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Bayesian models for area health and mortality variations; an overview Peter Congdon, Centre for Statistics and Dept of Geography, QMUL. Bayesian methods have played major role in recent developments in statistical models for spatial data.

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Bayesian models for range wellbeing and mortality varieties; an outline Peter Congdon, Center for Statistics and Dept of Geography, QMUL Bayesian strategies have assumed real part in late improvements in factual models for spatial information. Numerous Bayesian applications have happened in spatial the study of disease transmission – books by Lawson et al (1999), Elliott et al (2000). Other application regions: spatial econometrics – see Lesage (1999); and geostatistics - see Diggle et al (JRSS,1998), Banerjee et al (2004)

Discrete zone (\'cross section\') system prevails in malady mapping utilizing regulatory information As a part of geostatistics (consistent spatial structure) objective is frequently spatial addition (kriging) between readings at tested areas. My discussion focuses on discrete zones, wellbeing applications, & WINBUGS usage of discrete spatial models Consider issues, for example, identifiability, decisions for priors, demonstrate niggardliness, parameter interpretability, multivariate & communication impacts, legitimacy of generally made presumptions, inadequate information, and so forth

Model for information y=(y 1 ,..y n ) starts with probability p(y| ). Derivation about parameters consolidates information in earlier with confirmation from information; molding on information evident in back thickness for parameters, p(|y) p(y|)p() The earlier is decided to (a) condense existing learning; might be "noninformative" (b) be reasonable for type of parameter; e.g. gamma earlier for backwards changes (precisions) c) for accumulation of parameters (e.g. irregular zone impacts) earlier may indicate type of relationship or on the other hand autonomous "replaceable" parameters BAYESIAN METHOD

Bayesian Hierarchical Models In various leveled models including dormant information Z (e.g. irregular impacts, missing perceptions), the joint thickness has the frame p(y, ,Z)=p(y| ,Z)p( ,Z)= p(y| ,Z)p(Z| )p( ). Illustrations: models with spatially connected irregular impacts, multilevel models, longitudinal models Posterior is p( ,Z|y) unless arbitrary impacts coordinated out. Bayes approach by means of MCMC simpler if arbitrary impacts held to characterize finish information probability p(y| ,Z)

MCMC Independent examining from back thickness ( )=p( |y) or ( ,Z )=p( ,Z|y) not normally possible. Markov Chain Monte Carlo (MCMC) techniques produce subordinate draws (t) or { (t), Z (t) } by means of Markov chains characterized by the supposition p( (t) | (1) ,.. (t-1) ) = p( (t) | (t-1) ), so that exclusive going before state important to future state. Testing from such a Markov tie focalizes to stationary circulation if extra prerequisites on chain are fulfilled.

CANONICAL MODEL FOR LATTICE DATA Let Y i and E i indicate watched & expected ailment cases in territories i, i =1,..., n . On the off chance that infection is uncommon or region populaces are little, may accept Y i ~ Poisson( i ); i = E i i where i is relative sickness chance; E i is normal occasions (Implicitly expect multiplicative age & range impacts – no age x territory connections) To take into account known covariates X i and Poisson over‑dispersion, basic to speak to relative dangers in log‑linear shape.

Thus: log i = + X i + U i where speaks to log relative danger of ailment crosswise over whole locale; U i speaks to effect of idle (imperceptibly) chance elements Could make U spatially associated: however imports solid earlier conviction Or could make U autonomous of space ("interchangeable") over zones paying little heed to contiguousness examples or separations between territories

Compromise between spatial & non-spatial impacts (otherwise known as convolution earlier) Besag, York, Mollie (1991) or BYM model is trade off between these two outrageous potential outcomes. In this way U i =h i +s i h i catch unstructured heterogeneity, and s i are spatially reliant Assume h i ordinary with general zero mean h i N(0, 2 h )

For s i , BYM display expect contingent autoregressive (CAR) reliance over ranges. Earlier for region i is restrictive on residual impacts (territories 1,..i-1,i+1,..n) meant s [i] . So s i| s [i] ~ N( An i ,V i ) An i is weighted normal of outstanding impacts: An i = j c ij s j/ j c ij = j w ij s j Variances are V i = 2 s/ j c ij rely on upon contingent difference 2 s & on spatial association structure spoke to by c ij .

Spatial Interactions Typical structures for c ij are parallel contiguousness: c ij =1 if ranges i and j are neighbors, c ij =0 something else (and c ii =0) remove rot: c ij =exp(- d ij ) where >0, d ij are separations between region focuses

Problems with CAR earlier and convolution display Only aggregate of h i and s i is recognized. Apportioning of aggregate lingering change between unstructured & spatial variety might be influenced by earlier details. Priors on fluctuations { 2 s, 2 h } or their inverses particularly applicable CAR earlier does not indicate normal level of s i , just their disparities. Need to focus them amid MCMC examining Two arrangements of impacts not tightfisted in displaying terms Poor distinguishing proof with diffuse priors on fluctuations (close shamefulness in back)

