A Bayesian Reinterpretation of Cornfield-Type Sensitivity Analysis: From Thresholds to Probabilities

Abstract: Sensitivity analysis for unmeasured confounding in observational studies is commonly based on threshold quantities, such as the Cornfield condition or the E-value, which quantify how strong a confounder must be to explain away an observed association. However, these approaches do not address a fundamental inferential question: how plausible is it that such a confounder exists? In this work, we propose a Bayesian reformulation of Cornfield-type sensitivity analysis in which the strength of unmeasured confounding is treated as a random variable. Within this framework, the E-value is reinterpreted as a threshold, and the central inferential quantity becomes the posterior probability that confounding exceeds this threshold. This transforms sensitivity analysis from a descriptive diagnostic into a probabilistic assessment of robustness. We develop a simple generative model linking observed effect estimates to true causal effects and confounding bias, and we specify prior distributions reflecting plausible confounding mechanisms. The resulting framework yields posterior measures of evidential vulnerability that are directly interpretable and applicable to summary-level data. Illustrations based on empirical case studies show that the proposed approach preserves the interpretability of the E-value while providing a more nuanced and decision-relevant characterization of robustness. More broadly, the framework aligns sensitivity analysis with Bayesian principles of inference under uncertainty, offering a coherent alternative to purely threshold-based reasoning.