Andrew B. Lawson 2019 BMTRY 763

Transcription

1 BMTRY 763

2 FMD Foot and mouth disease (FMD) is a viral disease of clovenfooted animals and being extremely contagious it spreads rapidly. It reduces animal production, and can sometimes be fatal in young stock. Counts of new cases of FMD were available for parishes within the county of Cumbria, Northern England for the period February 2001-August The data were available as half monthly counts and so there were 13 time periods available for analysis.

3 FMD incidence NW England (Cumbria) 2001 Infected premises (IPs) are to be modeled Within parishes we have counts of IPs and we also have a record of the total number of premises which changes over time. 138 parishes and 13 half-monthly time periods (February 2001-August 2001)

4 FMD first 6 time periods case/pop ratios (row-wise)

5 Data example: FMD in Cumbria Foot and Mouth outbreak in 2001 in Cumbria, UK 138 parishes in area Time period: single biweekly count of Ips Files: FMD_case_parish_data.txt FMD_INLA_Rcode.txt FMD_spatial_worked_examples_Rcode_ABDM.txt

6 FMD (FMDonePERIOD_data.txt) FMDframe<-list(n=138, count=c(1,0,1,0,2,3,1,0,1,0,3,2,0,3,1,6,1,10,2,1,8,4,4,1,1, 3,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0, 0,0,0,1,1,1,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0)) FMDmap<-readSplus("FMD_splusMAP.txt") plot(fmdmap)

7 Polygon Plot

8 FMD data Counts in parishes (y) Population of premises in parishes (n) Crude rate (y/n) crude rate map [0,0.17)(131) [0.17,0.33)(1) [0.33,0.5)(3) [0.5,0.67)(2) [0.67,0.83)(0) [0.83,1](1)

9 Possible descriptive Model A binomial model may be assumed A logit link also y logit( p ) α v bin( n, p ) i i i i N τ i = α + 1 v 0 (0, ) N(0, τ ) v i

10 INLA code ind<-seq(1:138) formula1<-count~1+f(ind,model="iid,param=c(2,1)) res1<inla(formula1,family="binomial",data=fmdframe,ntrial s=pop,control.compute=list(dic=true)) summary(res1)

11 Results: UH component UH<-res1$summary.random$ind[,2] fillmap(fmdmap,"posterior mean UHcomponent",UH,n.col=4)

12 UH component posterior mean UH component [-0.94,0.64)(108) [0.64,2.23)(9) [2.23,3.81)(13) [3.81,5.4](7)

13 Results: residuals fit<-res1$summary.fitted.values$mean resid<-count-fit fillmap(fmdmap,"estimated crude residuals",resid,n.col=4)

14 Residuals estimated crude residuals [-0.01,2.45)(118) [2.45,4.92)(7) [4.92,7.38)(1) [7.38,9.84](3)

15 Results: infection probability linpred<-res1$summary.linear.predictor$mean prob<-exp(linpred)/(1+exp(linpred)) fillmap(fmdmap,"posterior mean infection probability",prob,n.col=4)

16 posterior mean infection probability Infection probability [0,0.12)(129) [0.12,0.24)(4) [0.24,0.36)(2) [0.36,0.48](3)

17 Results: DIC LOCdic<- res1$dic$local.dic fillmap(fmdmap,"local DIC",LOCdic,n.col=5)

18 DIC local DIC [0.21,1.35)(107) [1.35,2.5)(1) [2.5,3.65)(13) [3.65,4.8)(10) [4.8,5.94](6)

19 Space Time Modeling

20 Space and time considerations Often we need to consider both spatial and temporal effects in disease data Could have a location and date of diagnosis OR could have counts of disease within small areas and fixed time periods The second of these is more common and more aggregated I will only consider this latter situation here

21 Ohio county level respiratory cancer 10 years A well known dataset (full dataset 21 years ) shown here SIRs displayed

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23 Space-time (ST) Modeling Some notation Assume counts within fixed spatial and temporal periods: map evolutions Both space and time are subscripts in the analysis Consider separable models (with spatial and separate temporal terms) Also interaction effects

24 Notation outcome : y ; RRisk: ij θ ij expected count: e ij i = 1,..., m: small areas j = 1,..., J : time periods

25 Expected Counts Computation (simplest - overall average): e p. y / p = ij ij ij ij i j i j where p ij is the population of the ij th unit

26 Basic retrospective model Infinite population; small disease probability Poisson assumption y ~ Pois( e θ ) ij ij ij log( θ ) = α + S + T + ST S T i j ST ij 0 i j ij : spatial terms : temporal terms ij : interaction

