Bias Formulas for Estimating Direct and Indirect Effects When Unmeasured Confounding Is Present

Mediation analysis examines the influence of intermediate factors in the causal pathway between an exposure and an outcome. It yields estimates of the direct effect of the exposure on the outcome and of the indirect effect through the intermediate variable. Both estimates can be biased if the relationship between the mediator and the outcome is confounded. In this article, we study the effect of unmeasured confounding on direct and indirect effect estimates for a continuous mediator and an outcome that may be either binary, count, or continuous. We formulate the effect of the confounder on the intermediate and on the outcome directly in regression models, which makes the formulas intuitive to use by applied users. The formulas are derived under the assumption that the confounder follows a normal distribution. In simulations, the formulas for a linear response model performed well, also as it did when the unmeasured confounder was binary. For a rare binary outcome, the formulas for logistic regression performed well if the unmeasured confounder followed a normal distribution, but for a binary confounder the bias in the direct effect was overcorrected. We applied the formulas to data from a case-control study (Leiden Thrombophilia Study) on risk factors for venous thrombosis. This showed that unmeasured confounding can severely bias the estimates of direct and indirect effects.