized to (about) 1 at age 20 and 100
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Levetidsmodellering: SAINT
Spread parametrization
Level
Slope (r1)
Curvature (r2)
Age
20 30 40 50 60 70 80 90 100
-1.0
-0.5
0.0
0.5
1.0
Regressors
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Levetidsmodellering: SAINT
Estimation
First, we estimate spread parameters (at,bt,ct) for each year
by maximum likelihood based on the Poisson-model:
- trend is kept fixed
- estimates of (at,bt,ct) depend only on data for year t
Second, the VAR-parameters (A and Ω) are estimated based on the
estimated time-series of spread parameters (at,bt,ct)t=1933,…,2005:
)),())()(exp(),((Poiss~),( 21 xtExrcxrbaxtxtD DKtttINTDK
4/)1,1(),1()1,(),(),( xtxtxtxtxtINT
t t+1
x
x+1
),0(~,,,, 3 ,111 NeecbaAcba tt
t
ttt
t
ttt
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Levetidsmodellering: SAINT
Illustration of spread adjustment
Age
De
ath
ra
te
20 30 40 50 60 70 80 90 100
0.0
1%
0.1
%1
%1
0%
10
0%
Estimates
a2004= 21%
b2004= 5%
c2004=-19%
International trend
Danish data
Danish fit
Female mortality in 2004
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Levetidsmodellering: SAINT
Forecasting
Forecasting in the VAR-model based on conditional distributions
- where T is the last observation year, h is the forecasting horizon and
Mean forecast
- where we use the mean forecast
- trend is kept fixed
),(~),,(|),,( hht
TTTt
hThThT VmNcbacba
1
0)( ,),,(
h
i
tiih
tTTT
hh AAVcbaAm
))(~)(~~exp(),(ˆ),( 21 xrcxrbaxhTxhT hThThTINTDK
hhThThT mcba )~,~
,~(
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Levetidsmodellering: SAINT
1950 2000 2050 2100
-0.4
-0.2
0.0
0.2
0.4
Year
Long recovery period
Fitted at
Fitted bt
Fitted ct
Forecast
Estimated and forecasted spread
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Levetidsmodellering: SAINT
Danish mortality – fit and forecast
4050
60
70
80
90
100
3020
Danish female mortality and international trend
Age
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Denmark falling behind … and catching up again
Similar development in
old age mortality
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Levetidsmodellering: SAINT
Confidence intervals
95%-confidence intervals
- due to stationarity the variance has a finite limit as h tends to ∞,
i.e. the Danish deviation from the international trend is bounded (in probability)
95%-confidence intervals for the intensities
- where
The confidence intervals reflect the stochastic nature of the VAR-model itself
- parameter uncertainty is not taken into account
)diag(96.1),,(CI95% hhhThThT Vmcba
xhtxx
thINTDK rVrrmxhTxhT 96.1),(log)),((logCI95%
tx xrxrr ))(),(,1( 21
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Levetidsmodellering: SAINT
1950 2000 2050 2100
-0.4
-0.2
0.0
0.2
0.4
Year
Pointwise 95% confidence intervals
Fitted at
Fitted bt
Fitted ct
Forecast
Estimated and forecasted spread
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Levetidsmodellering: SAINT
Forecast uncertainty
Analytical methods
- only feasible for very few quantities of interest, e.g. the spread itself
Monte Carlo
- simulate N spread series and calculate mortality forecasts for each
- calculate quantity of interest, e.g. life expectancy, for each forecast
- compute uncertainty measures, e.g. 95%-confidence intervals
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
84 85 86 87 88
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Life expectancy
Females aged 60 in 2005
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
Year
De
ath
ra
te
1950 2000 2050 2100
0.0
1%
0.1
%1
%1
0%
10
0%
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Levetidsmodellering: SAINT
Robustness: Initial year of estimation period
SAINT Lee-Carter
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Levetidsmodellering: SAINT
