Bayesian modeling of inseparable space-time variation in disease risk

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1 Bayesian modeling of inseparable space-time variation in disease risk Leonhard Knorr-Held Laina Mercer Department of Statistics UW May 23, 2013

2 Motivation Area and time-specific disease rates Area and time-specific disease rates are of great interest for health care and policy purposes Facilitate effective allocation of resources and targeted interventions Sample size often too small at granular space-time scale for reliable estimates Bayesian approach to borrow strength over space and time to improve reliability 2

3 Motivation Ohio Lung Cancer Example Lung Cancer Mortality Rates 1972 [0,1.5) [1.5,3.01) [3.01,4.51) [4.51,6.01] 3

4 Motivation Ohio Lung Cancer Example Ohio Lung cancer mortality by county ages Raw Lung Cancer Deaths per Year 4

5 Background Previous approaches Previous works have used a Hierarchical Bayesian framework to expand purely spatial models by Besag, York, and Mollie (1991) to a space time framework. Bernardinelli et al. (1995) Area-specific intercept and temporal trends All temporal trends assumed linear Waller et al. (1997) Spatial model for each time point No spatial main effects Knorr-Held and Besag (1998) Included spatial and temporal main effects Does not allow for space time interactions 5

6 Motivation Ohio Lung Cancer Example Ohio lung cancer mortality by county Lung Cancer Deaths per 1, Year 6

7 Motivation Ohio Lung Cancer Example Ohio lung cancer mortality by county Lung Cancer Deaths per 1, Year 7

8 The Model The Set Up Address the situation where the disease variation cannot be separated into temporal and spatial main effects and spatio-temporal interactions become and important feature. n it - persons at risk in county i at time t. y it - cases or deaths in county i at time t y it π it Bin(n it, π it ) ( ) η it = log πit 1 π it 8

9 The Model Stage 1 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 9

10 The Model Stage 2 - Exchangeable Effects The exchangeable effects γ (for time) and φ (for space) we are assigned multivariate Gaussian priors with mean zero and precision matrix λk: ( ) p(γ λ γ ) N 0, 1 λ γ Kγ 1 p(φ λ φ ) N where K γ = I TxT and K φ = I kxk. ( ) 0, 1 λ φ K 1 φ 10

11 The Model Stage 2 - First Order Random Walk Temporally structured effect of time α t is assigned a random walk. ( ) 1 α t α t, λ α N 2 (α t 1 + α t+1 ), 1/(2λ α ) Also represented as p(α λ α ) exp ( λα 2 αt K α α ) where K α =

12 The Model Stage 2 - Intrinsic Autoregressive (ICAR) Spatially structured effect of area θ i is assigned θ i θ i, λ θ N 1 1 θ j, m i m i λ θ j:j i where m i is the # of neighbors. ( The improper ) joint distribution can be written as p(θ λ θ ) exp λ θ 2 θ T K θ θ, where m i i = j K θ,ij = 1 i j 0 otherwise 12

13 The Model Stage 2 - iid prior for δ Type I Independent multivariate Gaussian prior the joint distribution is δ it δ it, λ δ N (0, 1/λ δ ) ( ) p(δ λ δ ) exp λ δ 2 δ T K δ δ where K δ = K φ K γ = I kt kt. 13

14 The Model Stage 2 - Random Walk prior for δ Type II Temporal trends differ by area - first order random walk prior ( ) 1 δ it δ it, λ δ N 2 (δ i,t 1 + δ i,t+1 ), 1/(2λ δ ) The improper joint distribution can be expressed as ( ) p(δ λ δ ) exp λ δ 2 δ T K δ δ where K δ = K φ K α (rank k(t 1)). 14

15 The Model Stage 2 - ICAR prior for δ Type III Spatial trends differ over time - Intrinsic autoregression prior δ it δ it, λ δ N 1 1 δ jt, m i m i λ δ j:j i The improper joint distribution can be expressed as ( ) p(δ λ δ ) exp λ δ 2 δ T K δ δ Where K δ = K θ K γ (rank (k 1)T ). 15

16 The Model Stage 2 - Space-Time prior for δ Type IV Spatio-temporal interaction - conditional depends on first and second order neighbors δ it δ ( it, λ δ 1 N 2 (δ i,t 1 + δ i,t+1 ) + 1 m i δ jt 1 m i j:j i j:j i (δ j,t 1 + δ j,t+1 ), 1 2m i λ δ ) The improper joint distribution can be expressed as ( ) p(δ λ δ ) exp λ δ 2 δ T K δ δ Where K δ = K θ K α (rank (k 1)(T 1)). 16

