Bayesian distributed lag models: estimating effects of particulate matter air pollution on daily mortality

A distributed lag model (DLagM) is a regression model that includes lagged exposure variables as covariates; its corresponding distributed lag (DL) function describes the relationship between the lag and the coefficient of the lagged exposure variable. DLagMs have recently been used in environmental epidemiology for quantifying the cumulative effects of weather and air pollution on mortality and morbidity. Standard methods for formulating DLagMs include unconstrained, polynomial, and penalized spline DLagMs. These methods may fail to take full advantage of prior information about the shape of the DL function for environmental exposures, or for any other exposure with effects that are believed to smoothly approach zero as lag increases, and are therefore at risk of producing suboptimal estimates. In this article, we propose a Bayesian DLagM (BDLagM) that incorporates prior knowledge about the shape of the DL function and also allows the degree of smoothness of the DL function to be estimated from the data. We apply our BDLagM to its motivating data from the National Morbidity, Mortality, and Air Pollution Study to estimate the short-term health effects of particulate matter air pollution on mortality from 1987 to 2000 for Chicago, Illinois. In a simulation study, we compare our Bayesian approach with alternative methods that use unconstrained, polynomial, and penalized spline DLagMs. We also illustrate the connection between BDLagMs and penalized spline DLagMs. Software for fitting BDLagM models and the data used in this article are available online.