# Fit a multivariate spatial generalised linear mixed model to data, where the random effects are modelled by a multivariate conditional autoregressive model. Fit a multivariate spatial generalised linear mixed model to areal unit data, where the response variable can be binomial, Gaussian, multinomial or Poisson. The linear predictor is modelled by known covariates and a vector of random effects. The latter account for both spatial and between variable correlation, via a Kronecker product formulation. Spatial correlation is captured by the conditional autoregressive (CAR) prior proposed by Leroux et al. (2000), and between variable correlation is captured by a between variable covariance matrix with no fixed structure. This is a type of multivariate conditional autoregressive (MCAR) model. Further details are given in the vignette accompanying this package. Independent (over space) random effects can be obtained by setting rho=0, while the intrinsic MCAR model can be obtained by setting rho=1. 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 the multinomial model the first category in the multinomial data (first column of the response matrix) is taken as the baseline, and the covariates are linearly related to the log of the ratio (theta_j / theta_1) for j=1,...,J, where theta_j is the probability of being in category j. For a full model specification see the vignette accompanying this package. ```r MVS.CARleroux(formula, family, data=NULL, trials=NULL, W, burnin, n.sample, thin=1, n.chains=1, n.cores=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.nu2=NULL, prior.Sigma.df=NULL, prior.Sigma.scale=NULL, rho=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 the offset (if included) should be matrices of dimension K*J, where K is the number of spatial units and J is the number of different variables (categories in the multinomial model). The covariates should each be a K*1 vector, and different regression parameters are estimated for each of the J variables. Missing (NA) values are allowed in the response. - `family`: One of either "binomial", "gaussian", "multinomial", or "poisson", which respectively specify a binomial likelihood model with a logistic link function, a Gaussian likelihood model with an identity link function, a multinomial 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. - `trials`: Only used if family="binomial" or family="multinomial". For the binomial family it is a K*J matrix matrix the same dimension as the response. A the multinomial family it is a vector of length K. - `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.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.Sigma.df`: The degrees of freedom for the Inverse-Wishart prior formulation for the covariance matrix Sigma. This prior formulation follows the marginally weakly-informative specification proposed by Huang and Wand (2013). Defaults to 2, which corresponds to non-informative uniform priors on the interval [-1,1] being assigned to each correlation parameter within the Sigma matrix. - `prior.Sigma.scale`: The J times 1 vector of prior scales for the square roots of the diagonal elements of the covariance matrix Sigma. This prior formulation is the marginally weakly-informative prior specification proposed by Huang and Wand (2013). Thus, the jth element of this vector is the scale parameter for the zero centred half-t prior (with shape given by prior.Sigma.df) assumed for the standard deviation of the random effects corresponding to the jth outcome. Defaults to a vector of values of 100,000. - `rho`: The value in the interval [0, 1] that the spatial dependence parameter rho is fixed at if it should not be estimated. If this arugment is NULL then rho is estimated in the model. - `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" or family="multinomial". - `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 matrix of fitted values based on posterior means from the model. Each row of a matrix relates to an area and each column to a response (category). - **residuals**: A list with 2 elements, where each element is a matrix of a type of residuals. Each row of a matrix relates to an area and each column to a response (category). The types of residual 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**: NULL, for compatability with other models. - **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 Gelfand, A and Vounatsou, P (2003). Proper multivariate conditional autoregressive models for spatial data analysis, Biostatistics, 4, 11-25. Huang, A., and Wand, M (2013). Simple Marginally Noninformative Prior Distributions for Covariance Matrices. Bayesian Analysis, 8, 439-452. Kavanagh, L., D. Lee, and G. Pryce (2016). Is Poverty Decentralising? Quantifying Uncertainty in the Decentralisation of Urban Poverty, Annals of the American Association of Geographers, 106, 1286-1298. Leroux B, Lei X, Breslow N (2000). "Estimation of Disease Rates in SmallAreas: A New Mixed Model for Spatial Dependence." In M Halloran, D Berry (eds.), **Statistical Models in Epidemiology, the Environment and Clinical Trials**, pp. 179-191. Springer-Verlag, New York. ## 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) #### 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 K <- nrow(W) #### Generate the correlation structures Q.W <- 0.99 * (diag(apply(W, 2, sum)) - W) + 0.01 * diag(rep(1,K)) Q.W.inv <- solve(Q.W) Sigma <- matrix(c(1,0.5,0, 0.5,1,0.3, 0, 0.3, 1), nrow=3) Sigma.inv <- solve(Sigma) J <- nrow(Sigma) N.all <- K * J precision.phi <- kronecker(Q.W, Sigma.inv) var.phi <- solve(precision.phi) #### Generate the covariate component x1 <- rnorm(K) x2 <- rnorm(K) XB <- cbind(0.1 * x1 - 0.1*x2, -0.1 * x1 + 0.1*x2, 0.1 * x1 - 0.1*x2) #### Generate the random effects phi <- mvrnorm(n=1, mu=rep(0,N.all), Sigma=var.phi) #### Generate the response data lp <-as.numeric(t(XB)) + phi prob <- exp(lp) / (1 + exp(lp)) trials.vec <- rep(100,N.all) Y.vec <- rbinom(n=N.all, size=trials.vec, prob=prob) #### Turn the data and trials into matrices where each row is an area. Y <- matrix(Y.vec, nrow=K, ncol=J, byrow=TRUE) trials <- matrix(trials.vec, nrow=K, ncol=J, byrow=TRUE) #### Run the Leroux model formula <- Y ~ x1 + x2 ## Not run: model <- MVS.CARleroux(formula=formula, family="binomial", trials=trials, W=W, burnin=20000, n.sample=100000) ## End(Not run) #### Toy example for checking model <- MVS.CARleroux(formula=formula, family="binomial", trials=trials, W=W, burnin=10, n.sample=50) ```

MVS.CARleroux function