CARBayes (version 1.3)

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

Description

The function fits a binomial logistic random effects models to spatial data, where the random effects are modelled as independent. The model represents the logit transform of the mean function for the set of binomial responses 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

binomial.independent(formula, beta = NULL, theta = NULL, sigma2 = NULL, trials, 
burnin = 0, n.sample = 1000, thin=1, blocksize.beta = 5, blocksize.theta = 10, 
prior.mean.beta = NULL, prior.var.beta = NULL, prior.max.sigma2 = NULL)

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.

beta

A vector of starting values for the regression parameters (including the intercept term). If this argument is not specified the function will randomly generate starting values.

theta

A vector of starting values for the random effects. If this argument is not specified the function will randomly generate starting values.

sigma2

A starting value for the variance parameter of the random effects. If this argument is not specified the function will randomly generate a starting value.

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.

blocksize.theta

The size of the blocks in which to update the random effects in the MCMC algorithm. Defaults to 10.

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.max.sigma2

The maximum allowable value for the random effects variance sigma2 (a Uniform(0,M) prior is assumed). Defaults to M=1000.

Value

• formula
  The formula for the covariate and offset part of the model.
• samples.beta
  A matrix of MCMC samples for the regression parameters beta.
• samples.theta
  A matrix of MCMC samples for the random effects theta.
• samples.sigma2
  A matrix of MCMC samples for the random effects variance sigma2.
• 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.
• random.effects
  A summary matrix of the posterior distributions of the random effects 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.
• DIC
  The Deviance Information Criterion.
• p.d
  The effective number of parameters in the model.
• summary.results
  A summary table of the parameters.

Details

For further details about how to apply the function see the examples below and in the main CARBayes helpfile.

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
#### Let the function randomly generate starting values for the parameters
#### Use the default priors specified by the function (for details see the help files)
formula <- Y ~ x1 + x2
model <- binomial.independent(formula=formula, trials=trials, burnin=5000, 
n.sample=10000)
model <- binomial.independent(formula=formula, trials=trials, burnin=20, 
n.sample=50)