# Fit a spatio-temporal generalised linear mixed model to data, with a spatio-temporal autoregressive process and a piecewise constant intercept term. 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), which is the same as that 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, because some pairs of neighbouring areas or time periods can have similar values (same intercept) while other neighbouring pairs have very different values (different intercepts). Furter details are given in Lee and Lawson (2016) and in the vignette accompanying this package. Inference is conducted in a Bayesian setting using Markov chain Monte Carlo (MCMC) simulation. ```r ST.CARlocalised(formula, family, data=NULL, G, trials=NULL, W, burnin, n.sample, thin=1, n.chains=1, n.cores=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.delta=NULL, prior.tau2=NULL, MALA=TRUE, 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 (KN)*1, where K is the number of spatial units and N is the number of time periods. Each vector should be ordered so that the first K data points are the set of all K spatial locations at time 1, the next K are the set of spatial locations for time 2 and so on. The response must NOT contain missing (NA) values. - `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 and in the same order as the response containing the total number of trials for each area and time period. Only used if family="binomial". - `W`: A non-negative 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 matrix can be non-binary, but each row must contain at least one non-zero entry. - `burnin`: The number of MCMC samples to discard as the burn-in 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). - `n.chains`: The number of MCMC chains to run when fitting the model. Defaults to 1. - `n.cores`: The number of computer cores to run the MCMC chains on. Must be less than or equal to n.chains. 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 100,000. - `prior.delta`: The prior maximum M, in a Uniform(0,M) prior, for the intercept process 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 tau2. Defaults to c(1, 0.01). - `MALA`: Logical, should the function use Metropolis adjusted Langevin algorithm (MALA) updates (TRUE, default) or simple random walk (FALSE) updates for the regression parameters. Not applicable if family="gaussian". - `verbose`: Logical, should the function update the user on its progress. ## Returns - **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 matrix with 2 columns where each column is a type of residual and each row relates to an area and time period. The types are "response" (raw), and "pearson". - **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), the Watanabe-Akaike Information Criterion (WAIC) and its corresponding estimated number of effective parameters (p.w), and the loglikelihood. - **accept**: The acceptance probabilities for the parameters. - **localised.structure**: A vector giving the posterior mean of which intercept component (cluster) each data point is in. - **formula**: The formula (as a text string) for the response, covariate and offset parts of the model. - **model**: A text string describing the model fit. - **mcmc.info**: A vector giving details of the numbers of MCMC samples generated. - **X**: The design matrix of covariates. ## References Lee, D and Lawson, C (2016). Quantifying the spatial inequality and temporal trends in maternal smoking rates in Glasgow, Annals of Applied Statistics, 10, 1427-1446. ## Author(s) Duncan Lee ## Examples ```r ################################################# #### 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) N <- 10 N.all <- N * K #### set up spatial neighbourhood matrix W distance <- as.matrix(dist(Grid)) W <-array(0, c(K,K)) W[distance==1] <-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:N) { 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(2, 70), rep(4, 30)),N) LP <- jump + phi fitted <- exp(LP) Y <- rpois(n=N.all, lambda=fitted) #### Run the model ## Not run: model <- ST.CARlocalised(formula=Y~1, family="poisson", G=3, W=W, burnin=10000, n.sample=50000) ## End(Not run) #### Toy example for checking model <- ST.CARlocalised(formula=Y~1, family="poisson", G=3, W=W, burnin=10, n.sample=50) ```

ST.CARlocalised function