Causal inference necessarily relies upon untestable assumptions; hence, it is crucial to assess the robustness of obtained results to violations of identification assumptions. However, such sensitivity analysis is only occasionally undertaken in practice, as many existing methods only apply to relatively simple models and their results are often difficult to interpret. We take a more flexible approach to sensitivity analysis and view it as a constrained stochastic optimization problem. We focus on linear models with an unmeasured confounder and a potential instrument. We show how the $R^2$-calculus - a set of algebraic rules that relates different (partial) $R^2$-values and correlations - can be applied to identify the bias of the $k$-class estimators and construct sensitivity models flexibly. We further show that the heuristic "plug-in" sensitivity interval may not have any confidence guarantees; instead, we propose a boostrap approach to construct sensitivity intervals which perform well in numerical simulations. We illustrate the proposed methods with a real study on the causal effect of education on earnings and provide user-friendly visualization tools.

翻译：因果推断必然依赖于不可检验的假设；因此，评估所得结果在违背识别假设时的稳健性至关重要。然而，此类敏感性分析在实践中很少开展，因为现有许多方法仅适用于相对简单的模型，且其结果通常难以解释。我们采用一种更灵活的敏感性分析方法，将其视为一个受约束的随机优化问题。我们聚焦于存在未测量混杂变量和潜在工具变量的线性模型。我们展示了$R^2$演算——一组关联不同（偏）$R^2$值与相关系数的代数规则——如何应用于识别$k$类估计量的偏差，并灵活构建敏感性模型。我们进一步证明了启发式“代入”敏感性区间可能不具备任何置信度保证；相反，我们提出了一种自举法来构建敏感性区间，该方法在数值模拟中表现良好。我们通过一项关于教育对收入因果效应的实际研究，展示了所提出的方法，并提供了用户友好的可视化工具。