  Maternal exposure to air pollution during pregnancy has a substantial public
health impact. Epidemiological evidence supports an association between
maternal exposure to air pollution and low birth weight. A popular method to
estimate this association while identifying windows of susceptibility is a
distributed lag model (DLM), which regresses an outcome onto exposure history
observed at multiple time points. However, the standard DLM framework does not
allow for modification of the association between repeated measures of exposure
and the outcome. We propose a distributed lag interaction model that allows
modification of the exposure-time-response associations across individuals by
including an interaction between a continuous modifying variable and the
exposure history. Our model framework is an extension of a standard DLM that
uses a cross-basis, or bi-dimensional function space, to simultaneously
describe both the modification of the exposure-response relationship and the
temporal structure of the exposure data. Through simulations, we showed that
our model with penalization out-performs a standard DLM when the true
exposure-time-response associations vary by a continuous variable. Using a
Colorado, USA birth cohort, we estimated the association between birth weight
and ambient fine particulate matter air pollution modified by an area-level
metric of health and social adversities from Colorado EnviroScreen.
