####  WinBUGS code for implementing the Beta-Binomial Model with Conditionally-autoregressive spatial effects, when $\gamma(s) = \gamma$ 


model { 

#### means and variances 

mage<-mean(age[])
sdage <- sd(age[])
mfemale <- mean(female[])
sdfemale <- sd(female])
msmok <- mean(smoker[])
sdsmok <- sd(smoker[])
mbrf <- mean(brfl[])
sdbrf <- sd(brfl[])



#### Standardized parameters 

beta1t[1] <- beta1[1]/sdage
beta1t[2] <- beta1[2]/sdfemale
beta1t[3] <- beta1[3]/sdsmok
beta1t[4] <- beta1[4]/sdbrf




#### Intercept 
A <- Int - beta1t[1]*mage - beta1t[2]*mfemale - beta1t[3]*msmok - beta1t[4]*mbrf


for (i in 1:N) {  #100 patients

                   	 for (s in 1:M) { # 32 teeth
                                      
# Below is a way to code a BB model, W is the response (count)

                         W[i,s] ~ dbin(theta[i,s], n[s])
                         W1[i,s] ~ dbin(theta[i,s], n[s])

	
theta[i,s] ~ dbeta(a[i,s], b[i,s])I(0.001, 0.999)


		
#### Modeling parameters a and b with the new re-parametrization 
			
a[i,s] <-  max( mu[i,s]*tau, 0.001)
b[i,s] <-  max( (1-mu[i,s])*tau, 0.001)
			
			

#### Log-likelihood
LL[i,s] <- logfact(n[s]) - logfact(W[i,s]) - logfact(n[s]-W[i,s]) + W[i,s]*log(theta[i,s]) + (n[s] - W[i,s])*log(1- theta[i,s])
	
	
## to compute CPO's 

CPO.inv[i,s] <- exp(-LL[i,s])
		

#### This is done to remove instability 
			
mu[i,s]<-max(0.001,min(0.999,mu1[i,s]))
		   
						
#### modeling logit of mu (the mean) 

logit(mu1[i,s]) <- Int + beta1t[1]*(age[i]-mage) + beta1t[2]*(female[i]-mfemale) + beta1t[3]*(smoker[i]-msmok)  + beta1t[4]*(brfl[i] - mbrf)   + U[i,s] 

}


#### Proper CAR prior distribution for spatial random effects: 

U[i,1:M] ~ car.proper(thetam[], C[], adj[], num[], m[], prec, gamma)


 }


Deviance <- -2*sum(LL[,])


#### modeling tau.........the dispersion 

tau ~ dgamma(0.1, 0.01)




#### Setting up car.proper() 

	# Set up 'data' to define spatial dependence structure
	# =====================================
	for(i in 1 : M) { 
		m[i] <- 1/num[i]       # scaling factor for variance in each cell
		thetam[i] <- alpha1
		thetan[i] <- alpha2
	} 

#### This is copied from WinBUGS manual, note the changes done to create the C matrix 

	cumsum[1] <- 0
	for(i in 2:(M+1)) {
		cumsum[i] <- sum(num[1:(i-1)])
	}	
	for(k in 1 : sumNumNeigh) { 
		for(i in 1:M) {
			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] <-  sqrt(num[adj[k]]/ inprod(num[], pick[k,]))  # weight for each neighbour-pair (in BUGS manual)
		
               C[k] <-  1 / inprod(num[], pick[k,])    # Sudipto Banerjee's paper suggests this instead of the other one, this runs for our data                                    
	}
	epsilon <- 0.0001




### Some priors

alpha1~ dnorm(0, 0.01)
alpha2 ~ dnorm(0, 0.01)

prec ~ dgamma(0.1, 0.01)
spvar <- 1/prec

gamma ~ dbeta(4,1)

Int ~ dnorm(0, 0.01)

             
for (i in 1:4) { 
beta1[i] ~ dnorm(0,0.25) 
odds[i] <- exp(beta1t[i])
}

     
} ## Model ends here   



######   Initial values

list(alpha1 = 0, gamma= 0.9,  Int = 0.01, beta1 = c(0.1,0.1,0.1,0.1), 
theta = structure(.Data = c(0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,
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
.Dim = c(100,32))    )

