Levetidsmodellering: SAINT Indhold Levetidsmodellering: SAINT-modellen Hvad er SAINT? Gængse levetidsmodeller Dødeligheden i små populationer SAINT-modellen - trend - spread Dansk Demografisk Forening Søren Fiig Jarner Resultater 27. januar 2010 Esben Masotti Kryger Implementering i ATP www.atp.dk 2 Levetidsmodellering: SAINT Levetidsmodellering: SAINT ATP’s mortality model Mortality models Main methodologies SAINT = Spread Adjusted InterNational Trend - describes small population mortality as temporary deviations from underlying trend 1. Expert judgment (data free) Deterministic - developed in-house in 2007 2. Deterministic improvements Stochastic mortality model Stochastic mortality model 3. Lee-Carter family - produce a range of possible, future evolutions of mortality intensities, μ (t,x) Stochastic - the mean forecast is used to calculate the tariff and set the reserve for POM 4. Parametric time-series modelling - 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 www.atp.dk 4 Levetidsmodellering: SAINT Levetidsmodellering: SAINT Mortality modelling Mortality modelling Lee-Carter (1992) Parametric time-series modelling - log μ (t,x) = a(x) + b(x)k(t) + noise - assume functional form of (population) mortality, i.e. μ (t,x) = F( θ t ,x), e.g. Makeham or logistic period life tables - assumes age-specific, constant, relative rates of improvement - time series model for (low dimensional) parameter vector ( θ ) time-series model for (low-dimensional) parameter vector ( θ t ) - conceptually simple; improvements driven by single index t ll i l i t d i b i l i d - easy to fit and typically provides good description of data - projections overly confident when based only on index variability - no structural limitations to the shape of mortality rates; - provides no insight into what causes the drift in ( θ t ) 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 www.atp.dk 6 1
Levetidsmodellering: SAINT Levetidsmodellering: SAINT Forecasting principle Simple projections lack structure and robustness Danish female mortality ”In the absence of additional information the best one can do is to extrapolate past trends” 100% Reasonable short-term projections Age - sounds sensible, but what does it actually mean? 100 10% 90 90 Model 80 4% Death rate m(t) = a + b t + ε t 1% 70 3% Death rate (m) m(t) 1/2 = a + b t + ε t 60 0.1% 50 2% log m(t) = a + b t + ε t 0.01% 40 log m(t) = a + b t + c t 2 + ε t 30 20 1% 1990 1950 2000 2050 2100 Implausible long-term projections lacking (biological) structure Year www.atp.dk 1950 2000 2050 7 www.atp.dk 8 Levetidsmodellering: SAINT Levetidsmodellering: SAINT Small population mortality Data and terminology Human Mortality Database (www.mortality.org) Modelling challenge: Produce plausible, long-term forecasts reflecting both the general pattern and the ”wildness” seen in data Danish and international female mortality from 1933 to 2005 - General pattern - mortality increases with age - 19 countries in the international dataset: USA, Japan, West Germany, UK, France, Italy, Spain, Australia, Canada, Holland, Portugal, Austria, Belgium, France Italy Spain Australia Canada Holland Portugal Austria Belgium - age-specific death rates decline over time f Switzerland, Sweden, Norway, Finland, Iceland & Denmark. - rates of improvement decrease with age Death counts and exposures for each year and each age group - rates of improvement for old age groups increase over time - Deviations D(t,x) = number of deaths age - substantial deviations from the general pattern E(t,x) = exposure (”years lived”) - even periods with increasing mortality for some age groups x+1 Death rate, D(t,x)/E(t,x), is an estimate of (the The