  Developmental epidemiology commonly focuses on assessing the association
between multiple early life exposures and childhood health. Statistical
analyses of data from such studies focus on inferring the contributions of
individual exposures, while also characterizing time-varying and interacting
effects. Such inferences are made more challenging by correlations among
exposures, nonlinearity, and the curse of dimensionality. Motivated by studying
the effects of prenatal bisphenol A (BPA) and phthalate exposures on glucose
metabolism in adolescence using data from the ELEMENT study, we propose a
low-rank longitudinal factor regression (LowFR) model for tractable inference
on flexible longitudinal exposure effects. LowFR handles highly-correlated
exposures using a Bayesian dynamic factor model, which is fit jointly with a
health outcome via a novel factor regression approach. The model collapses on
simpler and intuitive submodels when appropriate, while expanding to allow
considerable flexibility in time-varying and interaction effects when supported
by the data. After demonstrating LowFR's effectiveness in simulations, we use
it to analyze the ELEMENT data and find that diethyl and dibutyl phthalate
metabolite levels in trimesters 1 and 2 are associated with altered glucose
metabolism in adolescence.