         

#########  This is the data


list(N=100, M = 32, 
adj = c(2,3, 32,
        1,3, 4, 31,
        1,2,4,5, 30,
        2, 3,5,6, 29,
        3,4,6,7, 28,
        4,5,7,8,
        5,6,8,9, 
        6,7, 9,10, 
        7,8,10,11,
        8,9,11,12,
        9,10,12,13,
        10,11,13,14,21,
        11,12,14,15,20,
        12,13,15,16,19,
        13,14,16,18,
        14,15,17,
        16, 18,19,
        17, 19,20,15,  
        17,18, 20,21,14,
        18,19, 21,22, 13,
        19,20, 22,23,12,  
        20,21, 23,24,  
        21,22, 24, 25,
        22,23, 25,26, 
        23,24, 26,27,   
        24,25, 27,28, 
        25,26, 28,29, 
        26,27, 29,30,5, 
        27,28, 30,31, 4,
        28,29, 31,32, 3, 
        29,30, 32, 2,
        30,31, 1),

num = c(3,4,5,5,5,4,4,4,4,4,4,5,5,5,4,3,3,4,5,5,5,4,4,4,4,4,4,5,5,5,4,3), 
sumNumNeigh = 136,

n = c(5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5),

W=structure(.Data = c(5,5,1,0,0,0,4,1,0,0,1,0,0,1,1,5,5,1,0,1,0,0,0,0,0,0,0,0,1,0,5,5,5,5,5,5,5,4,2,3,3,3,4,5,5,5,5,5,5,5,5,5,5,3,0,0,0,2,2,5,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,5,5,0,5,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,4,4,3,2,4,4,5,5,5,5,5,5,5,5,5,3,0,0,0,0,0,0,0,5,5,5,5,5,5,0,5,2,0,1,0,3,1,0,2,5,1,5,5,5,2,1,1,0,0,0,0,0,0,1,0,0,0,4,5,5,5,4,5,5,4,4,4,4,4,4,5,2,5,0,5,5,0,5,0,1,0,0,0,0,0,0,0,0,5,5,5,3,2,3,3,3,1,0,0,0,0,0,2,5,1,0,5,5,1,3,0,0,0,0,0,0,0,0,0,2,5,5,1,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,0,0,0,0,4,4,0,0,0,0,0,5,5,5,3,3,1,1,0,0,0,0,0,4,1,1,2,2,5,5,2,2,5,1,0,0,0,0,0,0,1,3,5,2,5,5,1,0,3,2,0,2,4,4,2,2,3,0,1,1,5,5,2,5,5,2,0,0,0,0,0,0,2,3,2,1,5,5,0,2,2,2,0,0,0,0,0,0,0,0,5,0,0,0,3,5,0,0,0,0,0,0,0,0,0,0,5,2,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,4,0,0,0,0,0,5,5,5,5,5,5,1,0,0,2,0,0,1,4,0,0,5,0,1,1,5,5,5,5,0,0,0,0,0,0,0,0,0,0,5,5,5,5,5,5,5,0,0,0,0,0,4,0,2,5,5,1,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,5,5,0,2,1,5,0,0,0,0,4,0,2,5,2,1,5,5,5,1,3,2,0,0,0,0,0,0,0,0,5,3,5,0,5,1,0,5,0,0,0,0,0,0,0,1,0,1,5,0,5,5,5,0,0,0,0,0,0,0,0,3,5,1,5,5,5,5,0,5,0,4,1,3,3,