Bounds on Direct E ects in the Presence of Confounded Intermediate Variables

This paper considers the problem of estimating the average con trolled direct e ect 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 e ect unidenti able even in cases where the total e ect is unconfounded hence identi able Kaufman et al applied a linear programming software to nd the minimum and maximum possible values of the ACDE for speci c numerical data In this paper we apply the symbolic Balke Pearl 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 clini cal experimenters to assess the direct e ect of treatment from observed data with minimum computational e ort and they further shed light on the sign of the direct e ect and the accuracy of the assessments