Optimization-based Sensitivity Analysis

Presentation in Mathematics
Statistics in general and causal inference in particular relies on untestable assumptions. Hence, it is crucial to assess the robustness of obtained results to potential violations of these 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 hard to interpret. This work takes a more flexible approach to sensitivity analysis by viewing it as a constrained stochastic optimization problem.
In the talk, I focus on the confounding influence of an unobserved covariate on estimating a causal effect of interest. In order to assess deviations from the typical identification assumptions of ordinary least squares (OLS) or two-stage least squares (TSLS) estimators, the causal effect is expressed in terms of partial correlations and standard deviations. These can either be estimated from the observed data or serve as sensitivity parameters. Practitioners can express their beliefs about the values of these sensitivity parameters by stipulating constraints on them. For instance, they may compare the influence of the unmeasured confounder on treatment or outcome with an observed variable in terms of R^2-values. The specified constraints render the causal effect partially identified. The range of feasible values can be estimated by solving a constrained optimization problem minimizing or maximizing over the sensitivity parameters. In order to address uncertainty in the sensitivity analysis, we construct confidence intervals for the partially identified range. Since a heuristic "plug-in" confidence interval may not have any frequentist guarantees, this work instead follows a bootstrap approach. This method will be illustrated with a real data example and several user-friendly visualization tools.