Hierarchical regression with misaligned spatial data: Relating ambient ozone and pediatric asthma ER visits in Atlanta

Recent work by the authors enabled a hierarchical modeling solution to the general change of support problem (COSP), where they seek to make inferences about the values of a variable at either points or regions different from those at which it has been observed. The approach presumes an underlying continuous stationary spatial process X(s) for locations s ∈ 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 ∈ T and again s ∈ D. In this article they 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 particular grid at particular times. They 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 fixed monitoring stations (point locations) while the sociodemographic covariates and response variable is collected by zip code (areal summaries). In fact, the air quality data are 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 many recent investigations 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 of Atlanta who use the ER to manage their asthma.