17 The Model Stage 3 - Hyperpriors Hyperparameters were all assigned λ Gamma(1, 0.01) resulting in a convenient posterior distribution. For example the full conditional for λ δ is: λ δ δ Gamma ( rank(k δ ), δ k δ δ ) where rank(k δ ) depends on the interaction type. 17

18 Model Evaluation Posterior Deviance To compare fit and complexity of each model the saturated deviance was calculated based on 2,500 samples from the posterior. D (s) = 2 where π (s) n i=1 t=1 ( T y itlog ( ) it = exp η (s) it ( 1+exp η (s) it ). y it n it π (s) it ) + (n it y it )log n it y ( it ) n it 1 π (s) it 18

19 The Model Implementation Knorr-Held (2000) employed Markov chain Monte Carlo to sample from the implied posterior distributions. Univariate Metropolis steps were applied for each parameter and hyperparameters were updated with samples from their full conditionals. MCMC in R: an update for every parameter in an interaction model takes s. Note: in INLA the main effects and interaction models type I-III all fit in less than 20min. 19

20 Ohio Lung Cancer Posterior Distribution of the Deviance Main Effects Posterior Deviance Type I Posterior Deviance Type II Posterior Deviance Type III Posterior Deviance Type IV Posterior Deviance 20

21 Ohio Lung Cancer Posterior Distribution of the Deviance Model Median Mean IQR SD Main Effects Type I Type II Type III Type IV Table: Laina s values in black and Knorr-Held (2000) in gray. 21

22 Ohio Lung Cancer Type I interaction Ohio lung cancer mortality by county Type I Interactions Lung Cancer Deaths per 1, Year 22

23 Ohio Lung Cancer Type II interaction Ohio lung cancer mortality by county Type II Interactions Lung Cancer Deaths per 1, Year 23

24 Knorr-Held (2000) Conclusions & Critique Provided and motivated a flexible approach for modeling space-time data. Was thin on MCMC details and diagnostics. Did not motivate use of deviance over DIC or p D for model selection. Focussed exclusively on non-parametric smoothing approaches. No discussion of incorporating covariates. 24

25 Thank you! Thank you all for your feedback and support throughout the quarter. Specifically, I would like to thank: William for suggesting a hair cut and shaded plots, Bob for suggesting enthusiasm, and Jon for suggesting this paper! 25

26 References Bernardinelli, L., D. Clayton, C. Pascutto, C. Montomoli, M. Ghislandi, and M. Songini (1995). Bayesian analysis of space-time variation in disease risk. Statistics in Medicine 14(21-22), Besag, J., J. York, and A. Mollie (1991). Bayesian image restoration, with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics 43, Knorr-Held, L. (2000). Bayesian modelling of inseperable space-time variation in disease risk. Statistics in Medicine 19, Knorr-Held, L. and J. Besag (1998). Modelling risk from a disease in time and space. Statistics in Medicine 17, Knorr-Held, L. and H. Rue (2002). On block updating in markov random field models for disease mapping. Scandinavian Journal of Statistics 29, Rue, H. and L. Held (2005). Gaussian Markov Random Fields Theory and Applications. Number 104 in Monographs on Statistics and Applied Probability. Chapman and Hall. Schrodle, B. and L. Held (2010). Spatio-temporal disease mapping using inla. Environmetrics. Waller, L. A., B. P. Carlin, H. Xia, and A. E. Gelfand (1997). Hierarchical spatio-temporal mapping of disease rates. Journal of the American Statistical Association 92(438),

27 Ohio Lung Cancer Type III interaction Ohio lung cancer mortality by county Type III Interactions Lung Cancer Deaths per 1, Year 27

28 Ohio Lung Cancer Type IV interaction Ohio lung cancer mortality by county Type IV Interactions Lung Cancer Deaths per 1, Year 28

29 Knorr-Held (2000) Next Steps A logical next step was to make fitting these models much faster. Knorr-Held and Rue (2002) introduced block updating Rue and Held (2005) great overview of Gaussian Markov Random Fields and more details on block updating Schrodle and Held (2010) describes (poorly) how to fit models from Knorr-Held (2000) in INLA. 29

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