Other Options Introduce additional parameter ("legitimate CAR" earlier) s i| s [i] ~ N( A i ,V i ) in range [0,1] Interpretation issues in regards to and with constraining case =0. Not a direct spatial relationship parameter

LLB Spatial Compromise Prior Leroux, Lei, Breslow (1999) r i | r [i] ~ N(a i ∑ j≠i r j ,V i ) an i =λ/(1-λ+λ∑ j≠i c ij ) V i = 2 r/(1-λ+λ∑ j≠i c ij ) Reduces to unstructured heterogeneity when =0. Under twofold nearness, c ij =1 if regions {i,j} adjoining, M i =number of regions alongside territory i, an i =λ/(1-λ+λM i ) V i = 2 r/(1-λ+λM i )

Joint earlier under LLB (r 1 ,… ,r n )~N n (0, 2 r D - 1 ) D=R+(1-)I r ii = - ∑ j≠i c ij , r ij =c ij (i ≠j) Adaptive adaptation (spatial reliance shifts over guide), logit( λ i ) typical with obscure mean and fluctuation D= Λ R+(I- Λ )I, Λ =diag( λ i )

Worked Example Elevated blood level readings y i among youngsters (under 72 months) tried in 133 areas of Virginia in 2000 - see http://www.vahealth.org/leadsafe/dataclpp.htm Numbers examined T i change extensively (from 1 to 3808). Spatial structure is parallel, with c ij =1 for intercounty separates under 50km and c ij =0 something else.

n=133, NN=1056

Compare BYM & LLB for these information Assume binomial testing with y i Bin(T i ,π i ), one alternative is BYM convolution demonstrate, logit(π i )=α+s i +h i restrictive change δ s for CAR spatial impacts s i ; difference δ h for unstructured impacts. Share of aggregate fluctuation because of unstructured change is δ h/[δ h +V s ]; V s =var(s i ) is minimal difference of spatial eeffects figured at every emphasis For BYM show, priors are 1/δ s ~Ga(1,0.001) and g=δ h/(δ h +V s )~U(0,1). Undifferentiated from uniform shrinkage earlier ( Daniels, Canadian J. Insights, 1999)

LLB demonstrate for binomial information Compared to Leroux et al (1999) display with logit(π i )=α+r i r i | r [i] ~ N(a i ∑ j≠i r j ,V i ) an i =λ/(1-λ+λM i ) U(0,1) Adaptive form additionally with logit( i )~N( μ λ , τ λ )

Fit & Model Checking Use aberrance data paradigm, curtailed DIC, and log pseudo peripheral probability, log(psML), to analyze models Model checking conceivable utilizing appraisals of restrictive prescient ordinates (CPOs) see Congdon, Bayesian Statistical Modeling 2 nd ed (Wiley, 2006) for additional on these measures

DIC Deviance Information Criterion (DIC) incorporated with WINBUGS (Spiegelhalter et al, 2002, JRSSB) Benefit of DIC approach is in giving measure of model unpredictability p e (otherwise known as "viable parameter number"). Additionally reflects level of heterogeneity in irregular impacts (if all impacts non-noteworthy p e will be low)

BYM Model Code demonstrate {for (i in 1:133) {y[i] ~dbin(p[i],T[i]) logit(p[i]) <- alph+s[i]+h[i] ; h[i] ~dnorm(0,inv.delta.h); # survey noteworthiness of individual spatial impacts p.sig[i] <- step(s[i]) # log-probability L[i] <- logfact(n[i])- logfact(y[i])- logfact(r[i])+y[i]*log(p[i])+r[i]*log(1-p[i]) # Estimate log(CPO) for every region by taking less logs of back method for reverse probabilities G[i] <- 1/exp(L[i]); r[i] <- T[i]-y[i]}

Rest of code for BYM display s[1:133]~ car.normal(adjcodes[], wneigh[], numneigh[], inv.delta.s) for (j in 1:1056) {wneigh[j] <- 1} # earlier on remaining var as propn of aggregate var g ~dunif(0,1); V.s <- pow(sd(s[]),2); delta.h <- g*V.s/(1-g); inv.delta.h <- 1/delta.h; inv.delta.s ~dgamma(1,0.001); alph ~dflat()}

BYM Results Inferences from cycles 1000-50,000 of 2 chain run. 15 from 133 s i parameters critical: back probabilities Pr(s i >0|y) > 0.95 or < 0.05. Normal aberrance (- 2*logLKD) is 501 DIC and p e are 577 and 76 individually. log pseudo minimal probability is - 315.4 CPOs differ from 0.001 to 0.945 g evaluated at 0.42.

LLB Model Code display {for (i in 1:133) {y[i] ~dbin(p[i],T[i]) p.sig[i] <- step(r[i]); logit(p[i]) <- alph+r[i] .. r[i] ~dnorm(R[i],inv.V[i]); inv.V[i] <- inv.delta * (1-lam+lam*M[i]) R[i] <- (lam/(1-lam+lam*M[i]))*sum(Wr[C[i]+1:C[i+1] ])} # blunder vector over neighbors for (i in 1:1056) {W