CARBayesST (version 2.0)

ST.CARlocalised: Fit a spatio-temporal generalised linear mixed model to data, with a spatio-temporal autoregressive process and a piecewise constant intercept term.

Description

Fit a spatio-temporal 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, a vector of random effects and a piecewise constant intercept process. The random effects follow the multivariate first order autoregressive time series process proposed by Rushworth et al.(2014)., and are the same as those used in the ST.CARar() function. The piecewise constant intercept component allows neighbouring areal units to have very different values if they are assigned to a different intercept component. This model allows for localised smoothness as some pairs of neighbouring areas or time periods can have similar values while other neighbouring pairs have different values. This is a spatio-temopral extension of Lee and Sarran (2015), and furter 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.CARlocalised(formula, family, data=NULL,  G, trials=NULL, W, burnin, n.sample, 
    thin=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.delta=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' 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.

G

The maximum number of distinct intercept terms (clusters) to allow in the model.

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

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.delta

The prior maximum for the cluster smoothing parameter delta. Defaults to 10.

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 variance 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.
• 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
  A vector giving the posterior median of which intercept component (group) each data point is in.
• 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

Lee, D and Sarran, C (2015). Controlling for unmeasured confounding and spatial misalignment in long-term air pollution and health studies, Environmetrics, to appear.

Rushworth, A., D. Lee, and R. Mitchell (2014). A spatio-temporal model for estimating the long-term effects of air pollution on respiratory hospital admissions in Greater London. Spatial and Spatio-temporal Epidemiology 10, 29-38.

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 neighbourhood matrix W
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 
        }    
    }


#### Simulate the elements in the linear predictor and the data
Q.W <- 0.99 * (diag(apply(W, 2, sum)) - W) + 0.01 * diag(rep(1,K))
Q.W.inv <- solve(Q.W)
phi.temp <- mvrnorm(n=1, mu=rep(0,K), Sigma=(0.1 * Q.W.inv))
phi <- phi.temp
    for(i in 2:T)
    {
    phi.temp2 <- mvrnorm(n=1, mu=(0.8 * phi.temp), Sigma=(0.1 * Q.W.inv))
    phi.temp <- phi.temp2
    phi <- c(phi, phi.temp)
    }
jump <- rep(c(rep(3, 70), rep(4, 30)),T)
LP <- jump + phi
fitted <- exp(LP)
Y <- rpois(n=N.all, lambda=fitted)

#### Run the model     
model1 <- ST.CARlocalised(formula=Y~1, family="poisson", G=3, W=W, burnin=10000,
n.sample=50000)