  Epidemiological research supports an association between maternal exposure to
air pollution during pregnancy and adverse children's health outcomes. Advances
in exposure assessment and statistics allow for estimation of both critical
windows of vulnerability and exposure effect heterogeneity. Simultaneous
estimation of windows of vulnerability and effect heterogeneity can be
accomplished by fitting a distributed lag model (DLM) stratified by subgroup.
However, this can provide an incomplete picture of how effects vary across
subgroups because it does not allow for subgroups to have the same window but
different within-window effects or to have different windows but the same
within-window effect. Because the timing of some developmental processes are
common across subpopulations of infants while for others the timing differs
across subgroups, both scenarios are important to consider when evaluating
health risks of prenatal exposures. We propose a new approach that partitions
the DLM into a constrained functional predictor that estimates windows of
vulnerability and a scalar effect representing the within-window effect
directly. The proposed method allows for heterogeneity in only the window, only
the within-window effect, or both. In a simulation study we show that a model
assuming a shared component across groups results in lower bias and mean
squared error for the estimated windows and effects when that component is in
fact constant across groups. We apply the proposed method to estimate windows
of vulnerability in the association between prenatal exposures to fine
particulate matter and each of birth weight and asthma incidence, and estimate
how these associations vary by sex and maternal obesity status, in a
Boston-area prospective pre-birth cohort study.
