  Often linear regression is used to perform mediation analysis. However, in
many instances, the underlying relationships may not be linear, as in the case
of placental-fetal hormones and fetal development. Although, the exact
functional form of the relationship may be unknown, one may hypothesize the
general shape of the relationship. For these reasons, we develop a novel
shape-restricted inference-based methodology for conducting mediation analysis.
This work is motivated by an application in fetal endocrinology where
researchers are interested in understanding the effects of pesticide
application on birth weight, with human chorionic gonadotropin (hCG) as the
mediator. We assume a practically plausible set of nonlinear effects of hCG on
the birth weight and a linear relationship between pesticide exposure and hCG,
with both exposure-outcome and exposure-mediator models being linear in the
confounding factors. Using the proposed methodology on a population-level
prenatal screening program data, with hCG as the mediator, we discovered that,
while the natural direct effects suggest a positive association between
pesticide application and birth weight, the natural indirect effects were
negative.