0,5,5,2,5,5,5,0,5,0,0,0,0,0,0,0,0,0,5,2,1,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,0,0,4,4,0,0,0,0,5,5,5,5,5,5,5,5,5,0,0,4,4,0,0,5,5,5,5,5,5,0,2,5,5,0,2,4,2,0,2,5,3,5,1,5,5,3,5,2,0,0,0,0,0,0,0,0,5,5,5,5,1,5,5,5,5,1,4,0,0,4,4,5,1,2,2,5,5,3,5,2,5,0,0,0,0,0,0,1,1,5,5,1,5,5,2,5,0,0,0,4,4,4,0,2,2,5,5,5,5,5,5,5,0,0,0,0,0,0,0,0,0,5,1,5,5,5,5,5,5,0,0,4,4,0,0,5,5,5,5,5,5,5,5,0,5,4,4,0,0,4,4,5,5,5,5,5,5,5,1,5,2,0,0,0,0,0,4,5,5,1,5,5,5,5,5,0,2,0,0,0,0,0,0,1,5,5,5,5,5,5,5,5,5,2,0,4,0,0,0,0,5,5,5,5,5,5,5,3,5,0,0,0,4,4,0,0,5,5,5,5,5,5,5,5,0,0,0,0,2,0,0,1,5,0,0,0,5,5,5,5,0,0,1,0,0,0,0,0,0,5,0,5,5,1,5,0,5,0,0,0,0,0,0,5,5,5,5,5,5,5,5,5,2,0,0,0,0,0,0,5,5,5,5,5,5,3,2,2,2,0,0,0,2,0,0,0,0,1,0,5,5,0,5,0,0,0,0,0,2,2,0,1,0,3,2,2,5,5,5,5,5,0,0,4,4,0,0,3,5,5,5,5,5,5,5,5,5,0,0,0,0,0,0,0,5,5,5,5,5,5,3,5,0,0,4,4,4,4,0,0,5,1,0,5,5,2,5,1,0,0,0,0,0,0,0,0,0,0,2,3,5,2,2,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,0,0,0,0,0,0,0,0,0,1,0,2,2,0,5,0,0,0,0,0,0,0,0,0,0,0,1,0,1,5,5,1,1,0,0,0,0,0,0,0,0,0,0,0,1,5,0,0,5,2,0,0,4,4,4,0,0,2,5,5,5,0,0,0,5,1,0,4,4,4,4,0,0,5,2,4,0,0,5,5,5,2,2,0,0,4,4,0,0,5,5,5,5,5,5,5,5,2,3,0,0,0,0,4,0,0,5,5,5,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,5,0,0,0,0,4,0,4,0,0,0,0,0,0,0,5,5,5,1,5,0,0,0,0,0,0,3,2,5,5,5,5,5,5,0,0,0,0,0,0,0,0,0,2,4,5,5,1,5,5,5,5,4,0,4,3,4,4,5,5,5,5,5,5,1,5,5,2,0,0,0,0,0,0,5,1,5,5,5,5,1,5,0,0,0,0,0,0,0,0,0,0,5,3,5,1,1,5,1,0,0,0,0,0,0,0,0,2,5,5,5,0,1,4,1,2,0,0,4,0,0,0,1,1,2,1,1,0,1,5,0,0,0,0,0,0,0,0,0,0,5,1,1,5,5,1,1,1,0,0,0,0,0,0,1,1,5,5,5,5,5,5,1,0,0,0,0,0,0,0,0,0,3,2,5,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,5,2,0,0,0,0,0,0,1,0,0,0,0,0,2,5,5,1,2,0,0,0,0,0,0,0,0,0,0,2,0,5,5,5,2,1,1,0,0,0,0,0,0,1,0,5,0,0,0,5,5,4,0,0,0,0,0,0,0,0,0,5,5,0,5,2,5,5,5,0,0,0,0,0,4,5,5,1,1,5,5,5,5,0,0,0,0,0,0,0,0,0,0,5,5,5,5,0,5,5,0,0,0,0,0,0,0,1,0,5,1,5,5,5,5,0,0,0,0,0,0,0,0,5,0,5,5,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,1,0,0,0,0,0,2,5,5,5,5,5,5,5,5,5,0,2,4,4,3,0,5,5,5,5,2,1,5,5,5,2,1,0,0,0,0,1,2,5,5,2,5,5,0,5,0,0,0,0,0,0,0,0,5,0,5,0,5,5,0,1,0,0,0,0,0,0,0,0,0,0,1,0,5,0,