# CAR-MM prior model ```r car_mm(d_list, ...) ``` ## Arguments - `d_list`: List of data inputs for the stan model. - `...`: Arguments passed to `rstan::sampling` (e.g. iter, chains). ## Returns An object of class `stanfit` returned by `rstan::sampling` ## Examples ```r set.seed(455) #---- Load data data(W_sel) ## Number of areas n <- nrow(W_sel) ## Number of memberships m <- 153 #---- Simulate covariates X <- cbind(rnorm(nrow(W_sel)), rnorm(nrow(W_sel))) ## Min-max normalisation X_cent <- apply(X, 2, function(x) (x - min(x))/diff(range(x))) #---- Simulate MM matrix w_ord <- c(.5, .35, .15) # Weight of each neighbours orders ord <- length(w_ord) - 1 # Order of neighbours to include H_sel_sim <- sim_MM_matrix( W = W_sel, m = m, ord = ord, w_ord = w_ord, id_vec = rep(1, nrow(W_sel)) ) #---- Simulate outcomes ## Linear term parameters gamma <- -.5 # Intercept beta <- c(1, .5) # Covariates coefficients ## CAR random effects phi_car <- sim_car(W = W_sel, alpha = .9, tau = 5) # Areal log relative risks l_RR <- X_cent %*% beta + phi_car ## Membership log relative risks l_RR_mm <- as.numeric(apply(H_sel_sim, 1, function(x) x %*% l_RR)) ## Expected rates exp_rates <- rpois(m, lambda = 20) ## Outcomes y <- rpois(m, lambda = exp_rates*exp(l_RR_mm)) #---- Create dataset for stan function d_sel <- list( # Number of areas n = nrow(W_sel), # Covariates k = ncol(X_cent), X_cov = X_cent, # Adjacency W_n = sum(W_sel) / 2, # Number of neighbour pairs W = W_sel, # Memberships m = nrow(H_sel_sim), H = H_sel_sim, # Outcomes y = y, log_offset = log(exp_rates), # Prior parameters ## Intercept (mean and sd of normal prior) mu_gamma = 0, sigma_gamma = 1, ## Covariates (mean and sd of normal prior) mu_beta = 0, sigma_beta = 1, ## Marginal precision gamma prior tau_shape = 2, tau_rate = 0.2 ) #---- HMC parameters niter <- 1E4 nchains <- 4 #---- Stan sampling fit <- car_mm( d_list = d_sel, # arguments passed to sampling iter = niter, chains = nchains, refresh = 500, control = list(adapt_delta = .99, max_treedepth = 15) ) ``` ## References Marco Gramatica. Silvia Liverani. Peter Congdon. "Structure Induced by a Multiple Membership Transformation on the Conditional Autoregressive Model." Bayesian Analysis Advance Publication 1 - 25, 2023. https://doi.org/10.1214/23-BA1370 Petrof, O, Neyens, T, Nuyts, V, Nackaerts, K, Nemery, B, Faes, C. On the impact of residential history in the spatial analysis of diseases with a long latency period: A study of mesothelioma in Belgium. Statistics in Medicine. 2020; 39: 3840– 3866. https://doi.org/10.1002/sim.8697

car_mm function