Reduced hierarchical models with application to estimating health effects of simultaneous exposure to multiple pollutants.

Hierarchical models (HM) have been used extensively in multisite time series studies of air pollution and health to estimate health effects of a single pollutant adjusted for other pollutants and other time-varying factors. Recently, Environmental Protection Agency (EPA) has called for research quantifying health effects of simultaneous exposure to many air pollutants. However, straightforward application of HM in this context is challenged by the need to specify a random-effect distribution on a high-dimensional vector of nuisance parameters. Here we introduce reduced HM as a general statistical approach for analyzing correlated data with many nuisance parameters. For reduced HM we first calculate the integrated likelihood of the parameter of interest (e.g. excess number of deaths attributed to simultaneous exposure to high levels of many pollutants), and we then specify a flexible random-effect distribution directly on this parameter. Simulation studies show that the reduced HM performs comparably to the full HM in many scenarios, and even performs better in some cases, particularly when the multivariate random-effect distribution of the full HM is misspecified. Methods are applied to estimate relative risks of cardiovascular hospital admissions associated with simultaneous exposure to elevated levels of particulate matter and ozone in 51 US counties during 1999-2005.