  Standard methods for inference about direct and indirect effects require
stringent no unmeasured confounding assumptions which often fail to hold in
practice, particularly in observational studies. The goal of this paper is to
introduce a new form of indirect effect, the population intervention indirect
effect (PIIE), that can be nonparametrically identified in the presence of an
unmeasured common cause of exposure and outcome. This new type of indirect
effect captures the extent to which the effect of exposure is mediated by an
intermediate variable under an intervention that holds the component of
exposure directly influencing the outcome at its observed value. The PIIE is in
fact the indirect component of the population intervention effect, introduced
by Hubbard and Van der Laan (2008). Interestingly, our identification criterion
generalizes Judea Pearl's front-door criterion as it does not require no direct
effect of exposure not mediated by the intermediate variable. For inference, we
develop both parametric and semiparametric methods, including a novel doubly
robust semiparametric locally efficient estimator, that perform very well in
simulation studies. Finally, the proposed methods are used to measure the
effectiveness of monetary saving recommendations among women enrolled in a
maternal health program in Tanzania.
