ST.CARadaptive: Fit a spatio-temporal generalised linear mixed model to data with a spatio-temporal autoregressive process that has an adaptive autocorrelation stucture.

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 a vector of random effects. The latter follows a multivariate first order autoregressive time series process, where spatial autocorrelation is modelled via the precision matrix which is based on an adjacency matrix W. The non-zero elements of W associated with each pair of geographically adjacent areal units are treated as random variables, which allows step changes to be identified in the random effects surface between geogrphically adjacent regions. The model is similar to that proposed by Rushworth et al. (2014). 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.CARadaptive(formula, family, data=NULL, trials=NULL, W, burnin, n.sample, thin=1,  
prior.mean.beta=NULL, prior.var.beta=NULL, prior.nu2=NULL, prior.tau2=NULL, 
prior.zeta2 = NULL, verbose=TRUE, rhofix = NULL, epsilon = 0)

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 n

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'.

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 o

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 variance tau2. Defaults to c(0.001, 0.001).

prior.zeta2

The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for the second level adjacency random effect variance zeta2. Defaults to c(0.001, 0.001).

verbose

Logical, should the function update the user on its progress.

rhofix

Value between 0 and 1 at which to fix the spatial correlation parameter. Setting to 1, reduces the random effects prior to the intrinsic CAR model. When set to NULL, (the default) the parameter is estimated with a uniform[0,1] prior.

epsilon

Diagonal ridge parameter to add to the random effects prior precision, only required when rhofix = 1, and the prior precision is improper. Defaults to 0.

Value

summary.resultsA summary table of the parameters.
samplesA list containing the McMC samples from the model.
fitted.valuesA vector of fitted values for each area and time period.
residualsA vector of residuals for each area and time period.
modelfitModel 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).
acceptThe acceptance probabilities for the parameters.
localised.structureA list with 2 K*K matrices, Wmn and W99 summarising the estimated adjacency relationships. Wmn contains the posterior median for each w_ij element estimated in the model for adjacent areal units, while W99 contains Prob(w_ij < 0.5|data). For both matrices, elements corresponding to non-adjacent pairs of areas have NA values.
formulaThe formula for the covariate and offset part of the model.
modelA text string describing the model fit.
XThe design matrix of covariates.

References

Rushworth, A., Lee, D., and Sarran, C. An adaptive spatio-temporal smoothing model for estimating trends and step changes in disease risk (2014) arXiv:1411.0924

Examples

Run this code

##################################################
#### 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.CARadaptive(formula=Y~1, family="poisson", W=W, burnin=10000,
n.sample=50000)

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