  Natural direct and indirect effects are mediational estimands that decompose
the average treatment effect and describe how outcomes would be affected by
contrasting levels of a treatment through changes induced in mediator values
(in the case of the indirect effect) or not through induced changes in the
mediator values (in the case of the direct effect). Natural direct and indirect
effects are not generally point-identifiable in the presence of a
treatment-induced confounder, however they may still be identified if one is
willing to assume monotonicity between a treatment and the treatment-induced
confounder. We argue that this assumption may be reasonable in the relatively
common encouragement-design trial setting where intervention is randomized
treatment assignment and the treatment-induced confounder is whether or not
treatment was actually taken/adhered to. We develop efficiency theory for the
natural direct and indirect effects under this monotonicity assumption, and use
it to propose a nonparametric, multiply robust estimator. We demonstrate the
finite sample properties of this estimator using a simulation study, and apply
it to data from the Moving to Opportunity Study to estimate the natural direct
and indirect effects of being randomly assigned to receive a Section 8 housing
voucher -- the most common form of federal housing assistance -- on risk
developing any mood or externalizing disorder among adolescent boys, possibly
operating through various school and community characteristics.
