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In randomized trials, once the total effect of the intervention has beenestimated, it is often of interest to explore mechanistic effects throughmediators along the causal pathway between the randomized treatment and theoutcome. In the setting with two sequential mediators, there are a variety ofdecompositions of the total risk difference into mediation effects. We derivesharp and valid bounds for a number of mediation effects in the setting of twosequential mediators both with unmeasured confounding with the outcome. Weprovide five such bounds in the main text corresponding to two differentdecompositions of the total effect, as well as the controlled direct effect,with an additional thirty novel bounds provided in the supplementary materialscorresponding to the terms of twenty-four four-way decompositions. We also showthat, although it may seem that one can produce sharp bounds by adding orsubtracting the limits of the sharp bounds for terms in a decomposition, thisalmost always produces valid, but not sharp bounds that can even be completelynoninformative. We investigate the properties of the bounds by simulatingrandom probability distributions under our causal model and illustrate how theyare interpreted in a real data example.

Sharp nonparametric bounds for decomposition effects with two binary mediators