  In studies of maternal exposure to air pollution a children's health outcome
is regressed on exposures observed during pregnancy. The distributed lag
nonlinear model (DLNM) is a statistical method commonly implemented to estimate
an exposure-time-response function when it is postulated the exposure effect is
nonlinear. Previous implementations of the DLNM estimate an
exposure-time-response surface parameterized with a bivariate basis expansion.
However, basis functions such as splines assume smoothness across the entire
exposure-time-response surface, which may be unrealistic in settings where the
exposure is associated with the outcome only in a specific time window. We
propose a framework for estimating the DLNM based on Bayesian additive
regression trees. Our method operates using a set of regression trees that each
assume piecewise constant relationships across the exposure-time space. In a
simulation, we show that our model outperforms spline-based models when the
exposure-time surface is not smooth, while both methods perform similarly in
settings where the true surface is smooth. Importantly, the proposed approach
is lower variance and more precisely identifies critical windows during which
exposure is associated with a future health outcome. We apply our method to
estimate the association between maternal exposure to PM$_{2.5}$ and birth
weight in a Colorado USA birth cohort.
