E-Value Sensitivity Analysis for Unmeasured Confounding

The E-Value is a summary measure of robustness to unmeasured confounding, introduced by Tyler J. VanderWeele and Peng Ding in a 2017 paper in the Annals of Internal Medicine. It answers a question that readers of observational research routinely pose: how strong would a hidden common cause of exposure and outcome have to be to account for the entire observed association? The E-value is the minimum strength of association, expressed as a risk ratio, that an unmeasured confounder would need to have with both the treatment and the outcome, conditional on the measured covariates, to shift the observed estimate to the null. Large values indicate that only an implausibly powerful confounder could negate the finding, whereas values near 1 indicate fragility. Because it requires no assumptions about the number, type, or distribution of the unmeasured confounders, the E-value has been adopted widely in epidemiology and clinical research as a standard companion to effect estimates from cohort studies.

In a randomized controlled trial, assignment is independent of baseline characteristics by design, so exposure groups are exchangeable in expectation. Observational studies lack this guarantee. Even after adjusting for measured covariates by regression, matching, or weighting, residual association between exposure and outcome may reflect common causes that were not recorded. Sources of this problem include confounding by indication, in which the reason for treatment predicts the outcome, and lifestyle factors that cluster with exposure. Classical remedies rely on subject-matter argument, negative controls, or instruments. Sensitivity analysis offers a complementary strategy: rather than asserting that no confounding exists, it asks how much would be required. The broader family of such techniques is surveyed in quantitative bias analysis.

The intellectual origin is the 1959 argument of Jerome Cornfield and colleagues in response to the suggestion by R. A. Fisher that a genetic factor might cause both smoking and lung cancer. Cornfield showed that, if smokers had a risk of lung cancer nine times that of nonsmokers, then any binary confounder that fully explained the association would need to be at least nine times more prevalent among smokers than nonsmokers. This is the Cornfield condition: the relative prevalence of the confounder in exposed versus unexposed individuals must be at least as large as the observed risk ratio. The result is notable because it makes no assumptions about the confounder's identity. Subsequent work by Schlesselman, Bross, and Rosenbaum and Rubin (1983) extended the approach, but those formulations typically required specifying parameters for a particular hypothetical confounder, which limited their uptake.

Ding and VanderWeele (2016) generalized Cornfield's result by deriving a bound valid for arbitrary confounders, whether binary, categorical, or continuous. Let RR_EU be the maximum risk ratio comparing the exposed with the unexposed for any category of the confounder U, and let RR_UD be the maximum risk ratio comparing any two categories of U with respect to the outcome, within exposure strata. The factor by which confounding can bias the observed risk ratio is at most the bounding factor

B = (RR_EU × RR_UD) / (RR_EU + RR_UD − 1).

The bounding factor is symmetric in its two arguments and increases with each. To explain away an observed risk ratio RR > 1 fully, the bias factor must satisfy B ≥ RR. Setting the two associations equal, RR_EU = RR_UD = E, gives E²/(2E − 1) = RR, which solves to

E = RR + √(RR(RR − 1)).

This is the E-value for a point estimate. For example, an observed RR of 2 yields E = 2 + √2 ≈ 3.41: an unmeasured confounder associated with both exposure and outcome by risk ratios of at least 3.41 each, above and beyond measured covariates, could explain away the association, whereas weaker confounding could not. For protective estimates with RR < 1, one first inverts the estimate to 1/RR and applies the same formula. The null value gives E = 1.

Statistical uncertainty is handled by applying the formula to the confidence limit closer to the null. If the lower limit of a 95% interval for a harmful exposure is 1.3, the E-value for the limit is 1.3 + √(1.3 × 0.3) ≈ 1.92, indicating the confounding strength needed to move the interval to include the null. If the interval already covers 1, the E-value for the limit is defined as 1. Reporting both E-values distinguishes robustness of the point estimate from robustness of statistical significance.

The derivation is on the risk ratio scale, so other measures require approximation or conversion.

Rare outcomes. When the outcome prevalence is below roughly 15 percent, the odds ratio and hazard ratio approximate the risk ratio, and the formula can be applied directly. This approximation underlies use of the E-value with results from logistic models and the Cox proportional hazards model.

Common outcomes. For odds ratios with common outcomes, the square root of the odds ratio is used as an approximate risk ratio before applying the formula. For hazard ratios with common outcomes, an analogous transformation based on the ratio of log survival probabilities is used.

Continuous outcomes. For standardized mean differences d, an approximate risk ratio of exp(0.91 × d) is employed, then the formula is applied.

Risk differences can be handled with additional baseline-risk assumptions. These conversions are approximations, and software such as the R package EValue and a web calculator automate them.

Rosenbaum's sensitivity analysis for matched observational studies uses a parameter Γ, the maximum odds ratio by which two matched subjects differing in an unmeasured covariate may differ in their odds of receiving treatment. For each Γ, the analysis computes a range of p-values; the Γ at which the p-value crosses a threshold, such as 0.05, measures the sensitivity of significance to hidden bias. Several differences distinguish the two approaches. Rosenbaum bounds are tied to a specific design (matching) and test statistic and concern the treatment-assignment model only, while the E-value is applicable to any estimate in risk ratio form and incorporates association of the confounder with the outcome. Rosenbaum bounds address inference on p-values under the sharp null; the E-value addresses the magnitude of the point estimate and its confidence limit. The scales also differ, so Γ and E are not directly interchangeable, though both give a calibrated sense of how much hidden bias is needed.

An E-value is not a probability that confounding exists, and a high E-value does not rule out bias from selection, misclassification, or reverse causation. Interpretation requires benchmarking against measured covariates: if the strongest observed confounder has risk ratios of 1.5 with exposure and outcome, an E-value of 3.4 suggests that an unmeasured factor would need to be much stronger than anything measured. Critics note that the bound assumes the worst-case confounder configuration, so it can be conservative; that a single number may encourage threshold thinking; and that the conversions for non-risk-ratio measures are approximate. VanderWeele and colleagues recommend reporting E-values alongside, not instead of, substantive discussion of plausible confounders. Related approaches, including the proximal causal inference framework, attempt to identify effects in the presence of unmeasured confounders by using auxiliary proxies.

1. VanderWeele, T. J., & Ding, P. (2017). Sensitivity analysis in observational research: Introducing the E-value. Annals of Internal Medicine, 167(4), 268–274. https://doi.org/10.7326/M16-2607
2. Ding, P., & VanderWeele, T. J. (2016). Sensitivity analysis without assumptions. Epidemiology, 27(3), 368–377. https://doi.org/10.1097/EDE.0000000000000457
3. Cornfield, J., Haenszel, W., Hammond, E. C., Lilienfeld, A. M., Shimkin, M. B., & Wynder, E. L. (1959). Smoking and lung cancer: Recent evidence and a discussion of some questions. Journal of the National Cancer Institute, 22(1), 173–203. https://doi.org/10.1093/jnci/22.1.173
4. Rosenbaum, P. R., & Rubin, D. B. (1983). Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. Journal of the Royal Statistical Society: Series B, 45(2), 212–218. https://doi.org/10.1111/j.2517-6161.1983.tb01242.x