# Fit a spatio-temporal generalised linear mixed model to data, with spatial and temporal main effects and a spatio-temporal interaction. 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 three vectors of random effects. The latter include spatial and temporal main effects and a spatio-temporal interaction. The spatial and temporal main effects are modelled by the conditional autoregressive (CAR) prior proposed by Leroux et al. (2000), while the spatio-temporal interaction random effects are independent. Due to the lack of identifiability between the interactions and the Gaussian errors, only main effects are allowed in the Gaussian model. Missing values are allowed in the response in this model, and are sampled from in the model using data augmentation. The model is similar to that proposed by Knorr-Held (2000), 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. ```r ST.CARanova(formula, family, data=NULL, trials=NULL, W, interaction=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, rho.S=NULL, rho.T=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 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 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. - `interaction`: TRUE or FALSE indicating whether the spatio-temporal interaction random effects should be included. Defaults to TRUE unless family="gaussian" in which case interactions are not allowed. - `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.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.S`: The value in the interval [0, 1] that the spatial dependence parameter rho.S is fixed at if it should not be estimated. If this arugment is NULL then rho.S is estimated in the model. - `rho.T`: The value in the interval [0, 1] that the temporal dependence parameter rho.T is fixed at if it should not be estimated. If this arugment is NULL then rho.T is estimated in the model. - `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**: NULL, for compatability with the 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 Knorr-Held, L. (2000). Bayesian modelling of inseparable space-time variation in disease risk. Statistics in Medicine, 19, 2555-2567. Leroux, B., X. Lei, and N. Breslow (2000). Estimation of disease rates in small areas: A new mixed model for spatial dependence, Chapter Statistical Models in Epidemiology, the Environment and Clinical Trials, Halloran, M and Berry, D (eds), pp. 135-178. Springer-Verlag, New York. ## 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 (W) and temporal (D) neighbourhood matrices distance <- as.matrix(dist(Grid)) W <-array(0, c(K,K)) W[distance==1] <-1 D <-array(0, c(N,N)) for(i in 1:N) { for(j in 1:N) { if(abs((i-j))==1) D[i,j] <- 1 } } #### Simulate the elements in the linear predictor and the data gamma <- rnorm(n=N.all, mean=0, sd=0.001) x <- rnorm(n=N.all, mean=0, sd=1) beta <- 0.1 Q.W <- 0.99 * (diag(apply(W, 2, sum)) - W) + 0.01 * diag(rep(1,K)) Q.W.inv <- solve(Q.W) phi <- mvrnorm(n=1, mu=rep(0,K), Sigma=(0.01 * Q.W.inv)) Q.D <- 0.99 * (diag(apply(D, 2, sum)) - D) + 0.01 * diag(rep(1,N)) Q.D.inv <- solve(Q.D) delta <- mvrnorm(n=1, mu=rep(0,N), Sigma=(0.01 * Q.D.inv)) phi.long <- rep(phi, N) delta.long <- kronecker(delta, rep(1,K)) LP <- 4 + x * beta + phi.long + delta.long + gamma mean <- exp(LP) Y <- rpois(n=N.all, lambda=mean) #### Run the model ## Not run: model <- ST.CARanova(formula=Y~x, family="poisson", interaction=TRUE, W=W, burnin=10000, n.sample=50000) ## End(Not run) #### Toy example for checking model <- ST.CARanova(formula=Y~x, family="poisson", interaction=TRUE, W=W, burnin=10, n.sample=50) ```

ST.CARanova function