Bayesian modeling of inseparable space-time variation in disease risk Leonhard Knorr-Held Laina Mercer Department of Statistics UW May, 013
Motivation Ohio Lung Cancer Example Lung Cancer Mortality Rates 197 [0,8.36) [8.36,16. [16.73,5 [5.09,33
Stage 1 Address the situation where the disease variation cannot be separated into temporal and spatial main effects and spatio-temporal interactions become and important feature. η it = µ + α t + γ t + θ i + φ i + δ it µ - overall risk level α t - temporally structured effect of time t γ t - independent effect of time t θ i - spatially structured effect of county i φ i - independent effect of county i δ it - space time interaction 3
Stage - Independent Priors Prior for γ : ) pγ λ γ ) exp λγ T t=1 γ t ) = exp λγ γt K γ γ where K γ = I TxT. Prior for φ: where K φ = I nxn. pφ λ φ ) exp = exp λ φ λ φ n φ t i=1 ) ) φt K φ φ 4
Stage - First Order Random Walk α t+1 α t N 0, λ 1 α ), t = 1,..., T 1 5
Stage - First Order Random Walk α t+1 α t N 0, λ 1 α ), t = 1,..., T 1 pα λ α ) λ T 1 exp λα t=1 α exp λα αt K α α ) T 1)/ α T 1)/ = λ α t+1 α t ) ) 5
Stage - First Order Random Walk α t+1 α t N 0, λ 1 α ), t = 1,..., T 1 Where, pα λ α ) λ and has rank T 1. T 1 exp λα t=1 α exp λα αt K α α ) T 1)/ α T 1)/ = λ 1 1 1 1 K α =......... 1 1 1 1 α t+1 α t ) ) 5
Stage - First Order Random Walk With we have that pα λ α ) λ α t α t, λ α N T 1)/ α exp λ ) α αt K α α ) 1 α t 1 + α t+1 ), 1/λ α ) 6
Stage - ICAR Prior θ i θ j N 0, λ 1 θ ), for neighbors i and j 7
Stage - ICAR Prior θ i θ j N 0, λ 1 θ ), for neighbors i and j pθ λ θ ) λ n 1)/ θ exp = λ n 1)/ θ exp λ θ i j λ θ θ T K θ θ θ i α j ) ) ) 7
Stage - ICAR Prior θ i θ j N 0, λ 1 θ ), for neighbors i and j pθ λ θ ) λ n 1)/ θ exp = λ n 1)/ θ exp λ θ i j λ θ θ T K θ θ Where our precision matrix has elements: n i i = j K θ,ij = 1 i j 0 otherwise and has rank n 1. θ i α j ) ) ) 7
Stage - ICAR Prior 1 0 1 An example K θ = 1 0 1 0 0 1 1 1 1 1 3 8
Stage - ICAR Prior With we have that pθ λ θ ) λ n 1)/ θ exp λ ) θ αt K θ θ θ i θ i, λ θ N 1 1 θ j, n i n i λ θ j:j i 9
Stage - iid prior for δ Type I Independent - Independent multivariate gaussian prior ) n T pδ λ δ ) exp λ δ δ it ) i=1 t=1) = exp λ δ δ T K δ δ Where K δ = K γ K φ = I nt nt. 10
Stage - Random Walk prior for δ Type II Temporal trends differ by area - first order random walk prior ) n pδ λ δ ) exp λ T 1 δ δ it δ i,t 1 ) i=1 t=1 ) = exp λ δ δ T K δ δ Where K δ = K α K φ rank nt 1)). 11
Stage - ICAR prior for δ Type III Spatial trends differ over time - Intrinsic autoregression prior ) pδ λ δ ) exp λ T δ δ it δ jt ) i j t=1 ) = exp λ δ δ T K δ δ Where K δ = K γ K θ rank n 1)T ). 1
Stage - Space-Time prior for δ Type IV Spatio-temporal interaction - conditional depends on first and second order neighbors pδ λ δ ) exp λ ) δ T i j t= δ it δ jt δ i,t 1 + δ j,t 1 ) ) = exp λ δ δ T K δ δ Where K δ = K α K θ rank n 1)T 1)). 13
Implementation Difficulties Overall level can be absorbed by both θ and α in main effects model and interaction model I. recenter θ and α after each iteration to have mean zero omit µ and recenter θ or α Additional constraints for interaction models II, III, and IV: II recenter δ it row-wise across time) III recenter δ it column-wise over space) IV recenter δ it row-wise and column-wise 14
Example Ohio Lung Cancer without smoothing Ohio Lung cancer mortality by county 1968 1988 ages 55 64 Raw Lung Cancer Deaths per 10,000 0 10 0 30 40 50 1968 1970 197 1974 1976 1978 1980 198 1984 1986 1988 Year 15
Example Ohio Lung Cancer with main effects smoothing Ohio Lung cancer mortality by county 1968 1988 ages 55 64 Smoothed Lung Cancer Deaths per 10,000 10 15 0 1968 1970 197 1974 1976 1978 1980 198 1984 1986 1988 Year 16
Example Ohio Lung Cancer with main effects + space-time interaction smoothing Ohio Lung cancer mortality by county 1968 1988 ages 55 64 Smoothed Lung Cancer Deaths per 10,000 10 15 0 1968 1970 197 1974 1976 1978 1980 198 1984 1986 1988 Year 17
What s next? Currently coding all 5 models on a very small 5 nodes and 5 time points) graph with To do Winbugs INLA MCMC tuning in an automated fashion for univariate Metropolis updating block updating 18
The End Questions? 19
References Besag, J., P. Green, D. Higdon, and K. Mengersen 1995). Bayesian computation and stochastic system. Statistical Science 101), 3 66. Besag, J. and C. Kooperberg 1995). On conditional and intrinsic autoregression. Biometrika 84), 733 746. Knorr-Held, L. 000). Bayesian modelling of inseperable space-time variation in disease risk. Statistics in Medicine 19, 555 567. Knorr-Held, L. and J. Besag 1998). Modelling risk from a disease in time and space. Statistics in Medicine 17, 045 060. Rue, H. and L. Held 005). Gaussian Markov Random Fields Theory and Applications. Number 104 in Monographs on Statistics and Applied Probability. Chapman and Hall. 0