  Path-specific effects are a broad class of mediated effects from an exposure
to an outcome via one or more causal pathways with respect to some subset of
intermediate variables. The majority of the literature concerning estimation of
mediated effects has focused on parametric models with stringent assumptions
regarding unmeasured confounding. We consider semiparametric inference of a
path-specific effect when these assumptions are relaxed. In particular, we
develop a suite of semiparametric estimators for the effect along a pathway
through a mediator, but not some exposure-induced confounder of that mediator.
These estimators have different robustness properties, as each depends on
different parts of the observed data likelihood. One of our estimators may be
viewed as combining the others, because it is locally semiparametric efficient
and multiply robust. The latter property is illustrated in a simulation study.
We apply our methodology to an HIV study, in which we estimate the effect
comparing two drug treatments on a patient's average log CD4 count mediated by
the patient's level of adherence, but not by previous experience of toxicity,
which is clearly affected by which treatment the patient is assigned to, and
may confound the effect of the patient's level of adherence on their virologic
outcome.
