Distributed lag interaction models with two pollutants

Distributed lag models (DLMs) have been widely used in environmental epidemiology to quantify the lagged effects of air pollution on a health outcome of interest such as mortality and morbidity. Most previous DLM approaches consider only one pollutant at a time. We propose a distributed lag interaction model to characterize the joint lagged effect of two pollutants. One natural way to model the interaction surface is by assuming that the underlying basis functions are tensor products of the basis functions that generate the main effect distributed lag functions. We extend Tukey's 1 degree-of-freedom interaction structure to the two-dimensional DLM context. We also consider shrinkage versions of the two to allow departure from the specified Tukey interaction structure and achieve bias-variance trade-off. We derive the marginal lag effects of one pollutant when the other pollutant is fixed at certain quantiles. In a simulation study, we show that the shrinkage methods have better average performance in terms of mean-squared error across various scenarios. We illustrate the methods proposed by using the 'National morbidity, mortality, and air pollution study' data to model the joint effects of particulate matter and ozone on mortality count in Chicago, Illinois, from 1987 to 2000.