Bounds on natural direct effects in the presence of confounded intermediate variables

In epidemiological studies we often want to learn about the direct effect of an exposure on an outcome, i.e. the effect that is not relayed by a specific intermediate variable. In the literature, there are two common definitions of direct effects; controlled and natural. When the intermediate variable and the outcome have common causes, neither the controlled nor the natural direct effect is identified. Cai et al. (Biometrics 2007; 64(3):695-701) derived bounds for the controlled direct effect under a set of hierarchical assumptions. In this paper we derive bounds on the natural direct effect under the same assumptions.