ST.CARclustrends {CARBayesST}

Fit a spatio-temporal generalised linear mixed model to data, with a clustering of temporal trend functions and a temporally common spatial surface.

Description

Fit a spatio-temporal generalised linear mixed model to areal unit data, where the response variable can be binomial or Poisson. The linear predictor is modelled by known covariates, a temoporally common spatial surface, and a mixture of temporal trend functions. The spatial component is modelled by the conditional autoregressive (CAR) prior proposed by Leroux et al. (2000). The temporal trend functions are user-specified and are fixed parametric forms (e.g. linear, step-change) or constrained shapes (e.g. monotonically increasing). Further details are given in Napier et al. (2018) and in the vignette accompanying this package. Inference is conducted in a Bayesian setting using Metropolis coupled Markov chain Monte Carlo (MCMCMC) simulation.

Usage

    ST.CARclustrends(formula, family, data=NULL, trials=NULL, W, burnin, n.sample,
    thin=1, trends=NULL, changepoint=NULL, knots=NULL, prior.mean.beta=NULL,
    prior.var.beta=NULL, prior.mean.gamma=NULL, prior.var.gamma=NULL,
    prior.lambda=NULL, prior.tau2=NULL, Nchains=4, verbose=TRUE)

Arguments

formula

A formula using the syntax of the lm() function. However, due to identifiability issues covariates are not allowed. So the only elements allowed on the right side of the formula are an intercept term and an offset term using the offset() function. The response variable and the offset (if specified) 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.

family

One of either "binomial" or"poisson", which respectively specify a binomial likelihood model with a logistic 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.

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).

trends

A vector containing the temporal trend functions to include in the model, which include: constant ("Constant""); linear decreasing ("LD"); linear increasing ("LI"); Known change point, where the trend can increase towards the change point before subsequently decreasing ("CP"); or decrease towards the change point before subsequently increasing ("CT"); and monotonic cubic splines which are decreasing ("MD") or increasing ("MI"). At least two trends have to be selected, with the constant trend always included. To avoid identifiability problems only one of "LI" or "MI" can be included at a given time (similarily for "LD" and "MD").

changepoint

A scalar indicating the position of the change point should one of the change point trend functions be included in the trends vector, i.e. if "CP" or "CT" is specified.

knots

A scalar indicating the number of knots to use should one of the monotonic cubic splines trend functions be included in the trends vector, i.e. if "MD" or "MI" is specified.

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.mean.gamma

A vector of prior means for the temporal trend parameters (Gaussian priors are assumed). Defaults to a vector of zeros.

prior.var.gamma

A vector of prior variances for the temporal trend parameters (Gaussian priors are assumed). Defaults to a vector with values 100,000.

prior.lambda

A vector of prior samples sizes for the Dirichlet prior controlling the probabilities that each trend function is chosen. The vector should be the same length as the trends vector and defaults to a vector of ones.

prior.tau2

The prior shape and scale in the form of c(shape, scale) for an Inverse-Gamma(shape, scale) prior for the random effect variances tau2. Defaults to c(1, 0.01).

Nchains

The number of parallel Markov chains to be used in the Metropolis coupled Markov chain Monte Carlo (MCMCMC) simulations. Defaults to 4.

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 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 containing two elements. The first is "trends", which is a vector the same length and in the same order as the number of areas. The kth element specifies which trend area k has been allocated to based on the posterior mode. The second element is "trend.probs", which is a matrix containing the probabilities associated with each trend for each areal unit.

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.

Author(s)

Gary Napier

References

Leroux, B., Lei, X., and Breslow, N. (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.

Napier, G., Lee, D., Robertson, C., and Lawson, A. (2019). A Bayesian space-time model for clustering areal units based on their disease trends, Biostatistics, 20, 681-697.

Examples

##################################################
#### Run the model on simulated data on a lattice
##################################################
#### Load libraries
library(truncnorm)
library(gtools)


#### 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 	


#### Create the spatial covariance matrix
Q.W <- 0.99 * (diag(apply(W, 2, sum)) - W) + 0.01 * diag(rep(1,K))
Q.W.inv <- solve(Q.W)
  
           
#### Simulate the elements in the linear predictor and the data
beta <- 0.01
gamma <- 0.7
phi <- mvrnorm(n=1, mu=rep(0,K), Sigma=(0.01 * Q.W.inv))

lambda <- rep(1/2, 2)
w <- t(rmultinom(K, 1, lambda))

Y <- matrix(NA, nrow = K, ncol = N)
for (i in 1:N)
{
  LP <- beta + w[, 2] * (gamma * i) + phi
  mean <- exp(LP)
  Y[, i] <- rpois(n=K, lambda=mean)
 }
Y <- as.numeric(Y)


#### Run the model
## Not run: model <- ST.CARclustrends(formula=Y~1, family="poisson", W=W, burnin=10000, 
n.sample=50000, trends=c("Constant", "LI"))
## End(Not run)


#### Toy example for checking
model <- ST.CARclustrends(formula=Y~1, family="poisson", W=W, burnin=10, 
n.sample=50, trends=c("Constant", "LI"))