\name{S.CARiar}
\alias{S.CARiar}


\title{
Fit a generalised linear model with a set of spatially autocorrelated random effect following the intrinsic conditional autoregressive (CAR) prior to spatial data.
}

\description{
The function fits a Bayesian hierarchical model with spatially correlated random effects to the data, whre the data likelihood can be binomial, Gaussian or Poisson. The random effects are modelled by the intrinsic conditional autoregressive (IAR) model (Besag et. al. 1991). The model represents the linear predictor by a combination of covariates and a sets of random effects. The latter are spatially correlated and come from the intrinsic CAR model. A set of offsets can also be included on the linear predictor scale. Inference is based on Markov chain Monte Carlo (McMC) simulation, using a combination of Gibbs sampling and Metropolis steps.
}


\usage{
S.CARiar(formula, family, data=NULL,  trials=NULL, W, burnin=0, n.sample=1000,
thin=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.nu2=NULL, 
prior.tau2=NULL, verbose=TRUE)
}

\arguments{
  \item{formula}{
A formula for the covariate part of the model, using the same notation as for the lm() function. The offsets should also be included here using the offset() function.
}
  \item{family}{
One of either 'binomial', 'gaussian' or 'poisson', which respectively specify a binomial likelihood model with a logistic link function, a Gaussian likelihood model with an identity link function, or a Poisson likelihood model with a log link function. 
}
  \item{data}{
A data.frame containing the  variables in the formula.
}
 \item{trials}{
A vector the same length as the response containing the total number of trials for each area.
}
   \item{W}{
A binary n by n neighbourhood matrix (where n is the number of spatial units). The jkth element equals one if areas (j, k) are spatially close (e.g. share a common border) and is zero otherwise. 
}
  \item{burnin}{
The number of MCMC samples to discard as the burnin period. Defaults to 0.
}
  \item{n.sample}{
The number of MCMC samples to generate. Defaults to 1,000.
}
  \item{thin}{
The level of thinning to apply to the MCMC samples to reduce their temporal autocorrelation. Defaults to 1.
}
  \item{prior.mean.beta}{
A vector of prior means for the regression parameters beta (Gaussian priors are assumed). Defaults to a vector of zeros.
}
  \item{prior.var.beta}{
A vector of prior variances for the regression parameters beta (Gaussian priors are assumed). Defaults to a vector with values 1000.
}  
     \item{prior.nu2}{
Only used for the Gaussian model. The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for nu2. Defaults to c(0.001, 0.001).  
}
  \item{prior.tau2}{
The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for tau2. Defaults to c(0.001, 0.001).  
}
  \item{verbose}{
Logical, should the function update the user on its progress.  
}
}

\details{
For further details about how to apply the function see the examples below and in the vignette.
}


\value{
\item{summary.results }{A summary table of the parameters.}
\item{samples }{A list containing the McMC samples from the model.}
\item{fitted.values }{A vector of fitted values for each area.}
\item{residuals }{A vector of residuals for each area.}
\item{modelfit }{Model fit criteria including the Deviance Information Criterion (DIC), the effective number of parameters in the model(p.d), and the Log Marginal Predictive Likelihood (LMPL).}
\item{accept }{The acceptance probabilities for the parameters.}
\item{localised.structure }{NULL, for compatability with the other models.}
\item{formula }{The formula for the covariate and offset part of the model.}
\item{model }{A text string describing the model fit.}
\item{X }{The design matrix of covariates.}
}

\references{
Besag, J., J. York, and A. Mollie (1991). Bayesian image restoration with two applications in spatial statistics. Annals of the Institute of Statistics and Mathematics 43, 1-59.
}

\author{
Duncan Lee
}




\examples{
##################################################
#### Run the model on simulated data on a lattice
##################################################

#### Set up a square lattice region
x.easting <- 1:10
x.northing <- 1:10
Grid <- expand.grid(x.easting, x.northing)
n <- nrow(Grid)

#### set up distance and neighbourhood (W, based on sharing a common border) matrices
distance <-array(0, c(n,n))
W <-array(0, c(n,n))
	for(i in 1:n)
	{
		for(j in 1:n)
		{
		temp <- (Grid[i,1] - Grid[j,1])^2 + (Grid[i,2] - Grid[j,2])^2
		distance[i,j] <- sqrt(temp)
			if(temp==1)  W[i,j] <- 1 
		}	
	}
	
	
#### Generate the covariates and response data
x1 <- rnorm(n)
x2 <- rnorm(n)
theta <- rnorm(n, sd=0.05)
phi <- mvrnorm(n=1, mu=rep(0,n), Sigma=0.4 * exp(-0.1 * distance))
logit <- x1 + x2 + theta + phi
prob <- exp(logit) / (1 + exp(logit))
trials <- rep(50,n)
Y <- rbinom(n=n, size=trials, prob=prob)


#### Run the IAR model
formula <- Y ~ x1 + x2
\dontrun{model <- S.CARiar(formula=formula, family="binomial", trials=trials,
W=W, burnin=5000, n.sample=10000)}
}