### The following OpenBUGS code is written for GMCAR full model, where phi_1|phi_2 ~ CAR(alpha_1, tau_1) ### with its conditional mean beta_1 + (eta_0 I + eta_1 W) (phi_2 - beta_2) and phi_2 ~ CAR(alpha_2, tau_2) with ### its mean beta_2. Here the conditioning order is phi_1 given phi_2 and phi_2. model { for( i in 1 : N ) { Y10[i] ~ dpois(mu1[i]) Y20[i] ~ dpois(mu2[i]) mu1[i]<-N10[i]*exp(phi1[i]) mu2[i]<-N20[i]*exp(phi2[i]) } for( i in 1 : N ) { mu_phi2[i]<-beta2 for ( j in 1:N) { count[i, j]<-W[i, j ]*(phi2[j]-beta2)} # mean for phi_2 mu_phi1[i]<- beta1+ eta_0*(phi2[i]-beta2)+ eta_1*sum(count[i, ]) # conditional mean for phi_1|phi_2 m[i]<-1/num[i] # scaling factor for variance in each county } beta1 ~ dnorm(0.0,1.0E-3) beta2 ~ dnorm(0.0,1.0E-3) eta_0 ~ dnorm(0, 0.01) eta_1 ~ dnorm(0, 0.01) tau2 ~ dgamma(1, 0.1) tau1 ~ dgamma(1,0.1) alpha[1] ~ dunif(0, 0.999) # avoid improper CAR to maintain identifiability alpha[2] ~ dunif(0, 0.999) phi2[1:N] ~ car.proper(mu_phi2[], C[], adj[], num[], m[], tau2, alpha[2]) phi1[1:N] ~ car.proper(mu_phi1[], C[], adj[], num[], m[], tau1, alpha[1]) ## set up the weight matrix C (Copy from winbugs manual ) cumsum[1] <- 0 for(i in 2:(N+1)) { cumsum[i] <- sum(num[1:(i-1)]) } for(k in 1 : sumNumNeigh) { for(i in 1:N) { pick[k,i] <- step(k - cumsum[i] - epsilon) * step(cumsum[i+1] - k) # pick[k,i] = 1 if cumsum[i] < k <= cumsum[i=1]; otherwise, pick[k,i] = 0 } C[k] <- 1 / inprod(num[], pick[k,]) # weight for each pair of neighbours } epsilon <- 0.0001 } #initial value list(beta1=0, beta2=0, tau1=1, tau2=1, eta_0=0.3, eta_1=0.3, alpha=c(0.5, 0.5), phi1=c(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, 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), phi2=c(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, 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)) #Data list(Y10=c(96, 673, 142, 126, 114, 41, 169, 100, 160, 124, 147, 67, 137, 161, 52, 15, 51, 239, 725, 48, 116, 73, 79, 151, 149, 28, 3797, 80, 105, 90, 232, 45, 56, 127, 41, 86, 45, 68, 24, 68, 53, 88, 86, 23, 35, 94, 66, 88, 113, 187, 34, 76, 73, 48, 296, 226, 52, 136, 54, 172, 63, 1945, 20, 76, 74, 158, 52, 62, 1000, 145, 128, 54, 350, 89, 35, 53, 113, 21, 93, 53, 67, 450, 65, 27, 193, 211, 62), N10=c(84, 582, 131, 126, 108, 37, 189, 131, 131, 152, 129, 72, 124, 181, 42, 21, 73, 234, 695, 62, 147, 96, 111, 166, 185, 39, 3528, 84, 80, 92, 192, 63, 57, 169, 32, 70, 51, 55, 21, 103, 44, 106, 136, 25, 53, 120, 100, 94, 131, 205, 54, 95, 101, 46, 346, 280, 61, 102, 57, 156, 63, 1741, 22, 92, 94, 176, 52, 60, 921, 162, 117, 72, 394, 126, 46, 61, 110, 28, 90, 68, 78, 402, 60, 36, 185, 225, 66), N=87 ) list(Y20=c(14, 52, 8, 16, 9, 0, 20, 12, 12, 