CARBayes (version 3.0)

independent.re: Fit the independent random effects model to spatial data

Description

The function fits a Bayesian hierarchical model with independent random effects to the data, whre the data likelihood can be binomial, Gaussian or Poisson. The model represents the linear predictor by a combination of covariates and a set of independent random effects. 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

independent.re(formula, family=NULL, data=NULL,  trials=NULL, burnin=0, 
n.sample=1000, thin=1, blocksize.beta=5, prior.mean.beta=NULL, prior.var.beta=NULL, 
prior.nu2=NULL, prior.sigma2=NULL, verbose=TRUE)

Arguments

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.

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

A 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.

burnin

The number of MCMC samples to discard as the burnin period. Defaults to 0.

n.sample

The number of MCMC samples to generate. Defaults to 1,000.

thin

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

blocksize.beta

The size of the blocks in which to update the regression parameters in the MCMC algorithm. Defaults to 5.

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

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).

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

• formula
  The formula for the covariate and offset part of the model.
• samples
  A list containing the MCMC samples from the model.
• fitted.values
  A summary matrix of the posterior distributions of the fitted values for each area. The summaries include: Mean, Sd, Median, and credible interval.
• residuals
  A summary matrix of the posterior distributions of the residuals for each area. The summaries include: Mean, Sd, Median, and credible interval.
• W.summary
  NULL.
• modelfit
  Model fit criteria including the Deviance Information Criterion (DIC), the effective number of parameters in the model(p.d), DIC3 and the Marginal Predictive Likelihood (MPL).
• summary.results
  A summary table of the parameters.
• model
  A text string describing the model fit.
• accept
  The acceptance probabilities for the parameters.

Details

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

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


#### Run the independent model
formula <- Y ~ x1 + x2
model <- independent.re(formula=formula, family="binomial", trials=trials, 
burnin=5000, n.sample=10000)