27 Full data set: 21 years of Ohio lung cancer 10 years of SMRs standardized with the statewide rate: Frequently analyzed Row wise from 1979

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29 Some Random Effect models model 1a: log( θ ) = α + v + u + βt model 1b: ij 0 i i j log( θ ) = α + v + u + γ model 2: ij 0 i i j log( θ ) = α + v + u + γ + γ model 3: ij 0 i i 1j 2 j log( θ ) = α + v + u + γ + ψ model 4: ij 0 i i 2 j ij log( θ ) = α + v + u + γ + γ + ψ ij 0 i i 1j 2 j ij model 5: variants of (3) with ψ ij

30 Simple separable models Spatial components (BYM) plus linear time trend model 1a: log( θ ) = α + v + u + βt t v j i ij 0 i i j : time of j th period τ 1 (0, v ) u N u τ n i N 1... ( δ, u / δ ) i i

31 Simple trend models Spatial components+ random time effect model 1b: log( ) θ α γ = + v + u + ij 0 i i j

32 Random Walk Prior distribution Model 1 b: we assume a random effect for the time element and this has a random walk prior distribution: j 1 1 γ γ N ( γ, τ ) j More generally an AR1 prior could be used: 1 < j 1 γ γ N ( λγ, τ ); 0 λ 1 j

33 Simple space-time models Two random time effects: model 2: log( θ ) = α + v + u + γ + γ γ 1 j ij 0 i i 1j 2 j N 1 (0, τ ) γ j N( 2 j 1, ) γ γ τ γ 2

34 Interaction models Separable models can be enhanced by using additive interaction terms Model 3: log( ) θ α γ ψ = + v + u + + ij 0 i i 2 j ij

35 Interaction priors A variety of priors for the interaction can be assumed (both correlated and non-separable) Knorr-Held (2000) first suggested dependent priors (see Lawson (2013) ch12) Two simple separable examples of possible priors are: ψ ij N(0, τ ) uncorrelated (model 3) ψ ~ N( ψ, τ ) random walk (model 5) ij ψ i, j 1 ψ

36 Interaction models Model 4, as for (3) but with added uncorrelated tiem effect Model 5, as for (3) but with a correlated interaction prior in time model 4: log( θ ) = α + v + u + γ + γ + ψ ij 0 i i 1j 2 j ij model 5: variants of (3) with ψ ij

37 Model fitting Results (WinBUGS) Model DIC pd 1a b

38 Interpretation The temporal trend model does not provide a better fit than the random walk (1a, 1b) The extra RE in model 2 is not needed The inclusion of the interaction in model 3 is significant but model 4 is not good Model 5 with the random walk interaction seems best as it has lowest DIC and smaller pd than model 3

39 Model fitting on INLA Data setup: often space time data is in the form of a matrix: y(i,j), e(i,j) Rows are small areas (eg counties, parishes etc) Columns are time units (eg years, or months days etc) Ohio data: i=1,,88; j=1,,10 Matrix of 88 x 10 dimension For INLA use, it is convenient to reformat the data so that an individual small area represents a row, but the repeated measurements on the small area are repeated rows Long form format

40 R Code For matrices read in with 88 x 10 structures (in R) then the code for conversion is yl<-rep(0,880) el<-rep(0,880) T=10 for (i in 1:88){ for (j in 1:10){ k<-j+t*(i-1) yl[k]<-y[i,j] el[k]<-e[i,j]}}

41 Index setup Indices can now be set up to address different effects: year<-rep(1:10,len=880) region<-rep(1:88,each=10) region2<-region ind2<-rep(1:880)

42 OHIO_Stmapping_INLA_Rcode.txt All models are described in this file. Data is stored in OHIO_STmapping_INLA_RcodeFAL.txt Paste contents of file into R This will yield y, e, and indicator variables in the data.frame called data

43 Models fitted on INLA Spatial only (UH) Spatial only (UH+CH) UH +CH +time trend model 1a UH+CH +time iid UH+CH+time rw1 model 1b UH+CH+time (iid, rw1) model 2 UH +time rw1 +Stint UH+CH+time rw1 +Stint model 3

44 DIC comparison INLA model Model DIC pd WB model 1 Spatial only (UH) UH+CH UH+CH+time trend a 4 UH+CH+time iid UH+CH+time RW1 6 UH+CH+time (iid, rw1) 7 UH+time rw1+st int 8 UH+CH+time rw1+stint b

