  Children's health studies support an association between maternal
environmental exposures and children's birth outcomes. A common goal is to
identify critical windows of susceptibility--periods during gestation with
increased association between maternal exposures and a future outcome. The
timing of the critical windows and magnitude of the associations are likely
heterogeneous across different levels of individual, family, and neighborhood
characteristics. Using an administrative Colorado birth cohort we estimate the
individualized relationship between weekly exposures to fine particulate matter
(PM$_{2.5}$) during gestation and birth weight. To achieve this goal, we
propose a statistical learning method combining distributed lag models and
Bayesian additive regression trees to estimate critical windows at the
individual level and identify characteristics that induce heterogeneity from a
high-dimensional set of potential modifying factors. We find evidence of
heterogeneity in the PM$_{2.5}$-birth weight relationship, with some
mother-child dyads showing a 3 times larger decrease in birth weight for an IQR
increase in exposure (5.9 to 8.5 $\mu g/m^3$ PM$_{2.5}$) compared to the
population average. Specifically, we find increased susceptibility for
non-Hispanic mothers who are either younger, have higher body mass index or
lower educational attainment. Our case study is the first precision health
study of critical windows.
