- A Multivariate Model for Spatio-temporal Health Outcomes with an Application to Suicide Mortality
This article considers models for multivariate mortality outcomes (e.g., bivariate, trivariate, or higher dimensional) observed over a set of areas and through time. The model outlined here allows for spatially structured and white noise errors and for their intercorrelation. It also includes possible temporal continuity in such types of error via structured temporal effects. An extension to spatially varying regression effects is considered, as well as the option of nonparametric specification of priors for spatial residuals and regression effects. Allowing for spatially correlated intercepts or regression effects may alter inferences regarding the changing impact on mortality of socioeconomic or environmental predictors. The modeling framework is illustrated by an application to male and female suicide mortality in London, focusing on the impact on suicide of deprivation and social fragmentation ("anomie") in the 33 London boroughs during three periods: 1979–83, 1984–88 and 1989–93. Suicide trends by age group are also considered and show considerable differences in the trends in impacts of deprivation and social fragmentation.
The evolution over time of interdependent and spatially defined health outcomes has relevance in several modeling contexts, including Bayesian smoothing and regression modeling of social and environmental risk factors. Authors such as Waller et al. (1997), Knorr-Held and Besag (1998), Gelfand et al. (1998), and Sun et al. (2000) have considered the benefits of a fully Bayesian estimation approach to spatio-temporal disease mapping, using general linear model techniques applied to a count variable with Poisson sampling. However, such applications have been confined to a single (i.e., univariate) outcome. Only recently has the possibility of spatio-temporal analyses that extend to multivariate outcomes been considered in Carlin and Banerjee (2003). The vast majority of existing models for mortality- and disease-count responses also do not allow for spatially varying regression effects that are a source of spatially correlated residuals (Fotheringham, Charlton, and Brunsdon 2002). The predominant Geographically Weighted Regression methodology is a fixed effects one that does not pool strength over areas and is not practical for multivariate longitudinal [End Page 234] applications. A related point is that the great majority of count regression models for disease assume binary adjacency to represent spatial interaction; less commonly, distance decay modeling has been used (e.g., Conlon and Waller 1999).
An elegant bivariate spatial model for spatial residuals has been outlined in a cross-sectional application by Kim, Sun, and Tsutakawa (2001), but its nonlinear form would be difficult to extend to J > 2 jointly dependent outcomes or to longitudinal applications. More flexible multivariate priors has been proposed by Gelfand and Vounatsou (2003) and Carlin and Banerjee (2003), but these do not consider interdependence between spatial errors and unstructured errors and consider only binary adjacency forms of spatial dependency. Gelfand and Vounatsou (2003) consider a cross-sectional application and spatial residuals only. Carlin and Banerjee (2003) consider both spatial errors and spatially varying regression effects and present a particular form of spatio-temporal model in terms of a multivariate CAR prior specific for outcome and patient cohort (i.e., time varying spatial correlation).1 Carlin and Banerjee do not consider structured (e.g., autoregressive in time) priors for time effects or space-time effects. A random effects spatial regression approach is also proposed by Gamerman, Moreira, and Rue (2003) but for metric rather than discrete outcomes, and in a cross-sectional context only.
By contrast, the current paper considers multivariate outcomes in a spatio-temporal context and allows for interdependence between spatial errors and unstructured errors, and between spatial errors and spatially varying regression effects. It considers how spatially correlated residuals may be unnecessary when spatial regression effects are introduced. It also considers both contiguity- and distance-based spatial dependency and discusses questions around structured temporal effects such as temporal lags in both spatial residuals and spatial regression effects, and cross-lags between outcomes in such effects.
Fully Bayesian techniques developing on the approach of Langford et al. (1999) are utilized to assess how inferences regarding the impact of socioeconomic predictors are affected by allowing for spatial, error, and regressor effects. The method as...