Organic direct and indirect effects with post-treatment common causes of mediator and outcome

Abstract: Most of the literature on direct and indirect effects assumes that there are no post-treatment common causes of the mediator and the outcome. In contrast to natural direct and indirect effects, organic direct and indirect effects, which were introduced in Lok (2016, 2020), can be extended to provide an identification result for settings with post-treatment common causes of the mediator and the outcome. This article provides a definition and an identification result for organic direct and indirect effects in the presence of post-treatment common causes of mediator and outcome. These new organic indirect and direct effects have interpretations in terms of intervention effects. Organic indirect effects in the presence of post-treatment common causes are an addition to indirect effects through multivariate mediators. Organic indirect effects in the presence of post-treatment common causes can be used e.g. 1. to predict the effect of the initial treatment if its side affects are suppressed through additional interventions or 2. to predict the effect of a treatment that does not affect the post-treatment common cause and affects the mediator the same way as the initial treatment.