  Greater understanding of the pathways through which an environmental mixture
operates is important to design effective interventions. We present new
methodology to estimate natural direct and indirect effects and controlled
direct effects of a complex mixture exposure on an outcome through a mediator
variable. We implement Bayesian Kernel Machine Regression (BKMR) to allow for
all possible interactions and nonlinear effects of (1) the co-exposures on the
mediator, (2) the co-exposures and mediator on the outcome, and (3) selected
covariates on the mediator and/or outcome. From the posterior predictive
distributions of the mediator and outcome, we simulate counterfactuals to
obtain posterior samples, estimates, and credible intervals of the mediation
effects. Our simulation study demonstrates that when the exposure-mediator and
exposure-mediator-outcome relationships are complex, BKMR--Causal Mediation
Analysis performs better than current mediation methods. We applied our
methodology to quantify the contribution of birth length as a mediator between
in utero co-exposure to arsenic, manganese and lead, and children's
neurodevelopmental scores, in a prospective birth cohort in Bangladesh. Among
younger children, we found a negative (adverse) association between the metal
mixture and neurodevelopment. We also found evidence that birth length mediates
the effect of exposure to the metal mixture on neurodevelopment for younger
children. If birth length were fixed to its $75^{th}$ percentile value, the
harmful effect of the metal mixture on neurodevelopment is attenuated,
suggesting nutritional interventions to help increase fetal growth, and thus
birth length, could potentially block the harmful effect of the metal mixture
on neurodevelopment.
