Bounds on direct effects in the presence of confounded intermediate variables

This article considers the problem of estimating the average controlled direct effect (ACDE) of a treatment on an outcome, in the presence of unmeasured confounders between an intermediate variable and the outcome. Such confounders render the direct effect unidentifiable even in cases where the total effect is unconfounded (hence identifiable). Kaufman et al. (2005, Statistics in Medicine 24, 1683-1702) applied a linear programming software to find the minimum and maximum possible values of the ACDE for specific numerical data. In this article, we apply the symbolic Balke-Pearl (1997, Journal of the American Statistical Association 92, 1171-1176) linear programming method to derive closed-form formulas for the upper and lower bounds on the ACDE under various assumptions of monotonicity. These universal bounds enable clinical experimenters to assess the direct effect of treatment from observed data with minimum computational effort, and they further shed light on the sign of the direct effect and the accuracy of the assessments.