Generalized Hierarchical Multivariate CAR Models for Areal Data

Abstract

In the fields of medicine and public health, a common application of areal data models is the study of geographical patterns of disease. When we have several measurements recorded at each spatial location (for example, information on p>/= 2 diseases from the same population groups or regions), we need to consider multivariate areal data models in order to handle the dependence among the multivariate components as well as the spatial dependence between sites. In this article, we propose a flexible new class of generalized multivariate conditionally autoregressive (GMCAR) models for areal data, and show how it enriches the MCAR class. Our approach differs from earlier ones in that it directly specifies the joint distribution for a multivariate Markov random field (MRF) through the specification of simpler conditional and marginal models. This in turn leads to a significant reduction in the computational burden in hierarchical spatial random effect modeling, where posterior summaries are computed using Markov chain Monte Carlo (MCMC). We compare our approach with existing MCAR models in the literature via simulation, using average mean square error (AMSE) and a convenient hierarchical model selection criterion, the deviance information criterion (DIC; Spiegelhalter et al., 2002, Journal of the Royal Statistical Society, Series B64, 583-639). Finally, we offer a real-data application of our proposed GMCAR approach that models lung and esophagus cancer death rates during 1991-1998 in Minnesota counties.

Modelling geographically referenced survival data with a cure fraction

“…When multiple parameter sets (such as frailties in cure fractions and the Weibull link) need to be modelled jointly, as opposed to independently, we resort to multivariate CAR models originally proposed by Mardia 34 ; see also the works of Carlin and Banerjee35 and Gelfand and Vounatsou.36 Let be a vector of p variables associated with the ith region. Collecting these effects into , the joint distribution can be written down as (5) where B R is an np × np matrix with block elements (B R ) ij = R i B ij and 0 as diagonals, R i and B ij are p × p matrices, and Γ is an np × np block diagonal matrix with block elements Γ i , i = 1,…, n. Although the Kronecker structure offers computational and interpretational simplicity, there has been much research on developing more general spatial covariances, notably by Kim et al, 37 Carlin and Banerjee 35 and Jin et al 20 The last work offers a Generalized Multivariate Conditionally Auto Regressive (GMCAR) model with emphasis on computational simplicity.…”

“…Writing this joint distribution as (6) 20 refer to such models as GMCAR (α 1 , α 2 , γ 0 , γ 1 , τ 1 , τ 2 ) and show that many existing MCAR models are special instances of the GMCAR.…”

“…We work with univariate and multivariate conditionally autoregressive (CAR) models that accrue computational benefits, while incorporating spatial effects. In particular, we investigate a Generalized multivariate CAR (GMCAR) distribution, recently investigated by Jin et al 20 that encompasses very rich multivariate spatial associations.…”

Spatial Data Analysis

“…The key problem here is to specify rich and flexible spatial distributions for the φ ij s. Carlin & Banerjee (11) and Gelfand & Vounatsou (21) generalized the univariate CAR (2) to a joint model for the random effects φ ij , which permits modeling of correlation among the p diseases while maintaining spatial dependence for each of the diseases. These models were subsequently subsumed by more general, and flexible, Bayesian hierarchical frameworks developed and implemented by Jin et al (27,28).…”

Default Bayesian analysis for multivariate generalized CAR models

“…Multiple response variables are available as indicators of health status, and as a result, models for multivariate spatial lattice data are an indispensable tool for analyzing health disparity data. Recently, Greco and Trivisano (2009), Zhang et al (2009), Jin et al (2007), Sain and Cressie (2007), Jin et al (2005), Gelfand and Vounatsou (2003), Carlin and Banerjee (2003) and Kim et al (2001) explored multivariate spatial models for lattice data, adopting the Bayesian framework as the natural inferential approach. The only exception, Sain (2009) developed the maximum likelihood estimation procedure for a special case, the multivariate gaussian conditional autoregressive (CAR) model of Sain and Cressie (2007).…”