E Values and Alternative Sensitivity Frameworks

The one idea: Robustness value asks how much residual variance a hidden confounder would need to explain, on a partial-R² scale, to erase your result — precise, but a hard number to say out loud in a meeting. The E-value asks the same underlying question on a scale anyone already understands: how strong a risk multiplier would an unmeasured confounder need to be — on the treatment, and on the outcome — to fully explain away what you found.

⚠️ The trap: "Our effect is an odds ratio of 2.0, so the E-value formula for risk ratios applies directly." That feels right and it's wrong. The E-value formula is built for risk ratios. Odds ratios, hazard ratios, and linear-model coefficients all need a conversion step first — plugging them in raw silently overstates or understates how robust the result actually is.

The omitted-variable-bias lesson gave you the robustness value: a rigorous, partial-R²-based answer to "how strong would an unmeasured confounder need to be to zero out this estimate." It's exact, it benchmarks against your actual covariates, and it lives inside the regression framework you already fit. Its one practical weakness is communication — "partial R² of 22% with both treatment and outcome" is not a sentence that survives contact with a room full of non-statisticians. The E-value, introduced by Tyler VanderWeele and Peng Ding in 2017, answers a nearly identical question in a unit everyone already has an intuition for: a risk multiplier.

The question, restated on a risk-ratio scale

Take an observed association between treatment and outcome, expressed as a risk ratio (RR) — how many times more likely the…