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Sensitivity analysis assesses whether conclusions remain robust under violations of unmeasured confounding. Methods (e.g., E-value, Rotnitzky-Robins bounds) quantify the minimum strength of unmeasured confounding needed to reverse or nullify an observed association. Sensitivity analysis supplements point estimates with assessments of hidden bias.
Calculate E-values for published observational findings across a range of reported effect sizes, and evaluate whether plausible unmeasured confounders could meet the threshold. This trains the intuition for what "robust" actually means quantitatively.
From the counterfactual framework, you know that valid causal inference from observational data requires exchangeability: conditional on measured covariates, the exposed and unexposed groups must be comparable in their potential outcomes. This is the assumption of no unmeasured confounding. In a randomized trial, it holds by design. In observational studies, it is always an untestable assumption — you cannot rule out that some unmeasured variable simultaneously predicts both who gets exposed and who gets the outcome. Sensitivity analysis does not test this assumption; it asks a sharper question: how strongly would unmeasured confounding have to act to explain away your finding?
The E-value (introduced by VanderWeele and Ding) is the most widely used tool for this purpose. It answers: what is the minimum strength, on the risk ratio scale, that an unmeasured confounder would need to have with *both* the exposure and the outcome — simultaneously — to fully explain away the observed association? If you observe a risk ratio of 3.0, the E-value tells you precisely how large the confounder-exposure and confounder-outcome associations would each need to be. The calculation is simple: E-value = RR + √(RR × (RR − 1)). For an RR of 3.0, E-value ≈ 5.2. This means any unmeasured confounder would need associations of at least 5.2-fold with both the exposure and the outcome to explain the result away. You then ask: is there any plausible confounder with that magnitude of association? If the strongest known predictors of the outcome all have associations below 5.2, the result is robust.
Rotnitzky-Robins bounds (and related Cornfield conditions) approach the same problem differently — they impose parameter constraints on what a confounding variable would have to look like (its prevalence, its association with exposure and outcome) and derive bounds on the true causal effect under those constraints. Rather than a single threshold, bounds analysis yields an interval: the true effect could be anywhere from [lower bound] to [upper bound] if unmeasured confounding exists at a specified level. This is more informative than a point estimate when uncertainty about confounding is substantial.
The practical workflow is to report the E-value alongside every point estimate in an observational study. A large E-value provides a degree of protection: you are saying "for this finding to be spurious, confounding would need to be implausibly extreme." A small E-value is a warning: the finding is fragile and could easily be explained by modest unmeasured confounding. Critics cannot simply assert "there might be confounding" — they must propose a specific confounder with the required magnitude, which is a much harder scientific argument to make. Sensitivity analysis thus converts a logical impossibility (proving the absence of unmeasured confounding) into a tractable empirical question (is there anything plausible with that effect size?), and in doing so it disciplines the rhetoric around causal claims from observational data.