CAR-MM Modelling in Stan.

CARME

Stan based functions to estimate CAR-MM models. These models allow to estimate Generalised Linear Models with CAR (conditional autoregressive) spatial random effects for spatially and temporally misaligned data, provided a suitable Multiple Membership matrix.

The main references are:

• Gramatica, M, Liverani, S, Congdon, P.
  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
• Gramatica, M, Congdon, P, Liverani, S.
  Bayesian Modelling for Spatially Misaligned Health Areal Data: A Multiple Membership Approach.
  Journal of the Royal Statistical Society Series C: Applied Statistics,
  Volume 70, Issue 3, June 2021, Pages 645–666.
  https://doi.org/10.1111/rssc.12480

Installation

# CRAN installation
install.packages("CARME")

# Or the development version from GitHub:
# install.packages("devtools")
devtools::install_github("markgrama/CARME")

Usage

We start by setting the number of membership we wish to use. In this case we take the adjacency matrix from the South East London (SEL) dataset used in Gramatica et al. (2023). It consists of $m = 153$ memberships and $n = 152$ areas.

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)))

Then, we simulate a $(153 \times 15)2$ Multiple Membership matrix, together with the observed outcomes generated using a poisson distribution and a CAR prior.

    #---- 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
)

Finally, we can run the model:

 #---- 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)
)