8, 10, 1, 8, 11, 1, 0, 3, 14, 51, 1, 14, 8, 9, 15, 10, 3, 319, 4, 9, 8, 25, 4, 9, 12, 0, 2, 6, 8, 4, 18, 1, 7, 6, 1, 3, 12, 8, 7, 12, 17, 1, 7, 3, 7, 33, 25, 6, 11, 7, 12, 3, 161, 1, 13, 8, 16, 4, 3, 98, 10, 3, 1, 20, 11, 1, 7, 7, 1, 10, 3, 4, 35, 5, 5, 19, 12, 6), N20=c(7, 49, 11, 11, 9, 3, 16, 11, 11, 13, 11, 6, 10, 15, 4, 2, 6, 20, 59, 5, 12, 8, 9, 14, 16, 3, 298, 7, 7, 8, 16, 5, 5, 14, 3, 6, 4, 5, 2, 9, 4, 9, 11, 2, 4, 10, 8, 8, 11, 17, 5, 8, 9, 4, 29, 23, 5, 9, 5, 13, 5, 147, 2, 8, 8, 15, 4, 5, 77, 14, 10, 6, 33, 11, 4, 5, 9, 2, 8, 6, 7, 34, 5, 3, 16, 19, 6)) #adjacency matrix list( adj = c(11, 31, 69, 9, 58, 33, 48, 18, 27, 71, 30, 13, 82, 62, 14, 54, 44, 15, 29, 80, 56, 45, 68, 39, 36, 31, 11, 29, 15, 57, 73, 49, 48, 71, 78, 75, 76, 37, 22, 46, 83, 8, 52, 40, 81, 83, 17, 64, 65, 52, 7, 1, 69, 58, 27, 70, 72, 43, 86, 29, 4, 31, 1, 18, 49, 77, 80, 37, 76, 34, 65, 87, 82, 2, 30, 58, 54, 3, 56, 84, 60, 57, 4, 29, 3, 44, 38, 32, 51, 64, 8, 83, 11, 1, 48, 49, 25, 66, 70, 27, 62, 82, 50, 74, 25, 55, 61, 26, 56, 77, 46, 7, 81, 24, 50, 55, 85, 28, 22, 81, 74, 50, 79, 55, 20, 66, 19, 78, 84, 56, 21, 75, 19, 70, 10, 86, 2, 62, 23, 85, 15, 4, 11, 80, 3, 2, 71, 48, 33, 13, 4, 36, 69, 1, 11, 53, 17, 46, 48, 1, 58, 30, 12, 76, 61, 73, 47, 65, 68, 45, 39, 4, 31, 69, 6, 12, 87, 69, 16, 68, 4, 36, 81, 7, 52, 72, 70, 66, 87, 42, 59, 41, 87, 64, 51, 10, 72, 65, 47, 86, 54, 60, 15, 3, 35, 68, 4, 57, 60, 32, 83, 7, 22, 34, 73, 86, 43, 65, 5, 49, 18, 1, 33, 30, 71, 77, 11, 18, 48, 5, 73, 24, 20, 55, 23, 53, 59, 42, 64, 17, 7, 8, 65, 72, 40, 67, 51, 32, 60, 44, 3, 14, 79, 85, 23, 50, 20, 25, 84, 14, 3, 80, 77, 21, 26, 45, 4, 15, 63, 60, 13, 33, 1, 9, 41, 51, 67, 45, 57, 63, 15, 44, 54, 76, 75, 21, 73, 34, 27, 2, 82, 19, 60, 57, 17, 51, 42, 87, 65, 8, 43, 72, 52, 8, 64, 87, 12, 34, 47, 74, 81, 40, 70, 19, 25, 59, 53, 35, 39, 45, 4, 36, 31, 1, 9, 38, 66, 40, 72, 10, 27, 19, 86, 73, 5, 48, 30, 2, 65, 43, 10, 70, 40, 52, 86, 47, 34, 61, 77, 49, 5, 71, 24, 81, 66, 20, 6, 78, 26, 61, 76, 12, 6, 75, 61, 34, 21, 56, 80, 11, 49, 73, 84, 26, 75, 6, 85, 55, 25, 56, 3, 29, 11, 77, 24, 22, 7, 40, 66, 74, 19, 62, 2, 13, 46, 17, 8, 7, 14, 56, 26, 78, 28, 23, 55, 79, 47, 73, 71, 27, 10, 43, 37, 12, 65, 64, 42, 41 ), num = c(8,6,7,9,4,4,7,6,3,5,8,5,4,4,6,1,5,4,6,4,4,4,4,4,5,5, 6,2,5,5,5,3,4,6,2,4,3,2,3,6,3,4,5,4,5,4,5,7,6,4,5, 5,3,4,6,7,5,4,3,6,5,4,2,6,9,6,2,4,5,6,6,6,8,4,5,5,6, 4,3,5,6,4,4,4,4,6,6), sumNumNeigh = 418) # adjacency matrix 87 x 87 list(W=structure(.Data=c(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 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, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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