CARBayes (version 4.2)

S.CARlocalised: Fit a spatial generalised linear mixed model to data, where the random effects have conditional autoregressive prior and are augmented with a piecewise constant intercept term.

Description

Fit a spatial generalised linear mixed model to areal unit data, where the response variable can be binomial or Poisson. Note, a Gaussian likelihood is not allowed because of a lack of identifiability among the parameters. The linear predictor is modelled by known covariates, a vector of random effects and a piecewise constant intercept process. The random effects are modelled by an intrinsic CAR prior, while the piecewise constant intercept process was proposed by Lee and Sarran (2015), and allow neighbouring areas to have very different values. Further details are given in the vignette accompanying this package. Inference is conducted in a Bayesian setting using Markov chain Monte Carlo (McMC) simulation.

Usage

S.CARlocalised(formula, exposure=NULL, family, data=NULL, G, trials=NULL, W,
burnin, n.sample, thin=1, prior.mean.beta=NULL, prior.var.beta=NULL,
prior.mean.alpha=NULL, prior.var.alpha=NULL, prior.tau2=NULL,prior.delta=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.

exposure

This optional argument allows a single covariate to have variation in its values within an areal unit if family=`poisson'. The argument is a list object, and each element of the list corresponds to a single areal unit. Each element in the list is a 2 col

family

One of either `binomial' or `poisson', which respectively specify a binomial likelihood model with a logistic link function, or a Poisson likelihood model with a log link function.

data

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

G

The maximum number of distinct intercept terms (clusters) to allow in the model.

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. The matrix can b

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.mean.alpha

The prior mean for the regression parameter alpha (Gaussian priors are assumed) multiplying the covariate with within area variation in its values. Defaults to zero.

prior.var.alpha

The prior variance for the regression parameter alpha (Gaussian priors are assumed) multiplying the covariate with within area variation in its values. Defaults to 1000.

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).

prior.delta

The prior maximum for the cluster smoothing parameter delta. Defaults to 10.

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 vector giving the posterior median of which cluster (group) each area is in.
• 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 Sarran, C (2015). Controlling for unmeasured confounding and spatial misalignment in long-term air pollution and health studies, Environmetrics, to appear.

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)


#### Run the localised smoothing model
formula <- Y ~ 1
model <- S.CARlocalised(formula=formula, family="binomial", G=2,  
W=W,burnin=20000, n.sample=100000)