CARBayesST (version 2.0)

ST.CARanova: Fit a spatio-temporal generalised linear mixed model to data, with spatial and temporal main effects and a spatio-temporal interaction.

Description

Fit a spatio-temporal generalised linear mixed model to areal unit data, where the response variable can be binomial, Gaussian or Poisson. The linear predictor is modelled by known covariates and three vectors of random effects. The latter include spatial and temporal main effects and a spatio-temporal interaction. The spatial and temporal main effects are modelled by the conditional autoregressive (CAR) prior proposed by Leroux et al. (1999), while the spatio-temporal interaction random effects are independent. Due to the lack of identifiability of the interactions and the Gaussian errors, only main effects are allowed in the Gaussian model. The model is similar to that proposed by Knorr-Held (2000) 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.

Usage

ST.CARanova(formula, family, data=NULL,  trials=NULL, W, interaction=TRUE, burnin, 
                n.sample, thin=1, prior.mean.beta=NULL, prior.var.beta=NULL,
                prior.nu2=NULL, prior.tau2=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 (KT)*1, where K is the numb

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

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 and time period. 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 othe

interaction

TRUE or FALSE indicating whether the spatio-temporal interaction random effects should be included. Defaults to TRUE unless family=`gaussian' in which case interactions are not allowed.

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.nu2

The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for the Gaussian error variance nu2. Defaults to c(0.001, 0.001) and only used if family=`Gaussian'.

prior.tau2

The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for the random effect variances tau2. 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. The `tau2' element of this list has columns (tau2.phi, tau2.delta, tau2.gamma) (the latter if interaction=TRUE). Similarly, the `rho' element of this list has columns (rho.phi, rho.delta).
• fitted.values
  A vector of fitted values for each area and time period.
• residuals
  A vector of residuals for each area and time period.
• 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.

References

Knorr-Held, L. (2000). Bayesian modelling of inseparable space-time variation in disease risk. Statistics in Medicine, 19, 2555-2567.

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 the regular lattice    
x.easting <- 1:10
x.northing <- 1:10
Grid <- expand.grid(x.easting, x.northing)
K <- nrow(Grid)
T <- 10
N.all <- T * K

#### set up spatial (W) and temporal (D) neighbourhood matrices
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
			if(temp==1)  W[i,j] <- 1 
		}	
	}
	
D <-array(0, c(T,T))
for(i in 1:T)
{
    for(j in 1:T)
    {
        if(abs((i-j))==1)  D[i,j] <- 1 
    }	
}



#### Simulate the elements in the linear predictor and the data
gamma <- rnorm(n=N.all, mean=0, sd=0.001)
x <- rnorm(n=N.all, mean=0, sd=1)
beta <- 0.1

Q.W <- 0.99 * (diag(apply(W, 2, sum)) - W) + 0.01 * diag(rep(1,K))
Q.W.inv <- solve(Q.W)
phi <- mvrnorm(n=1, mu=rep(0,K), Sigma=(0.01 * Q.W.inv))

Q.D <- 0.99 * (diag(apply(D, 2, sum)) - D) + 0.01 * diag(rep(1,T))
Q.D.inv <- solve(Q.D)
delta <- mvrnorm(n=1, mu=rep(0,T), Sigma=(0.01 * Q.D.inv))


phi.long <- rep(phi, T)
delta.long <- kronecker(delta, rep(1,K))
LP <- 4 +  x * beta + phi.long +  delta.long + gamma
mean <- exp(LP)
Y <- rpois(n=N.all, lambda=mean)


#### Run the model
model <- ST.CARanova(formula=Y~x, family="poisson", interaction=TRUE, 
W=W, burnin=10000, n.sample=50000)