Bounds on controlled direct effects under monotonic assumptions about mediators and confounders

Adjusting for intermediate variables is a common analytic strategy for estimating a direct effect. Even if the total effect is unconfounded, the direct effect is not identified when unmeasured variables affect the intermediate and outcome variables. Therefore, some researchers presented bounds on the controlled direct effects via linear programming. They applied a monotonic assumption about treatment and intermediate variables and a no-interaction assumption to derive narrower bounds. Here, we improve their bounds without using linear programming and hence derive a bound under the monotonic assumption about an intermediate variable only. To improve the bounds, we further introduce the monotonic assumption about confounders. While previous studies assumed that an outcome is a binary variable, we do not make that assumption. The proposed bounds are illustrated using two examples from randomized trials.