Robust Distributed Lag Models with Multiple Pollutants using Data Adaptive Shrinkage

Abstract

There is growing interest in investigating the short-term delayed lag effects of environmental pollutants (e.g. air particulate matter and ozone) on a health outcome of interest measured at a certain time (e.g. daily mortality counts). Previous studies have shown that not only the current level of the exposure but exposure levels up to past few days may be associated with health event/outcome measured on current day. Distributed lag model (DLM) has been used in environmental epidemiology to characterize the lag structure of exposure effects. These models assume that the coefficients corresponding to exposures at different lags follow a given function of the lags. Under mis-specification of this function, DLM can lead to seriously biased estimates. In this dissertation, we first explore different methods to make the traditional DLM more robust. We then extend the single pollutant DLM to multi-pollutant scenarios. We illustrate the proposed methods using air pollution data from the National Morbidity, Mortality and Air Pollution Study (NMMAPS) and a dataset from Brigham and Women's Hospital (BWH) prospective birth cohort study.

In the first project, we propose three classes of shrinkage methods to combine an unconstrained DLM estimator and a constrained DLM estimator and achieve a balance between robustness and efficiency. The three classes of methods can be broadly described as (1) empirical Bayes-type shrinkage, (2) hierarchical Bayes, and (3) generalized ridge regression. A two-step double shrinkage approach that enforces the effect estimates approach zero at larger lags is also considered. A simulation study shows that all four approaches are effective in trading off between bias and variance to attain lower mean squared error with the two-step approaches having edge over others.

In the second project, we extend DLM to two-pollutant scenarios and focus on characterizing pollutant-by-pollutant interaction. We first consider to model the interaction surface by assuming the underlying basis functions are tensor products of the basis functions that generate the main-effect distributed lag functions. We also extend Tukey's one-degree-of-freedom interaction model to two-dimensional DLM context as a parsimonious way to model the interaction surface between two pollutants. Data adaptive approaches to allow departure from the specified Tukey's structure are also considered. A simulation study shows that shrinkage approach Bayesian constrained DLM has the best average performance in terms of relative efficiency.

In the third project, we extend DLM to a truly multi-dimensional space and focus on identifying important pollutants and pairwise interactions associated with a health outcome. Penalization-based approaches that induce sparsity in solution are considered. We propose a Hierarchical integrative Group LASSO (HiGLASSO) approach to perform variable selection at a group level while maintaining strong heredity constraints. Empirically, HiGLASSO identifies the correct set of important variables more frequently than other approaches. Theoretically, we show that HiGLASSO enjoys Oracle properties including selection and estimation consistency.