# Fit a spatial generalised linear mixed model to data, where a set of spatially smooth random effects are augmented with a piecewise constant intercept process. Fit a spatial generalised linear mixed model to areal unit data, where the response variable can be binomial or Poisson. Note, a Gaussian likelihood is not allowed because of a lack of identifiability among the parameters. The linear predictor is modelled by known covariates, a vector of random effects and a piecewise constant intercept process. The random effects are modelled by an intrinsic CAR prior, while the piecewise constant intercept process was proposed by Lee and Sarran (2015), and allow neighbouring areas to have very different values. Further details are given in the vignette accompanying this package. Inference is conducted in a Bayesian setting using Markov chain Monte Carlo (MCMC) simulation. Missing (NA) values are not allowed in this model. For a full model specification see the vignette accompanying this package. ```r S.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.tau2=NULL,prior.delta=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, offset and each covariate is a vector of length K*1.The response cannot 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 (groups) to allow in the model. - `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 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 in each chain. - `n.sample`: The overall number of MCMC samples to generate in each chain. - `thin`: The level of thinning to apply to the MCMC samples in each chain to reduce their 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.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). - `prior.delta`: The prior maximum for the cluster smoothing parameter delta. Defaults to 10. - `MALA`: Logical, should the function use Metropolis adjusted Langevin algorithm (MALA) updates (TRUE, default) or simple random walk updates (FALSE) for the regression parameters. - `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**: The fitted values based on posterior means from the model. - **residuals**: A matrix with 2 columns where each column is a type of residual and each row relates to an area. 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 median of which intercept group each area 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 Sarran, C (2015). Controlling for unmeasured confounding and spatial misalignment in long-term air pollution and health studies, Environmetrics, 26, 477-487. ## Author(s) Duncan Lee ## Examples ```r ################################################# #### Run the model on simulated data on a lattice ################################################# #### Load other libraries required library(MASS) #### Set up a square lattice region x.easting <- 1:10 x.northing <- 1:10 Grid <- expand.grid(x.easting, x.northing) K <- nrow(Grid) #### Split the area into two groups between which there will be a boundary. groups <-rep(1, K) groups[Grid$Var1>5] <- 2 #### set up distance and neighbourhood (W, based on sharing a common border) matrices distance <- as.matrix(dist(Grid)) W <-array(0, c(K,K)) W[distance==1] <-1 #### Generate the response data phi <- mvrnorm(n=1, mu=groups, Sigma=0.2 * exp(-0.1 * distance)) logit <- phi prob <- exp(logit) / (1 + exp(logit)) trials <- rep(50,K) Y <- rbinom(n=K, size=trials, prob=prob) #### Run the localised smoothing model formula <- Y ~ 1 ## Not run: model <- S.CARlocalised(formula=formula, family="binomial", trials=trials, G=2, W=W,burnin=20000, n.sample=100000) ## End(Not run) #### Toy example for checking model <- S.CARlocalised(formula=formula, family="binomial", trials=trials, G=2, W=W,burnin=10, n.sample=50) ```

S.CARlocalised function