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,5,5,0,0,0,0,0,0,0,0,0,3,0,0,5,5,0,5,3,0,0,0,0,0,0,5,3,2,5,5,1,5,5,5,2,0,0,0,0,0,0,0,5,5,5,0,5,0,5,0,0,0,0,0,0,0,0,0,0,0,0,5,5,0,5,5,0,0,0,0,0,0,0,0,0,0,0,5,5,5,5,5,1,4,4,4,4,4,4,1,1,2,5,5,5,5,5,5,0,0,0,0,0,0,0,1,5,5,5,5,0,0,3,3,2,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,3,2,0,0,1,2,1,2,2,1,1,1,1,1,1,0,1,5,1,2,3,5,1,1,0,1,3,4,0,0,0,5,5,5,5,5,5,2,1,0,0,0,0,0,0,0,0,1,1,1,5,5,5,2,1,0,0,0,0,0,0,0,0,0,3,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,5,1,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,5,5,5,0,0,4,4,4,2,4,0,0,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,0,0,4,4,4,0,5,5,5,5,5,5,1,5,2,0,0,0,0,0,0,0,3,2,5,3,5,5,5,5,5,5,4,4,4,0,0,0,0,0,5,1,5,5,3,5,5,5,0,0,0,0,0,0,0,1,5,0,5,5,1,5,5,0,0,4,4,4,0,4,5,5,5,5,5,5,5,5,0,5,0,0,0,0,0,0,0,5,5,3,5,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,1,0,0,0,0,0,0,0,0,0,0,1,5,0,5,5,1,1,0,0,0,3,0,0,0,1,1,5,1,5,5,5,1,0,0,0,0,0,0,0,0,0,5,5,5,5,5,0,1,0,0,0,0,0,0,0,0,0,0,0,0,5,5,5,5,0,0,0,0,0,0,0,0,0,0,0,5,5,5,5,5,5,5,4,4,4,4,4,0,5,5,0,5,5,5,5,5,1,5,0,4,4,4,4,0,1,2,5,5,5,5,5,5,1,4,3,2,3,3,0,0,5,5,1,5,5,5,5,5,3,5,0,0,0,0,0,3,0,5,3,5,5,5,5,5,5,5,4,4,4,4,4,4,5,5,5,3,5,5,5,2,5,2,0,0,0,4,4,4,5,5,5,5,5,5,1,2,5,5,0,0,0,4,0,0,1,1,5,1,5,5,2,5,0,0,0,0,0,0,0,0,1,1,2,1,1,1,5,5,1,1,0,0,0,0,0,0,1,0,5,5,0,0,5,5,0,0,0,0,0,0,0,0,0,0,5,5,1,5,1,5,0,5,1,3,0,2,0,0,0,5,5,1,5,5,5,5,0,0,0,0,0,0,4,0,0,0,5,5,1,5,1,5,1,5,1,0,1,1,0,0,2,5,5,5,5,1,5,5,5,0,0,0,0,0,0,0,0,0,5,5,5,5,0,1,1,0,0,0,0,0,0,0,0,0,0,0,5,5,1,1,0,0,0,0,0,0,0,0,0,0,1,1,5,5,1,2,0,0,0,0,0,0,0,0,0,0,0,1,5,5,0,5,1,0,0,0,0,0,0,0,1,1,1,5,1,5,5,5,5,5,4,4,4,4,4,0,5,5,5,5,5,5,5,5,0,0,0,0,0,0,0,0,0,0,5,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,1,1,5,5,0,2,0,0,0,0,0,0,0,0,0,0,0,2,5,0,5,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,5,5,0,5,5,0,0,0,0,0,0,0,1,0,5,5,5,1,5,5,0,0,0,0,0,0,0,0,1,5,5,5,0,5,1,1,1,0,0,4,0,0,0,1,1,1,5,1,1,5,5,1,0,0,0,0,0,0,0,0,1,5,5,5,0,1,0,0,0,0,0,0,0,0,0,0,1,0,5,5,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,5,5,5,5,5,0,0,0,2,0,0,0,0,5,5,5,5,0,5,0,0,0,0,0,0