# Fit a spatial generalised linear mixed model to data, where the random effects have a localised conditional autoregressive prior. Fit a spatial 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 are modelled by the localised conditional autoregressive prior proposed by Lee and Mitchell (2012), 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. Missing (NA) values are allowed in the response, and posterior predictive distributions are created for the missing values using data augmentation. These are saved in the "samples" argument in the output of the function and are denoted by "Y". For a full model specification see the vignette accompanying this package. ```r S.CARdissimilarity(formula, family, data=NULL, trials=NULL, W, Z, W.binary=TRUE, burnin, n.sample, thin=1, n.chains=1, n.cores=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.nu2=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, offset and each covariate is a vector of length K*1. The response can contain missing (NA) values. - `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 log link function. - `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. 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. For this model only the matrix must be binary. - `Z`: A list, where each element is a K by K matrix of non-negative dissimilarity metrics. - `W.binary`: Logical, should the estimated neighbourhood matrix have only binary (0,1) values. - `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.nu2`: The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for nu2. Defaults to c(1, 0.01) 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 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 updates (FALSE) 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**: 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 list containing two matrices: W.posterior contains posterior medians for each element w_kj of the K by K neighbourhood matrix W; W.border.prob contains posterior probabilities that each w_kj element of the K by K neighbourhood matrix W equals zero. This corresponds to the posterior probability of a boundary in the random effects surface. The latter is only present if W.binary=TRUE, otherwise it is missing (NA). In all cases W elements that correspond to two non-neighbouring areas have NA values. - **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 R. Mitchell (2012). Boundary detection in disease mapping studies. Biostatistics, 13, 415-426. ## 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) #### Generate a dissimilarity metric dissimilarity <- cbind(groups) + rnorm(K, sd=0.2) dissimilarity.matrix <- as.matrix(dist(cbind(dissimilarity, dissimilarity), method="manhattan", diag=TRUE, upper=TRUE)) * W/2 Z <- list(dissimilarity.matrix=dissimilarity.matrix) #### Run the localised smoothing model formula <- Y ~ 1 ## Not run: model <- S.CARdissimilarity(formula=formula, family="binomial", trials=trials, W=W, Z=Z, W.binary=TRUE, burnin=20000, n.sample=100000) ## End(Not run) #### Toy example for checking model <- S.CARdissimilarity(formula=formula, family="binomial", trials=trials, W=W, Z=Z, W.binary=TRUE, burnin=10, n.sample=50) ```

S.Cissimilarity.Rd function