Hierarchical Regression with Misaligned Spatial Data Relating Ambient Ozone and Pediatric Asthma ER Visits in Atlanta

Recent w ork (Gelfand, Zhu and Carlin, 2001) enables a hierarchical modeling solution to the general change of support problem (COSP), where we s e e k t o m a k e inferences about the values of a variable at either points or regions diierent from those at which it has been observed. The approach presumes an underlying continuous stationary spatial process X(s) for locations s 2 D, a region of interest. It is straightforwardly extended to the spatio-temporal setting by assuming a space-time separable covariance function for the X(s t) process, t 2 T and again s 2 D. In this paper we take up the associated misaligned regression problem, where, for example, the explanatory variable is envisioned from the X(s t) process, while the response is available only as areal summaries over a The authors are grateful to Profs. Paige Tolbert and James Mulholland for providing and permitting analysis of the Atlanta ozone and asthma dataset. 1 particular grid at particular times. We illustrate using a dataset relating several air quality indicators (ozone, particulate matter, nitrogen oxides, etc.) and a range of sociodemographic variables (age, gender, race, and a socioeconomic status surrogate) to the response, pediatric emergency room (ER) visit counts for asthma in Atlanta, Georgia. Here the air quality data is collected at xed monitoring stations (point locations) while the sociodemographic covariates and response variable is collected by zip code (areal summaries). In fact, the air quality data is available as daily averages at each monitoring station, and the response is available as daily counts of visits in each zip code. As is now common in hierarchical Bayesian spatial applications, computing is implemented via a carefully tailored Metropolis-Hastings algorithm, with map summaries created using a geographic information system (GIS). Like m a n y recent i n vestigations of the ozone-asthma link, our results suggest a small positive association, though our data can permit this conclusion only for those children living in the city o f A tlanta who use the ER to manage their asthma.