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Bounds on Direct and Indirect Effects of Treatment on a Continuous Endpoint

Direct effect of a treatment variable on an endpoint variable and indirect effect through a mediate variable are important concepts for understanding a causal mechanism. However, the randomized assignment of treatment is not sufficient for identifying the direct and indirect effects, and extra assumptions and conditions are required, such as the sequential ignorability assumption without unobserved confounders or the sequential potential ignorability assumption. But these assumptions may not be credible in many applications. In this article, we consider the bounds on controlled direct effect, natural direct effect, and natural indirect effect without these extra assumptions. Cai et al. [2008] presented the bounds for the case of a binary endpoint, and we extend their results to the general case for an arbitrary endpoint.