E-Value (Epidemiology)

E-value (epidemiology)—short for Evidence-for-Causality Value—is a quantitative sensitivity-analysis metric in epidemiology, biostatistics, and public health introduced by Tyler J. VanderWeele and Peng Ding in 2017 to evaluate the robustness of an observational association to potential unmeasured confounding. Defined on the risk ratio (RR) scale conditional on all measured covariates L, the E-value is the minimum strength of association that an unmeasured confounder U (whether a single binary variable, a continuous variable, or a high-dimensional vector of multiple unmeasured confounders) would need to exhibit with both the exposure (RR_EU) and the outcome (RR_UD) to shift an observed exposure–outcome risk ratio (RR_obs) all the way down to the null value of RR_true = 1.0 (or to shift the confidence interval bound closest to the null across 1.0). Because the E-value is derived from sharp mathematical bounding inequalities that make zero assumptions about the prevalence, distribution, or dimensionality of the unmeasured confounders, it provides a standardized, assumption-free benchmark accompanying observational point estimates and 95% confidence intervals.

R. A. Fisher's Constitutional Hypothesis and Jerome Cornfield's Inequality

The conceptual ancestor of the E-value arose during the 1950s debate over whether cigarette smoking causes lung cancer. Prominent statistician Ronald Fisher argued that the observed 9-fold relative risk (RR_obs ≈ 9.0) between cigarette smoking and lung cancer in observational cohorts and case-control studies could be explained by an unmeasured germline genotype (U, Fisher's "constitutional hypothesis") that predisposed individuals both to crave nicotine and to develop bronchogenic carcinoma.

In a foundational 1959 paper in the Journal of the National Cancer Institute, biostatistician Jerome Cornfield and colleagues proved mathematically that for a binary unmeasured confounder U to completely explain away an observed relative risk of RR_obs (RR_true = 1.0), U must increase the risk of the outcome by at least RR_obs (RR_UD ≥ 9.0) and be at least RR_obs times more prevalent among smokers than among non-smokers (RR_EU ≥ 9.0). Because no known genetic polymorphism or environmental factor exhibited a 9-fold association with both smoking behavior and lung cancer, Cornfield's inequality refuted the constitutional hypothesis and launched modern sensitivity analysis.

The Ding–VanderWeele (2016) Sharp Confounding Bound

Cornfield's classical conditions required both RR_EU ≥ RR_obs and RR_UD ≥ RR_obs separately (a necessary condition), meaning that if RR_EU = RR_UD = RR_obs exactly, the maximum confounding bias factor is still strictly smaller than RR_obs. In 2016, Peng Ding and Tyler VanderWeele proved a universal, sharp worst-case bounding theorem that holds for any categorical, continuous, or multivariate unmeasured confounder U without requiring rare-disease or no-interaction assumptions.

Define two maximal confounder risk ratios conditional on measured covariates L:

• RR_EU = max_u [ Pr(U = u | E = 1, L) / Pr(U = u | E = 0, L) ] (the maximum relative propensity of any level of U comparing exposed to unexposed).
• RR_UD = max_{e ∈ {0,1}} [ max_u Pr(D = 1 | E = e, U = u, L) / min_u Pr(D = 1 | E = e, U = u, L) ] (the maximum outcome risk ratio across levels of U within either exposure stratum).

Ding and VanderWeele (2016) proved that the maximum confounding bias factor (BF_max = RR_obs / RR_true) that U can generate is bounded above by:

BF_max = ( RR_EU × RR_UD ) / ( RR_EU + RR_UD − 1 )

Consequently, the true causal risk ratio RR_true is bounded below by RR_obs / BF_max.

Formula for Protective and Risk-Increasing Exposures

To find the E-value, we set the worst-case lower bound equal to the null (RR_obs / BF_max = 1, so BF_max = RR_obs) and solve for the equal-magnitude point where RR_EU = RR_UD = E:

E² / (2E − 1) = RR_obs ⟹ E² − (2 · RR_obs)E + RR_obs = 0

Solving this quadratic equation for the root greater than 1 yields the VanderWeele–Ding E-value formula for any observed risk ratio RR_obs ≥ 1:

E-value = RR_obs + [ RR_obs × (RR_obs − 1) ]^1/2

When the observed exposure is protective (RR_obs < 1, such as a vaccine or therapeutic drug), the investigator first takes the inverse RR^* = 1 / RR_obs > 1 and substitutes RR^* into the exact same equation:

