  In causal mediation analysis, nonparametric identification of the pure
(natural) direct effect typically relies on, in addition to no unobserved
pre-exposure confounding, fundamental assumptions of (i) so-called
"cross-world-counterfactuals" independence and (ii) no exposure- induced
confounding. When the mediator is binary, bounds for partial identification
have been given when neither assumption is made, or alternatively when assuming
only (ii). We extend existing bounds to the case of a polytomous mediator, and
provide bounds for the case assuming only (i). We apply these bounds to data
from the Harvard PEPFAR program in Nigeria, where we evaluate the extent to
which the effects of antiretroviral therapy on virological failure are mediated
by a patient's adherence, and show that inference on this effect is somewhat
sensitive to model assumptions.
