CARBayes (version 4.5)

MVS.CARleroux: Fit a multivariate spatial generalised linear mixed model to data, where the random effects are modelled by a multivariate conditional autoregressive model.

Description

Fit a multivariate spatial generalised linear mixed model to areal unit data, where the response variable can be binomial or Poisson. The linear predictor is modelled by known covariates and a vector of random effects. The latter account for both spatial and between variable correlation, via a Kronecker product formulation. Spatial correlation is captured by the conditional autoregressive (CAR) prior proposed by Leroux et al. (1999), and between variable correlation is captured by a between variable covariance matrix with no fixed structure. This is a type of multivariate conditional autoregressive (MCAR).Further details are given in the vignette accompanying this package. Independent (over space) random effects can be obtained by setting (fix.rho=TRUE, rho=0), in which case the neighbourhood matrix W is not part of the model. In this case enter a fake W matrix that is a K by K matrix of zeros, where K is the number of spatial units. Similarly, the intrinsic MCAR model can be obtained by setting (fix.rho=TRUE, rho=1). 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

MVS.CARleroux(formula, family, data=NULL,  trials=NULL, W, burnin, 
n.sample, thin=1, prior.mean.beta=NULL, prior.var.beta=NULL,
prior.Sigma.df=NULL, prior.Sigma.scale=NULL, fix.rho=FALSE, rho=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 and each covariate should be vectors of length (KJ)*1, where K is the number of spatial units and

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'.

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 matri

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 (no thinning).

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.Sigma.df

The prior degrees of freedom for the Inverse-Wishart prior for Sigma. Defaults to J+1.

prior.Sigma.scale

The prior J times J scale matrix for the Inverse-Wishart prior for Sigma. Defaults to the identity matrix.

fix.rho

Logical, should the spatial dependence parameter rho be fixed rather than being estimated in the model.

rho

The value that the spatial dependence parameter is set to if fix.rho=TRUE. Must be in the interval [0, 1], where 0 corresponds to independence while 1 corresponds to fitting the intrinsic CAR prior. Note, if rho=0 then the W matrix is not included in the

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) and its corresponding estimated effective number of parameters (p.d), the Log Marginal Predictive Likelihood (LMPL), and the Watanabe-Akaike Information Criterion (WAIC) and its corresponding estimated number of effective parameters (p.w).
• 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.

References

Gelfand, A and Vounatsou, P (2003). Proper multivariate conditional autoregressive models for spatial data analysis, Biostatistics, 4, 11-25.

Leroux, B., X. Lei, and N. Breslow (1999). Estimation of disease rates in small areas: A new mixed model for spatial dependence, Chapter Statistical Models in Epidemiology, the Environment and Clinical Trials, Halloran, M and Berry, D (eds), pp. 135-178. Springer-Verlag, New York.

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)

#### 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 
    }	
}
K <- nrow(W)

#### Generate the correlation structures
Q.W <- 0.99 * (diag(apply(W, 2, sum)) - W) + 0.01 * diag(rep(1,K))
Q.W.inv <- solve(Q.W)

Sigma <- matrix(c(1,0.5,0, 0.5,1,0.3, 0, 0.3, 1), nrow=3)
Sigma.inv <- solve(Sigma)
J <- nrow(Sigma)

precision.phi <- kronecker(Q.W, Sigma.inv)
var.phi <- solve(precision.phi)

#### Generate data
N.all <- K * J
x1 <- rnorm(N.all)
x2 <- rnorm(N.all)
phi <- mvrnorm(n=1, mu=rep(0,N.all), Sigma=var.phi)

lp <- 0.1 * x1 - 0.1 * x2 + phi
prob <- exp(lp) / (1 + exp(lp))
trials <- rep(100,N.all)
Y <- rbinom(n=N.all, size=trials, prob=prob)

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