# Fit a spatio-temporal generalised linear mixed model to data, with a spatio-temporal autoregressive process that has an adaptive autocorrelation stucture. 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 a CAR type structure and a neighbourhood (adjacency) matrix W. The non-zero elements of W associated with each pair of geographically adjacent areal units are treated as random variables with ranges in the unit interval, which allows step changes to be identified in the random effects surface between geographically adjacent regions. The model is similar to that proposed by Rushworth et al. (2017). Further details are given in the vignette accompanying this package. Inference is conducted in a Bayesian setting using Markov chain Monte Carlo (MCMC) simulation. ```r 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, rho = NULL, epsilon = 0, 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", "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 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. For this model the matrix must be binary. - `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). - `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). - `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. Setting rho=1, reduces the random effects prior to the intrinsic CAR model but does require epsilon>0. - `epsilon`: Diagonal ridge parameter to add to the random effects prior precision matrix, only required when rho = 1, and the prior precision is improper. Defaults to 0. - `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 list with 2 K*K matrices, Wmedian and W99 summarising the estimated adjacency relationships. Wmedian contains the posterior median for each w_ij element estimated in the model for adjacent areal units, while W99 contains binary indicator variables for whether Prob(w_ij < 0.5|data)>0.99. For both matrices, elements corresponding to non-adjacent pairs of 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. - **X**: The design matrix of covariates. ## References Rushworth, A., Lee, D., and Sarran, C (2017). An adaptive spatio-temporal smoothing model for estimating trends and step changes in disease risk. Journal of the Royal Statistical Society Series C, 66, 141-157. ## Author(s) Alastair Rushworth ## 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.CARadaptive(formula=Y~1, family="poisson", W=W, burnin=10000, n.sample=50000) ## End(Not run) #### Toy example for checking model <- ST.CARadaptive(formula=Y~1, family="poisson", W=W, burnin=10, n.sample=50) ```

ST.CARadaptive function