Levetidsmodellering: SAINT-modellen Dansk Demografisk Forening Søren Fiig Jarner 27. januar 2010 Esben Masotti Kryger
Levetidsmodellering: SAINT Indhold Hvad er SAINT? Gængse levetidsmodeller Dødeligheden i små populationer SAINT-modellen - trend - spread Resultater Implementering i ATP www.atp.dk 2
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 www.atp.dk 3
Levetidsmodellering: SAINT Mortality models Main methodologies 1. Expert judgment (data free) Deterministic 2. Deterministic improvements 3. Lee-Carter family Stochastic 4. Parametric time-series modelling www.atp.dk 4
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 www.atp.dk 5
Levetidsmodellering: SAINT Mortality modelling Parametric time-series modelling assume functional form of (population) mortality, i.e. μ(t,x) = F(θ t ,x), - e.g. Makeham or logistic period life tables time-series model for (low- dimensional) parameter vector (θ t ) - - easy to fit and typically provides good description of data provides no insight into what causes the drift in (θ t ) - www.atp.dk 6
Levetidsmodellering: SAINT 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 4% m(t) = a + b t + ε t 3% Death rate (m) m(t) 1/2 = a + b t + ε t 2% log m(t) = a + b t + ε t log m(t) = a + b t + c t 2 + ε t 1% www.atp.dk 1950 2000 2050 7
Levetidsmodellering: SAINT Simple projections lack structure and robustness Danish female mortality 100% Age Reasonable short-term projections 100 10% 90 80 Death rate 1% 70 60 0.1% 50 0.01% 40 30 20 1990 1950 2000 2050 2100 Year Implausible long-term projections lacking (biological) structure www.atp.dk 8
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 www.atp.dk 9
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 age E(t,x) = exposure (”years lived”) x+1 Death rate, D(t,x)/E(t,x), is an estimate of (the x average of) underlying intensity, μ(t,x) Death probability, q(t,x) = 1-e - ∫μ(t,x) ≈ ∫μ(t,x) t+1 t time www.atp.dk 10
Levetidsmodellering: SAINT Danish fluctuations around stable international trend Danish and international female mortality 100% Age Danish life expectancy 100 Small improvements among the highest in at the highest ages 90 the world 10% 80 Death rate 70 1% 60 Is this the beginning 50 of a catch up period? 0.1% 40 30 20 Denmark falling behind the international trend 1940 1950 1960 1970 1980 1990 2000 Year www.atp.dk 11
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 www.atp.dk 12
Levetidsmodellering: SAINT Homogeneous cohort – no selection x ( x ) e Gompertz-Makeham intensity: 1000 25% 20% 800 Population size 15% Intensity ( μ ) 600 10% 400 200 5% 0% 0 0 20 40 60 80 100 www.atp.dk 13 (x) Age
Levetidsmodellering: SAINT Selection effects within a cohort x x ( ) E ( | ) Individual: ( x ) ; z z e Cohort: x Z x e 1000 25% ( x , 2 ) 20% 800 ( x , 1 ) Population size 15% 600 Intensity ( μ) ( x ) 1 ( x , ) 10% 400 2 200 5% 0% 0 0 20 40 60 80 100 www.atp.dk 14 (x) Age
Levetidsmodellering: SAINT Trend model Generalize Gompertz-Makeham intensity to allow for time-dependent cumulative and instant factors Underlying individual intensities t ( , ; ) ( ) exp( ( , ) ) ( ) t x z z t g s s t x ds t t x ”treatment” level ( ) exp( ( )) t t t 1 2 0 ”wear - out” rate ( , ) ( ) ( ) g t x t t x x 1 2 0 3 0 ( ) exp( ( )) t t t ”accident” rate 1 2 0 mean 1 and variance σ 2 Γ -distributed frailties, z - This yields an 8-parameter trend model for population intensity 1 t u t Previous values of κ (and g) g g 2 ( , ) ( ) 1 ( ) ( ) t x e t e u du t t x t x are ”remembered” by the t x www.atp.dk population due to selection 15
Levetidsmodellering: SAINT Structure of individual intensity Common and separate components of individual intensities t t x 1 ( , ; ) ( ) exp( ) ( ) t x z z t g g t 2 2 2 2 t t 2 1 t x ( , ; ) ( ) exp( ) ( ) t x z z t g t 1 1 1 1 t 1 t t t 2 1 www.atp.dk 16
Levetidsmodellering: SAINT Rate of improvement Individual and population intensity t g ( , ; ) ( ) ( ) t x z z t e t t x t g ( , ) [ | , ] ( ) ( ) t x E Z t x t e t t x Senescent component of rate of improvement t g ) log( ( , ) ( )) log( [ | , ]) log( ( ) t x t E Z t x t e t x ( , ) t x s t t t for x t 2 2 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) www.atp.dk 17
Levetidsmodellering: SAINT Improvement rates – international trend Female ( σ =0.43, β 2 lille) Male ( σ =0.26, β 2 stor) www.atp.dk 18
Levetidsmodellering: SAINT Trend estimated from international data Maximum likelihood estimation based on Poisson-model x+1 ( , ) ~ Poiss ( ( , ) ( , )) D t x t x E t x INT INT INT 4 x ( , ) ( , ) ( , 1 ) ( 1 , ) ( 1 , 1 ) / t x t x t x t x t x INT t+1 t Estimates (t 0 =2000, x 0 =60) σ α 1 α 2 β 1 β 2 β 3 γ 1 γ 2 Female -8.78e0 9.90e-2 -1.18e1 -8.90e-2 4.29e-1 -1.85e-2 4.79e-6 1.31e-3 Male -1.06e1 1.06e-1 -7.52e0 -2.50e-2 2.62e-1 -1.78e-2 8.37e-5 5.59e-5 www.atp.dk 19
Levetidsmodellering: SAINT Trend – fit and forecast International female mortality 100% Increasing old age rate Age of improvement 10% 100 90 Death rate 1% 80 70 0.1% 60 50 0.01% 40 Early, young age rate General, long-term rate 30 of improvement = 9.1% of improvement = 1.8% 20 1950 2000 2050 2100 www.atp.dk 20 Year
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