CARBayes (version 4.3)

S.independent: Fit a spatial generalised linear mixed model to data, where the random effects are independent.

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 and a vector of random effects. The latter are modelled as independent and identically distributed, 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. Missing (NA) values are allowed in the response, and posterior predictive distributions are created for the missing values for predictive purposes. These are s aved in the `samples' argument in the output of the function and are denoted by `Y'.

Usage

S.independent(formula, family, data=NULL,  trials=NULL, burnin, n.sample, thin=1, 
prior.mean.beta=NULL, prior.var.beta=NULL, prior.sigma2=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' 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.

trials

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

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.sigma2

The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for sigma2. 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
  NULL, for compatability with the other models.
• 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.

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)
K <- nrow(Grid)
	
	
#### Generate the covariates and response data
x1 <- rnorm(K)
x2 <- rnorm(K)
theta <- rnorm(K, sd=0.05)
logit <- x1 + x2 + theta
prob <- exp(logit) / (1 + exp(logit))
trials <- rep(50,K)
Y <- rbinom(n=K, size=trials, prob=prob)


#### Run the independent model
formula <- Y ~ x1 + x2
model <- S.independent(formula=formula, family="binomial", trials=trials, 
burnin=20000, n.sample=100000)