45 Results: INLA model 7 UH [-0.6,-0.36)(2) [-0.36,-0.12)(16) [-0.12,0.12)(46) [0.12,0.36](23)

46 Results: INLA model 7 time effect yearr time

47 Break

48 FMD modeling in space-time Load data: FMD_case_parish_data.txt INLA code in: FMD_INLA_ST_Rcode.txt

49 Model Development Data is counts within parishes in 13 time periods We also have current population of premises A binomial model may be appropriate y Bin( p, n ) ij ij ij logit( p ) = f ( y ) + f ( predictors) + f (R Es) ij 1 i, j A Poisson model could also be considered if we modulate the Poisson mean with population y ij Pois( μ ) ij log( μ ) = f ( n ) + f ( y ) + f ( predictors and REs) ij 1 ij 2 i, j 1 3

50 Simpler Poisson Model We will look at the Poisson model Also only use random effects Lagged dependencies can be accommodated using the copying facility in INLA. We will only have lagged REs

51 Models Purely spatial (UH+CH) pl offset Purely spatial (UH+CH) pl predictor UH+CH+ pl predictor+ time RW1 UH+CH+ pl offset+ time RW1 UH+CH+ pl predictor+ time RW1+STint

52 Model Description DIC pd Results 1a UH+CH+ pl offset DICs 1b UH+CH+ pl predictor UH+CH+ pl predictor + time RW UH+CH+ pl offset + time RW UH+CH+ pl predictor + time RW1+ STint

53 Model 4 This model has spatial and temporal effects with a pl predictor and ST interaction y ij Pois( μ ) ij μ = exp{ α + α n + v + u + γ + ψ } ij 0 1 ij i i j ij NB lagged effects in count and population not included BUT a lagged dependency random effect is

54 Results UH [-0.73,0.05)(76) [0.05,0.82)(47) [0.82,1.6)(13) [1.6,2.38](1)

55 Results CH [-1.45,-0.62)(31) [-0.62,0.2)(56) [0.2,1.03)(35) [1.03,1.85](16)

56 Results time<-seq(1:13) plot(time,yearr) lines(time,yearr) yearr time

57 Results ST interaction period 1 ST interaction period 1 [-1.07,-0.45)(15) [-0.45,0.17)(93) [0.17,0.8)(10) [0.8,1.42)(6) [1.42,2.05)(4) [2.05,2.67](9)

58 Results: St interaction period 2 ST interaction period 2 [-1.13,-0.44)(13) [-0.44,0.26)(89) [0.26,0.95)(13) [0.95,1.64)(11) [1.64,2.34)(6) [2.34,3.03](6)

59 Finally Other INLA features Measurement error in predictors ( mec, meb) Missingness in outcomes (copy facility) Geographically weighted regression e.g. f(ind,x1,model= besag,graph= ) Smoothed predictors e.g. f(x1,model= rw1 ) Modeling point processes via SPDE facilities (LGCP)

60 INLA OpenBUGS Finally INLA versus OpenBUGS Runs on R x Only through Brugs or R2WinBUGS Large datasets Mixtures x x Posterior functionals? x Special spatial Models Missingness X some: LGCP for point processes Only outcomes in general, but can handle drop-out models X GeoBUGS +CAR models Can handle a range of missingness

61 Cautionary references A study has found that the default prior distributions for precisions in INLA can lead to very inaccurate precision estimates for random effects Carroll, R., Lawson, A. B., Faes, C., Kirby, R. S. and Aregay, M. (2015) Comparing INLA and OpenBUGS for hierarchical Poisson modeling in disease mapping. Spatial and Spatio-temporal Epidemiology, 14-15, See also Taylor, B. and Diggle, P. (2014) INLA or MCMC? A tutorial and comparative evaluation for spatial prediction in log-gaussian Cox Processes. Journal of Statistical Computation and Simulation, 84, 10, Teng, M., Nathoo, F.S., Johnson, T.D. (2017). Bayesian Computation for Log Gaussian Cox Processes: A Comparative Analysis of Methods. Journal of Statistical Computation and Simulation DOI: / So be warned!

62 Conclusions Thanks for your attention! Contact address: INLA examples given in Appendix D of Lawson, A. B. (2013) Bayesian Disease Mapping: hierarchical modeling in spatial epidemiology. 2 nd Ed CRC Press, New York And also chapter 15 of Lawson (2018) 3 rd Ed Full 2 x 1 x 2 day courses on BDM (including WinBUGS, INLA, CARBayes and Nimble) given in MUSC (March) Contacts: MUSC courses June Watson watsonju@musc.edu

63 Andrew B Lawson 2019