  Mediation analysis seeks to understand the mechanism by which a treatment
affects an outcome. Count or zero-inflated count outcome are common in many
studies in which mediation analysis is of interest. For example, in dental
studies, outcomes such as decayed, missing and filled teeth are typically zero
inflated. Existing mediation analysis approaches for count data assume
sequential ignorability of the mediator. This is often not plausible because
the mediator is not randomized so that there are unmeasured confounders
associated with the mediator and the outcome. In this paper, we develop causal
methods based on instrumental variable (IV) approaches for mediation analysis
for count data possibly with a lot of zeros that do not require the assumption
of sequential ignorability. We first define the direct and indirect effect
ratios for those data, and then propose estimating equations and use empirical
likelihood to estimate the direct and indirect effects consistently. A
sensitivity analysis is proposed for violations of the IV exclusion restriction
assumption. Simulation studies demonstrate that our method works well for
different types of outcomes under different settings. Our method is applied to
a randomized dental caries prevention trial and a study of the effect of a
massive flood in Bangladesh on children's diarrhea.