E-value (for RR_obs < 1) = (1 / RR_obs) + [ (1 / RR_obs) × ( (1 / RR_obs) − 1 ) ]^1/2

Dual Reporting: Point Estimate E-Value and Confidence Interval Limit E-Value

VanderWeele and Ding mandate reporting two E-values for every observational finding:

1. E-value for the Point Estimate (E_est): Quantifies the confounder strength required to reduce the observed point estimate RR_obs to 1.0.
2. E-value for the 95% Confidence Interval Limit Closest to the Null (E_CI): Letting LL denote the lower bound of the 95% CI when RR_obs > 1 (or UL denote the upper bound when RR_obs < 1), E_CI quantifies the confounder strength required to shift the 95% confidence interval so that it includes 1.0 (rendering the association statistically non-significant at α = 0.05). If the 95% CI already crosses 1.0, E_CI = 1.00 by definition.

Because epidemiologists fit logistic regressions, Cox proportional hazards models, and linear regressions depending on study design, VanderWeele and Ding (2017) derived validated E-value conversions across all standard effect measures:

1. Odds Ratios (OR) and Hazard Ratios (HR) for Rare Outcomes (< 15% incidence): Because OR and HR closely approximate RR when the outcome is rare (as in a nested case-control study), OR or HR can be plugged directly into the E-value formula.
2. Odds Ratios (OR) for Common Outcomes (> 15% incidence): Because the odds ratio overstates the risk ratio when prevalence is high, applying the square-root transformation RR ≈ OR^1/2 yields:

E_common-OR ≈ OR^1/2 + [ OR^1/2 × (OR^1/2 − 1) ]^1/2

1. Hazard Ratios (HR) for Common Outcomes (> 15% incidence): Using the survival approximation RR ≈ (1 − 0.5^HR1/2) / (1 − 0.5^HR−1/2) converts the hazard ratio to the risk-ratio scale before computing the E-value.
2. Standardized Mean Differences (d) in Continuous Outcomes: Dividing the regression difference by the pooled outcome standard deviation (d = β̂ / s_Y) and applying the Chinn–VanderWeele log-linear approximation RR ≈ exp(0.91 × d) enables E-value computation for continuous biomarkers.

As emphasized by John P. A. Ioannidis, Maya B. Mathur, and VanderWeele (2019, 2020), an E-value is not an automated mechanical pass/fail cutoff (unlike p < 0.05); rather, its scientific meaning depends entirely on what covariates L have already been adjusted for in the primary model:

• An E-value of E = 2.10 in a crude unadjusted study that failed to control for age, smoking, and socioeconomic status provides negligible evidence of causality, because smoking and age routinely exhibit risk ratios exceeding 2.5.
• Conversely, an E-value of E = 2.10 in a study that has already controlled for 25 measured clinical, laboratory, and socioeconomic confounders L means that a residual unmeasured confounder U must exhibit a 2.10-fold association with both E and D above and beyond (conditional on) all 25 measured covariates, which is substantially harder for a residual factor to achieve. Best practice therefore pairs the E-value with observed covariate benchmarking (computing the E-values or bias factors that would change if known measured confounders such as smoking or BMI were omitted from the model) and complements it with full quantitative bias analysis when exposure misclassification or collider bias is suspected.

Can the E-value shift an observed association to a non-null target value (RR_true ≠ 1.0)?

Yes. To compute the minimum confounder strength required to attenuate an observed risk ratio RR_obs > 1 down to a specified clinically meaningful threshold RR_target > 1 (rather than all the way to the null 1.0), one simply replaces RR_obs in the E-value formula with the ratio RR_ratio = RR_obs / RR_target:

E(RR_target) = (RR_obs / RR_target) + [ (RR_obs / RR_target) × ( (RR_obs / RR_target) − 1 ) ]^1/2

Does a high E-value rule out selection bias or measurement error?

In its standard formulation, the E-value measures sensitivity to unmeasured confounding, though Louisa H. Smith and Tyler VanderWeele (2019) derived analogous bounding E-values for selection bias (OR_sel) and differential misclassification. However, when validation data exist on diagnostic sensitivity and specificity, investigators should perform a full quantitative bias analysis or evaluate a negative control rather than relying solely on worst-case E-value bounds.

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. VanderWeele, T. J., & Mathur, M. B. (2020). Commentary: Developing best-practice guidelines for the reporting of E-values. International Journal of Epidemiology, 49(5), 1495–1497. https://doi.org/10.1093/ije/dyaa094