SAINT model structure x average of) underlying intensity, μ (t,x) mortality = international trend + spread Death probability, q(t,x) = 1-e - ∫μ (t,x) ≈ ∫μ (t,x) t+1 t time www.atp.dk 9 www.atp.dk 10 Levetidsmodellering: SAINT Levetidsmodellering: SAINT Danish fluctuations around stable international trend Trend modelling concepts Danish and international female mortality Population dynamics 100% Age - Ensure consistent intensity surfaces over time and ages by Danish life expectancy aggregating individual intensities to population level 100 Small improvements among the highest in at the highest ages 90 - Individuals living in the same period of time are influenced by the world 0% 10 common as well as individual factors ll i di id l f t 80 Death rate - Factors have either a cumulative or an instant effect on mortality 70 1% Frailty (unobservable) 60 Is this the beginning 50 - People are genetically different. Only the more robust of a catch up period? 0.1% individuals will attain very high ages 40 30 - Lack of historic improvements among the very old may be due 20 Denmark falling behind to selection effects. In the future the frailty composition at old the international trend ages will change 1940 1950 1960 1970 1980 1990 2000 Year www.atp.dk 11 www.atp.dk 12 2
Levetidsmodellering: SAINT Levetidsmodellering: SAINT Homogeneous cohort – no selection Selection effects within a cohort ( x ) x Gompertz-Makeham intensity: e Individual: ( ; ) x Cohort: ( ) E ( | ) x x z z e x Z x e 1000 25% 1000 25% ( x , 2 ) 20% 20% 800 800 ( x , 1 ) 8 8 2 2 Population size Population size 15% Intensity ( μ ) 15% 600 600 ( x ) Intensity ( μ ) 1 ( x , ) 10% 2 10% 400 400 200 200 5% 5% 0% 0% 0 0 0 20 40 60 80 100 0 20 40 60 80 100 www.atp.dk 13 www.atp.dk 14 Age (x) Age (x) Levetidsmodellering: SAINT Levetidsmodellering: SAINT Trend model Structure of individual intensity Generalize Gompertz-Makeham intensity to allow for Common and separate components of individual intensities time-dependent cumulative and instant factors Underlying individual intensities t t x 1 t ( , ; ) ( ) exp( ) ( ) t x z z t g g t ( , ; ) ( ) exp( ( , ) ) ( ) 2 t x z z t g s s t x ds t 2 2 2 t x t t 2 1 ”treatment” level ( ) exp( ( )) t t t 1 2 0 ( , ) ( ) ( ) ”wear-out” rate g t x t t x x 1 2 0 3 0 t x ( , ; ) ( ) exp( ) ( ) ( ) exp( ( )) t x z z t g t t t t ”accident” rate 1 1 2 0 1 1 1 t 1 mean 1 and variance σ 2 Γ -distributed frailties, z - This yields an 8-parameter trend model for population intensity t t t 1 2 1 t u t g g Previous values of κ (and g) ( , ) ( ) 1 2 ( ) ( ) t x e t x t e t x u du t are ”remembered” by the t x www.atp.dk population due to selection 15 www.atp.dk 16 Levetidsmodellering: SAINT Levetidsmodellering: SAINT Rate of improvement Improvement rates – international trend Female ( σ =0.43, β 2 lille) Male ( σ =0.26, β 2 stor) Individual and population intensity 0.015 t 0.015 g ( , ; ) ( ) ( ) 0.010 t x z z t e t x t 0.010 t g 0.005 ( , ) [ | , ] ( ) ( ) t x E Z t x t e t x t 0.005 0.0 Senescent component of rate of improvement 0.000 0.000 x=40 x=80 x=40 x=80 x=60 x=100 x=60 x=100 t ) g log( ( , ) ( )) log( [ | , ]) log( ( ) 1600 1700 1800 1900 2000 2100 2200 1600 1700 1800 1900 2000 2100 2200 t x t E Z t x t e t x ( , ) t x Year Year s t t t 0.016 for x t 2 2 0.015 0.012 Two opposite effects in rate of improvement: 0.010 - more frail people become old (i.e. first term is negative - and vanishing) 0.008 0.005 - general mortality improvements (i.e. second term is positive) 0.004 t=1900 t=2100 t=1900 t=2100 t=2000 t=2200 t=2000 t=2200 www.atp.dk 17 www.atp.dk 20 40 60 80 100 20 40 60 80 100 18 Age Age 3
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