Smoothing and Forecasting Mortality Rates with P-splines. Iain Currie. Data and problem. Plan of talk

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1 Smoothing and Forecasting Mortality Rates with P-splines Iain Currie Heriot Watt University Data and problem Data: CMI assured lives : 20 to 90 : 1947 to 2002 Problem: forecast table to 2046 London, June 2006 Gompertz model Plan of talk P-splines in 1-dimension P-splines in 2-dimensions Lee-Carter model -Period-Cohort model 2-d P-splines Gompertz: log m =

2 Simple Gompertz Quadratic Gompertz 70 log m = log m = log m = ^2 Quadratic forecast Quadratic fit: linear & quadratic forecasts Quadratic forecast Linear forecast

3 In ordinary regression we use basis like Regression basis Linear basis: {1, x} Quadratic basis: {1, x, x 2 } Polynomial basis: {1, x, x 2,..., x p }. Means and fitted values are linear combinations of the basis functions log µ = a + bx, log ˆµ = â + ˆbx log µ = a + bx + cx 2, log ˆµ = â + ˆbx + ĉx 2 log µ = a + b 1 x b p x p, log ˆµ = â + ˆb 1 x ˆb p x p Generalised linear models Model fitting uses the Generalised Linear Model framework. d x P(Exµ c x ) and log Exµ c x = log Ex c + a + bx or log Exµ c x = log Ex c + a + bx + cx 2. A single cubic B spline B-spline basis A B-spline regression basis uses local basis functions. B-spline basis: {B 1 (x), B 2 (x),..., B p (x)} where B 1 (x), B 2 (x),..., B p (x) are B-splines. Means and fitted values work the same way p p log µ = θ j B j (x), log ˆµ = ˆθ j B j (x) 1 1 B spline

4 Cubic B spline basis B spline regression B spline log(mortality) 70 Number of B splines: B spline regression Penalties Eilers & Marx (1996) imposed penalties on differences between adjacent coefficients. log(mortality) 70 Number of B splines: 23 P 2 (θ) = (θ 1 2θ 2 + θ 3 ) (θ p 2 2θ p 1 + θ p ) 2 = θ D 2D 2 θ where D 2 is a difference matrix. P 2 (θ) is a roughness penalty. Estimation is via penalised likelihood P L(θ) = L(θ) 1 2 λθ D 2D 2 θ where λ is the smoothing parameter which balances fit and smoothness. Interpolation (λ = 0) Linear regression (λ = )

5 P spline regression Forecasting to 2046 log(mortality) 70 Number of B splines: 23 Quadratic penalty log(mortality) Forecast to 2046 Quadratic penalty: linear forecast Lee-Carter model Forecasting to 2046 Lee & Carter (1992) log(mortality) Delta knots = 2.75 Delta knots = 11 log µ ij = α i + β i κ j, 20 i 90, 1947 j 2002

6 Alpha Beta Kappa : 70 : 50 : : Lee Carter forecasts to : 80 Discrete Lee-Carter model log µij = αi + βiκj, 20 i 90, 1947 j 2002 or in table form log M = α1 + βκ Smooth Lee-Carter α Baa, β Bab, κ Byk

7 Alpha Beta : 70 Kappa Smooth Lee Carter: ages 40 to Period-Cohort model log µij = αi + βj + γi j, 20 i 90, 1947 j 2002 Parameter redundancy 2na + 2ny 1 = 253 parameters 2na + 2ny 4 = 250 free parameters : 50 : : 70 :

8 : 50 : : 70 : Discrete -Period-Cohort model log µij = αi + βj + γi j, 20 i 90, 1947 j 2002 Smooth -Period-Cohort model α Baa, β Byb, γ Bcc component Planar component Components of Improvement = 0.015/annum Cohort component component Cohort

9 Components of Components of Cohort Improvement = 0.7/10years Improvement = 0.29/10years Cohort 2-d modelling of mortality tables Let B a, 71 13, be a 1-d B-spline basis for modelling a single age. Let B y, 56 10, be a 1-d B-spline basis for modelling a single year. Can we combine the marginal bases B a and B y to make a 2-d basis?

10 Regression matrix 2-d modelling of mortality tables The Kronecker product organises the multiplication of the marginal bases to give the regression matrix B = B y B a, : : 50 Penalties in 2-d Each regression coefficient is associated with the summit of one of the hills. Smoothness is ensured by penalizing the coefficients in rows and columns : : 70 2 d mortality: ages 40 to Summary of results DEV TR BIC Rank 2-d APC smooth LC smooth APC LC

11 Summary for age 60 APC 2 d LC SmoothLC Summary for age 70 APC 2 d LC SmoothLC Summary for age 80 APC 2 d LC SmoothLC Summary for ages 60, 70, 80 APC 2 d LC SmoothLC

12 Conclusions Forecasting a mortality table depends on Model choice Forecasting method Parameter uncertainty Stochastic uncertainty References Lee-Carter models Lee & Carter (1992) J American Statistical Association, 87, Brouhns, Denuit & Vermunt (2002) Insurance: Mathematics & Economics, 31, Period-Cohort models Clayton & Schlifflers (1987) Statistics in Medicine, 6, Clayton & Schlifflers (1987) Statistics in Medicine, 6, Penalized spline models Eilers & Marx (1996) Statistical Science, 11, Currie, Durban & Eilers (2004) Statistical Modelling, 4, Currie, Durban & Eilers (2006) Journal of the Royal Statistical Society, Series B, 68,