,0,0,0,0,1,1,1,5,5,5,5,1,1,4,4,4,1,0,2,2,5,5,5,5,5,5,5,5,0,0,0,0,1,0,1,2,5,5,5,5,1,1,0,0,4,0,0,2,0,0,1,1,1,2,5,5,5,3,5,1,0,0,0,0,0,0,0,1,2,5,2,5,5,1,1,5,0,0,1,1,0,0,1,0,0,0,5,5,0,5,1,5,0,0,0,0,0,0,0,0,1,2,5,5,1,5,0,0,0,0,0,0,0,0,0,0,1,5,5,1,2,2,5,0,0,0,0,0,0,0,0,5,2,2,2,5,0,5,0,0,0,0,0,4,4,0,5,5,1,0,5,1,0,5,0,0,0,0,0,0,0,0,1,0,0,5,5,5,5,5,0,0,0,0,0,0,0,0,0,0,5,5,5,5,5,0,0,0,0,0,0,0,0,0,0,5,3,5,5,5,5,5,5,5,1,0,0,0,0,2,2,5,2,5,5,5,5,5,5,0,0,0,0,0,0,0,0,5,2,3,5,5,0,5,0,0,0,0,0,0,0,0,0,0,0,1,5,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,5,3,5,5,5,0,0,4,4,4,4,5,3,4,5,5,5,5,5,3,2,4,1,0,0,1,4,5,2,5,5,5,2,2,3,0,0,0,0,0,0,0,0,1,0,2,2,5,5,5,5,5,0,0,4,0,0,0,0,0,5,5,5,2,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,5,2,2,2,0,0,0,0,0,0,0,0,0,1,2,3,0,5,0,5,0,0,0,0,0,0,0,0,0,0,0,0,5,0,5,5,5,0,0,0,0,0,0,0,0,0,2,5,2,0,0,5,5,5,4,4,4,4,4,0,5,5,5,5,0,5,5,5,5,5,0,0,0,0,0,0,0,0,5,5,5,5,2,4,3,5,2,4,0,4,4,4,3,4,3,1,1,5,5,5,2,3,0,0,0,0,0,0,2,3,5,5,5,1,1,2,0,0,0,0,0,0,0,0,0,0,5,1,1,0,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,5,5,2,1,0,0,0,0,0,0,0,0,5,5,5,5,0,1,5,0,0,0,0,0,0,0,0,0,2,0,0,0,5,1,2,1,1,0,0,0,0,0,0,0,2,2,1,5,2,1,5,1,0,0,0,0,0,0,0,0,5,5,1,2,5,0,1,5,5,0,0,0,2,0,0,2,3,3,0,0,1,5,5,1,2,0,0,0,0,0,0,0,2,5,5,5,5,5,5,5,5,4,0,0,0,0,0,3,5,5,5,5,5,5,5,3,3,0,0,0,0,0,0,1,2,5,5,5,5,1,0,0,0,0,4,0,4,0,0,0,0,0,1,5,5,1,1,0,0,0,0,0,0,0,0,0,0,1,1,5,5,1,5,5,0,0,0,0,0,0,0,1,5,0,2,5,5,5,5,2,2,0,0,0,0,0,0,0,2,0,5,5),
.Dim = c(100,32)),

age=c(37,65,61,55,48,61,57,52,57,59,55,62,45,48,42,56,50,44,62,53,55,56,47,58,51,54,52,47,63,62,50,56,34,65,40,56,59,55,44,36,59,48,58,71,59,68,46,39,56,52,61,66,53,51,34,53,73,68,58,65,42,41,48,71,70,62,40,47,42,40,63,55,62,27,34,60,34,29,45,43,40,41,46,51,39,48,64,51,63,49,64,51,54,48,38,38,51,67,45,61),

female=c(0,0,1,1,1,1,0,1,1,1,0,1,0,0,0,1,1,1,1,1,1,1,0,1,0,1,1,1,1,0,0,1,0,0,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,1,1,1,1,0,1,1,1,0,1,1,1,1,1,1,1,0,1),

smoker = c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1),

brfl = c(0,0,0,0,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,0)


)