CARBayes (version 5.2.5)

S.CARmultilevel: Fit a spatial generalised linear mixed model to multi-level areal unit data, where the spatial random effects have a Leroux conditional autoregressive prior and there are also individual or small group level random effects.

Description

Fit a spatial generalised linear mixed model to multi-level areal unit data, where the response variable can be binomial, Gaussian or Poisson. The data are structured with individuals within areal units, and different numbers of individuals are allowed within each areal unit. The linear predictor is modelled by known covariates (either individual or areal level) and two vectors of random effects. The latter include areal level effects modelled by the conditional autoregressive prior proposed by Leroux et al. (2000). Independent random effects can be obtained by setting rho=0, while the intrinsic CAR model can be obtained by setting rho=1. The second set of random effects are determined by a single effect for each level of a categorical covariate. These random effects are independent with a common variance sigma2. 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 a full model specification see the vignette accompanying this package.

Usage

S.CARmultilevel(formula, family, data=NULL,  trials=NULL, W, ind.area, 
    ind.re=NULL, burnin, n.sample, thin=1, prior.mean.beta=NULL, prior.var.beta=NULL,
    prior.nu2=NULL, prior.tau2=NULL, prior.sigma2=NULL, rho=NULL, MALA=FALSE,
    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, offset and each covariate are vectors with length equal to the number of individuals. 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 as the response containing the total number of trials for each individual. 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.

ind.area

A vector of integers the same length as the number of data points (individuals) giving which spatial unit (nunmbered from 1 to K to align with the rows of the W matrix) each individual belongs to.

ind.re

A categorical variable where each level will be assigned a different independent random effect. If NULL these random effects are omitted.

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 100000.

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).

prior.sigma2

The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for sigma2. 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.

MALA

Logical, should the function use Metropolis adjusted Langevin algorithm (MALA) updates (TRUE) or simple random walk (FALSE, default) updates for the regression parameters. Not applicable if family="gaussian".

verbose

Logical, should the function update the user on its progress.

Value

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.

residuals

A matrix with 2 columns where each column is a type of residual and each row relates to an area. 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.

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.

X

The design matrix of covariates.

Examples

# NOT RUN {
#################################################
#### Run the model on simulated data on a lattice
#################################################
    
#### 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 	

#### Generate the number of individuals per area and which individuals to which areas
n <- sample(5:30, K, replace=TRUE)
n.total <- sum(n)
ind.area.temp <- rep(1:K, n)
ind.area <- sample(ind.area.temp, n.total, replace=FALSE)

#### Generate a categorical variable
n.levels <- n.total / (0.5*1+0.36*2+0.14*3)
m2 <- round(n.levels * 0.36)
m3 <- round(n.levels * 0.14)
levels.temp <- c(kronecker(1:m3, rep(1,3)), kronecker((m3+1):(m3+m2), rep(1,2)))
m1 <- n.total-length(levels.temp)
levels.temp2 <- c(levels.temp, (max(levels.temp)+1):(max(levels.temp)+m1))
ind.re <-sample(levels.temp2)
q <- m1+m2+m3

#### Generate the covariates and response data
x1 <- rnorm(n.total)
x2 <- rnorm(n.total)
phi <- mvrnorm(n=1, mu=rep(0,K), Sigma=0.4 * exp(-0.1 * distance))
phi.extend <- phi[ind.area]
zeta <- rnorm(n=q, mean=0, sd=0.1)
zeta.extend <- zeta[ind.re] 
logit <- x1 + x2 + phi.extend + zeta.extend
prob <- exp(logit) / (1 + exp(logit))
trials <- rep(50,n.total)
Y <- rbinom(n=n.total, size=trials, prob=prob)
    
    
#### Run the model
formula <- Y ~ x1 + x2
# }
# NOT RUN {
model <- S.CARmultilevel(formula=formula, family="binomial", ind.area=ind.area,
                ind.re=as.factor(ind.re), trials=trials, W=W, burnin=20000, n.sample=100000)
# }
# NOT RUN {
                
#### Toy example for checking
model <- S.CARmultilevel(formula=formula, family="binomial", ind.area=ind.area,
                ind.re=as.factor(ind.re), trials=trials, W=W, burnin=10, n.sample=50)
# }