  Causal mediation analysis aims at disentangling a treatment effect into an
indirect mechanism operating through an intermediate outcome or mediator, as
well as the direct effect of the treatment on the outcome of interest. However,
the evaluation of direct and indirect effects is frequently complicated by
non-ignorable selection into the treatment and/or mediator, even after
controlling for observables, as well as sample selection/outcome attrition. We
propose a method for bounding direct and indirect effects in the presence of
such complications using a method that is based on a sequence of linear
programming problems. Considering inverse probability weighting by propensity
scores, we compute the weights that would yield identification in the absence
of complications and perturb them by an entropy parameter reflecting a specific
amount of propensity score misspecification to set-identify the effects of
interest. We apply our method to data from the National Longitudinal Survey of
Youth 1979 to derive bounds on the explained and unexplained components of a
gender wage gap decomposition that is likely prone to non-ignorable mediator
selection and outcome attrition.
