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. As the most popular method in empirical mediation analysis, the Baron-Kenny approach estimates the indirect and direct effects of the exposure on the outcome based on linear structural equation models. However, when the exposure and the mediator are not randomized, the estimates may be biased due to unmeasured confounding among the exposure, mediator, and outcome. Building on Cinelli and Hazlett (2020), we 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 to mediation 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 for mediation analysis. They measure the partial correlation between the unmeasured confounder and the exposure, mediator, outcome, respectively. With the sensitivity parameters, we 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 point estimate or confidence interval for the direct and indirect effects. Importantly, we prove that all our sensitivity bounds are attainable and thus sharp.

翻译：中介分析用于评估暴露因素通过中介变量间接影响结局的程度，以及通过其他途径直接影响结局的程度。作为实证中介分析中最常用的方法，Baron-Kenny 方法基于线性结构方程模型估计暴露对结局的间接效应和直接效应。然而，当暴露和中介变量未随机化时，由于暴露、中介变量和结局之间存在未测量的混杂，估计结果可能存在偏倚。基于 Cinelli 和 Hazlett (2020) 的研究，我们推导了具有向量响应变量和回归变量的线性回归中一般性的遗漏变量偏倚公式。随后，利用这些公式，我们开发了一种针对存在未测量混杂时 Baron-Kenny 中介分析方法的敏感性分析方法。为确保可解释性，我们将敏感性参数设计为对应于中介分析有向无环图联合分布的自然分解。这些参数分别衡量未测量混杂与暴露、中介变量和结局之间的偏相关系数。基于这些敏感性参数，我们提出了一种新指标，称为"中介分析的稳健性值"（简称为"稳健性值"），用于评估 Baron-Kenny 方法在存在未测量混杂时结果的稳健性。直观而言，稳健性值衡量为了推翻直接效应和间接效应的点估计或置信区间结果，未测量混杂对暴露、中介变量和结局所需解释的最大变异比例的最小值。重要的是，我们证明了所有敏感性界都是可达的，因此是尖锐的。