MVST.CARar {CARBayesST}

Fit a multivariate spatio-temporal generalised linear mixed model to data, with a multivariate spatio-temporal autoregressive process.

Description

Fit a multivariate spatio-temporal generalised linear mixed model to multivariate areal unit data, where the response variable can be binomial, Gaussian or Poisson. The linear predictor is modelled by known covariates and a vector of random effects. The latter allows for correlations over: (i) K areal units; (ii) N time periods; and (iii) J outcomes. These random effects are modelled by either a multivariate first order autoregressive time series process or a multivariate second order autoregressive time series process. In both cases the spatial and between outcome correlation is modelled via the precision matrix, and the spatial correlation is represented by the conditional autoregressive (CAR) prior proposed by Leroux et al. (2000). In contrast, the between outcome correlation structure is estimated from the data, and no prior form is assumed. Missing values are allowed in the response in this model, and are sampled from in the model using data augmentation. Further details are given in the vignette accompanying this package. Inference is conducted in a Bayesian setting using Markov chain Monte Carlo (MCMC) simulation.

Usage

MVST.CARar(formula, family, data=NULL,  trials=NULL, W, burnin, n.sample, thin=1, 
n.chains=1,  n.cores=1, prior.mean.beta=NULL, prior.var.beta=NULL, prior.nu2=NULL, 
prior.Sigma.df=NULL, prior.Sigma.scale=NULL, AR=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 the offset (if included) should be matrices of dimension (KN)*J, where K is the number of spatial units, N is the number of time periods and J is the number of different variables. Each column of the response and offset matrices relates to a different outcome variable. The values in each column of these matrices 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 covariates should each be a (KN)*1 vector, and different regression parameters are estimated for each of the J variables. 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 (KN)*J matrix of the same dimension as the response. 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).

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_j for each outcome j. Defaults to c(1, 0.01) and only used if family="Gaussian".

prior.Sigma.df

The degrees of freedom for the Inverse-Wishart prior formulation for the covariance matrix Sigma. This prior formulation follows the marginally weakly-informative specification proposed by Huang and Wand (2013). Defaults to 2, which corresponds to non-informative uniform priors on the interval [-1,1] being assigned to each correlation parameter within the Sigma matrix.

prior.Sigma.scale

The J times 1 vector of prior scales for the square roots of the diagonal elements of the covariance matrix Sigma. This prior formulation is the marginally weakly-informative prior specification proposed by Huang and Wand (2013). Thus, the jth element of this vector is the scale parameter for the zero centred half-t prior (with shape given by prior.Sigma.df) assumed for the standard deviation of the random effects corresponding to the jth outcome. Defaults to a vector of values of 100,000.

AR

The order of the autoregressive time series process that must be either 1 or 2.

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

Whether to fix or estimate the temporal dependence parameter(s) rho.T in the model. If this arugment is NULL then they are estimated in the model. If you want to fix them and AR=1 then it must be a single value. If AR=2 then it must be a vector of length two with the first and second order autoregressive coefficients.

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.

Value

summary.results

A summary table of the parameters.

samples

A list containing the MCMC samples from the model.

fitted.values

A matrix of fitted values for each area, time period and response variable in the same order as the response variable.

residuals

A list with 2 elements, where each element is a matrix of a type of residual. Each row of a matrix relates to an area and time period and each column to a response (category). The types of residual 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.

Author(s)

Duncan Lee

References

Gelfand, A and Vounatsou, P (2003). Proper multivariate conditional autoregressive models for spatial data analysis, Biostatistics, 4, 11-25.

Huang, A., and Wand, M (2013). Simple Marginally Noninformative Prior Distributions for Covariance Matrices. Bayesian Analysis, 8, 439-452.

Lee D, Robertson C, and Marques, D (2022). Quantifying the small-area spatio-temporal dynamics of the Covid-19 pandemic in Scotland during a period with limited testing capacity, Spatial Statistics, https://doi.org/10.1016/j.spasta.2021.100508.

Leroux B, Lei X, Breslow N (2000). "Estimation of Disease Rates in SmallAreas: A New Mixed Model for Spatial Dependence." In M Halloran, D Berry (eds.), Statistical Models in Epidemiology, the Environment and Clinical Trials, pp. 179-191. Springer-Verlag, New York.

Examples

#################################################
#### Run the model on simulated data on a lattice
#################################################
#### Set up a square lattice region
x.easting <- 1:8
x.northing <- 1:8
Grid <- expand.grid(x.easting, x.northing)


#### Set up the coordinate dimensions
K <- nrow(Grid)
N <- 15
J <- 2
N.all <- N * K * J


#### 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 	


#### Set up the spatial covariance matrix
Q.W <- 0.8 * (diag(apply(W, 2, sum)) - W) + 0.2 * diag(rep(1,K))
Q.W.inv <- solve(Q.W)


#### Set up the multivariate outcome covariance matrix
Sigma <- 0.01 * array(c(1, 1, 1, 2), c(2,2))
Sigma.inv <- solve(Sigma)


#### Spatial and between outcome covariance
QSig.prec <- kronecker(Q.W, Sigma.inv)
QSig.var <-solve(QSig.prec)


#### Generate the covariate
x1 <- rnorm(n=N * K, mean=0, sd=1)
lp.regression.mat <- cbind(0.1 + 0.1 * x1, 0.1 - 0.1*x1)
lp.regression <- as.numeric(t(lp.regression.mat))


#### Spatio-temporal random effects
phi.temp <- mvrnorm(n=1, mu=rep(0,(J*K)), Sigma=QSig.var)
phi <- phi.temp
    for(i in 2:N)
    {
    phi.temp2 <- mvrnorm(n=1, mu=(0.8 * phi.temp), Sigma=QSig.var)
    phi.temp <- phi.temp2
    phi <- c(phi, phi.temp)
    }
phi <- phi - mean(phi)
phi.true <- matrix(phi, ncol=2, byrow=TRUE)


#### Generate the binomial counts
lp <- lp.regression + phi
p <- exp(lp) / (1+exp(lp))
trials <- rpois(N.all, lambda=100)
Y <- rbinom(n=N.all, size=trials, prob=p)
Y.mat <- matrix(Y, nrow=(K*N), ncol=J, byrow=TRUE)
trials.mat <- matrix(trials, nrow=(K*N), ncol=J, byrow=TRUE)
formula <- Y.mat~x1

#### Run the model
formula <- Y.mat ~ x1
## Not run: mod <- MVST.CARar(formula=formula, family="binomial", trials=trials.mat, W=W, 
burnin=10000, n.sample=50000, AR=1, MALA=FALSE)
## End(Not run)

#### Toy example for checking
mod <- MVST.CARar(formula=formula, family="binomial", trials=trials.mat, W=W, 
burnin=10, n.sample=50, AR=1, MALA=FALSE)