CARBayes (version 4.3)

S.CARdissimilarity: Fit a spatial generalised linear mixed model to data, where the random effects have a localised conditional autoregressive prior.

Description

Fit a spatial generalised linear mixed model to areal unit data, where the response variable can be binomial, Gaussian or Poisson. The linear predictor is modelled by known covariates and a vector of random effects. The latter are modelled by the localised conditional autoregressive prior proposed by Lee and Mitchell (2012), and further details are given in the vignette accompanying this package. Inference is conducted in a Bayesian setting using Markov chain Monte Carlo (McMC) simulation. Missing (NA) values are allowed in the response, and posterior predictive distributions are created for the missing values for predictive purposes. These are saved in the `samples' argument in the output of the function and are denoted by `Y'.

Usage

S.CARdissimilarity(formula, family, data=NULL,  trials=NULL, W, 
Z, burnin, n.sample, thin=1, prior.mean.beta=NULL, 
prior.var.beta=NULL, prior.nu2=NULL, prior.tau2=NULL, verbose=TRUE)

Arguments

formula

A formula for the covariate part of the model using the syntax of the lm() function. Offsets can be included here using the offset() function. The response can contain missing (NA) values.

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.

data

An optional data.frame containing the variables in the formula.

trials

A vector the same length as the response containing the total number of trials for each area. Only used if family=`binomial'.

W

A K by K neighbourhood matrix (where K is the number of spatial units). Typically a binary specification is used, where the jkth element equals one if areas (j, k) are spatially close (e.g. share a common border) and is zero otherwise. For this model o

Z

A list, where each element is a K by K matrix of non-negative dissimilarity metrics.

burnin

The number of McMC samples to discard as the burnin period.

n.sample

The number of McMC samples to generate.

thin

The level of thinning to apply to the McMC samples to reduce their temporal autocorrelation. Defaults to 1.

prior.mean.beta

A vector of prior means for the regression parameters beta (Gaussian priors are assumed). Defaults to a vector of zeros.

prior.var.beta

A vector of prior variances for the regression parameters beta (Gaussian priors are assumed). Defaults to a vector with values 1000.

prior.nu2

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) and only used if family=`Gaussian'.

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).

verbose

Logical, should the function update the user on its progress.

Value

• summary.results
  A summary table of the parameters.
• samples
  A list containing the McMC samples from the model.
• fitted.values
  A vector of fitted values for each area.
• residuals
  A vector of residuals for each area.
• 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).
• accept
  The acceptance probabilities for the parameters.
• localised.structure
  A list containing two matrices: W.posterior contains posterior medians for each element w_kj of the K by K neighbourhood matrix W; W.border.prob contains posterior probabilities that each w_kj element of the K by K neighbourhood matrix W equals zero. This corresponds to the posterior probability of a boundary in the random effects surface. In both cases elements which correspond to two non-neighbouring areas have NA values.
• formula
  The formula for the covariate and offset part of the model.
• model
  A text string describing the model fit.
• X
  The design matrix of covariates.

References

Lee, D. and R. Mitchell (2012). Boundary detection in disease mapping studies. Biostatistics, 13, 415-426.

Examples

###########################################################
#### Run the model on simulated data - localised CAR model
###########################################################

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

#### Split the area into two groups between which there will be a boundary.
groups <-rep(1, K) 
groups[Grid$Var1>5] <- 2

#### set up distance and neighbourhood (W, based on sharing a common border) matrices
distance <-array(0, c(K,K))
W <-array(0, c(K,K))
  for(i in 1:K)
	{
		for(j in 1:K)
		{
		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 response data
phi <- mvrnorm(n=1, mu=groups, Sigma=0.2 * exp(-0.1 * distance))
logit <- phi
prob <- exp(logit) / (1 + exp(logit))
trials <- rep(50,K)
Y <- rbinom(n=K, size=trials, prob=prob)


#### Generate a dissimilarity metric
dissimilarity <- cbind(groups) + rnorm(K, sd=0.2)
dissimilarity.matrix <- as.matrix(dist(cbind(dissimilarity, dissimilarity), 
method="manhattan", diag=TRUE, upper=TRUE)) * W/2

Z <- list(dissimilarity.matrix=dissimilarity.matrix)

#### Run the localised smoothing model
formula <- Y ~ 1
model <- S.CARdissimilarity(formula=formula, family="binomial",
trials=trials, W=W, Z=Z, burnin=20000, n.sample=100000)