Robustness: Final year of estimation period
SAINT Lee-Carter
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Levetidsmodellering: SAINT
The SAINT model for ATP
ATP mortality (gender specific)
The spread parameters are estimated from ATP mortality data
ATP data only dates back to 1998
ATP time-series of spread parameters too short to estimate VAR-
parameteres, we therefore use the Danish VAR-parameters (A and Ω)
))()(exp(),(),( 21 xrcxrbaxtxt ATPt
ATPt
ATPtINTATP
),0(~,,,, 3 ,111 NeecbaAcba tt
tATPt
ATPt
ATPt
tATPt
ATPt
ATPt
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Levetidsmodellering: SAINT
Danish and ATP female spread (at)
0
0.05
0.1
0.15
0.2
0.25
0.3
1980 1990 2000 2010 2020
DK spread (level) DK forecast
ATP spread (level) ATP forecast
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Levetidsmodellering: SAINT
Output
Cellwise constant mortality surface
)62,2009(1-year survival probability
2009 2010 2011
62
63
)61,2010(
)62,2010(
)61,2009(
61 time
m))(n,exp(--1 m)q(n,
age
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Levetidsmodellering: SAINT
Remaining life expectancy
Female Male
0 years 65 years 0 years 65 years
G82 76.5 17.8 72.7 15.1
ATP2000 79.9 18.4 75.2 15.2
HMD2000 79.1 18.2 74.4 15.2
ATP2006 81.0 19.1 76.3 16.7
HMD2006 80.5 19.0 75.9 16.1
SAINT.DK (ALDER I 2006) 95.0 21.6 85.0 17.7
SAINT.ATP (ALDER I 2006) 95.0 21.4 85.0 17.7
SAINT.INT (ALDER I 2006) 95.1 22.7 84.9 17.4
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Levetidsmodellering: SAINT
2005 2010 2015 2020 2025 2030 2035
80
85
90
95
10
0
Year
Exp
ecte
d life
tim
e
Cohort lifetimes
40
60
0
20
40
60
200
AgeExpected lifetimes of Danish females
SAINT projection
No future improvements
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www.atp.dk 39
Levetidsmodellering: SAINT
Operation
The ATP mortality experience for a given year is available for
analysis in June the following year
ATP1: Calculate updated period life table (reserving: age - ½ year)
ATP2: Calculate new spread parameters, e.g. calculate
for males and females in June 2009
- make new forecast of
- make new forecast of entire mortality surface
- trend and VAR-parameters are kept fixed
- if the model is right the annual update will not cause systematic
changes in the mortality forecast, nor the reserve
Eventually trend and VAR-parameters should be reestimated
ATPATPATP cba 200820082008 ,,
,~,~
,~,~,~
,~201020102010200920092009ATPATPATPATPATPATP cbacba
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Levetidsmodellering: SAINT
Forecasts of level parameter (at) for different jump-off years
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
2000 2005 2010 2015 2020 2025 2030
Female 2007 2006 2005 2004 2003
Male 2007 2006 2005 2004 2003
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Levetidsmodellering: SAINT
Forecasts for remaining life expectancy in 2007 (ATP)
Jump-off
year
Cohort life expectancy Period life expectancy
Female Male Female Male
0 year 65 year 0 year 65 year 0 year 65 year 0 year 65 year
2003 95.20 21.63 85.13 17.54 81.07 19.19 76.34 16.32
2004 95.19 21.57 85.14 17.67 80.99 19.13 76.35 16.40
2005 95.21 21.59 85.19 17.75 81.08 19.15 76.61 16.45
2006 95.21 21.59 85.13 17.68 81.06 19.14 76.12 16.25
2007 95.20 21.56 85.12 17.76 80.77 18.91 75.87 16.19
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Levetidsmodellering: SAINT
Model structure
Stable underlying trend
- parsimonious parametric model estimated from international data
- frailty component give rise to changing improvement rates
Spread describes the deviations from the trend
- stationary, i.e. deviations are effectively bounded
- allows short- to medium-term fluctuations, but long-term
behaviour is determined by the trend
SAINT in summary
mortality = international trend + spread