Interpretable sensitivity analysis for the Baron-Kenny approach to mediation with unmeasured confounding

Abstract: Mediation analysis assesses the extent to which the exposure affects the outcome indirectly through a mediator and the extent to which it operates directly through other pathways. The popular Baron-Kenny approach estimates the indirect and direct effects of the exposure on the outcome based on linear regressions. However, when the exposure and the mediator are not randomized, the estimates may be biased due to unmeasured confounding. We first derive general omitted-variable bias formulas in linear regressions with vector responses and regressors. We then use the formulas to develop a sensitivity analysis method for the Baron-Kenny approach in the presence of unmeasured confounding. To ensure interpretability, we express the sensitivity parameters to correspond to the natural factorization of the joint distribution of the direct acyclic graph. They measure the partial correlation between the unmeasured confounder and the exposure, mediator, and outcome, respectively. We further propose a novel measure called the "robustness value for mediation" or simply the "robustness value", to assess the robustness of results based on the Baron-Kenny approach with respect to unmeasured confounding. Intuitively, the robustness value measures the minimum value of the maximum proportion of variability explained by the unmeasured confounding, for the exposure, mediator, and outcome, to overturn the results of the direct and indirect effect estimates. Importantly, we prove that all our sensitivity bounds are attainable and thus sharp.