Bayesian Sensitivity Analysis for Missing Data Using the E-value

Abstract: Sensitivity Analysis is a framework to assess how conclusions drawn from missing outcome data may be vulnerable to departures from untestable underlying assumptions. We extend the E-value, a popular metric for quantifying robustness of causal conclusions, to the setting of missing outcomes. With motivating examples from partially-observed Facebook conversion events, we present methodology for conducting Sensitivity Analysis at scale with three contributions. First, we develop a method for the Bayesian estimation of sensitivity parameters leveraging noisy benchmarks(e.g., aggregated reports for protecting unit-level privacy); both empirically derived subjective and objective priors are explored. Second, utilizing the Bayesian estimation of the sensitivity parameters we propose a mechanism for posterior inference of the E-value via simulation. Finally, closed form distributions of the E-value are constructed to make direct inference possible when posterior simulation is infeasible due to computational constraints. We demonstrate gains in performance over asymptotic inference of the E-value using data-based simulations, supplemented by a case-study of Facebook conversion events.