  Causal mediation analysis is an important statistical method in social and
medical studies, as it can provide insights about why an intervention works and
inform the development of future interventions. Currently, most causal
mediation methods focus on mediation effects defined on a mean scale. However,
in health-risk studies, such as alcohol or risky sex, outcomes are typically
count data and heavily skewed. Thus, mediation effects in these setting would
be natural on a rate ratio scale, such as in Poisson and negative binomial
regression methods. Existing methods also mainly rely on the assumption of no
unmeasured confounding between mediator and outcome. To allow for potential
confounders between the mediator and outcome, we define the direct and mediator
effects on a new scale and propose a multiplicative structural mean model for
mediation analysis with count outcomes. The estimator is compared with both
Poisson and negative binomial regression methods assuming sequential
ignorability using a simulation study and a real world example about an
alcohol-related intervention study. Mediation analyses using the new methods
confirm the study hypothesis that the intervention decreases drinking by
decreasing individual's normative perceptions of alcohol use.
