How is the VanderWeele E-value calculated from RR, HR, or OR?

For an observed risk ratio RR > 1, the E-value is RR + sqrt(RR * (RR - 1)). If RR 15%), VanderWeele & Ding (Ann Intern Med 2017) recommend approximating RR = sqrt(OR) before computing the E-value.

How do I respond to a peer reviewer asking about unmeasured confounding when PSM was not performed?

While PSM only balances measured baseline covariates, the VanderWeele E-value directly quantifies how strong an unmeasured confounder would have to be (in its associations with both treatment and outcome) to explain away the observed multivariable-adjusted effect.

CAUSAL INFERENCE · VANDERWEELE & DING (ANN INTERN MED 2017)

VanderWeele E-value Calculator & Reviewer Rebuttal Generator

Quantify how strong an unmeasured confounder must be to explain away an observed treatment–outcome association in observational cohort studies. Instant calculation, bias threshold plotting, and ready-to-copy manuscript & rebuttal paragraphs.

1. Input Effect Estimate & 95% CI

Effect Measure Type

Point Estimate

95% CI Lower

95% CI Upper

2. E-value Results & Bias Threshold Curve

E-value for Point Estimate

2.04

E-value for 95% CI Limit

1.63

Manuscript Results Paragraph (English / Chinese)

Peer-Review Rebuttal Template (Response to Reviewer)

Methodology: How VanderWeele’s E-value Quantifies Unmeasured Confounding

In observational clinical cohort studies (retrospective EHR cohorts, registries, and real-world evidence studies), reviewers frequently question whether an observed association could be explained away by unmeasured confounders (e.g., frailty, socioeconomic status, or undocumented medication adherence). According to VanderWeele and Ding (Annals of Internal Medicine, 2017;167(4):268–274), the E-value is defined as the minimum strength of association, on the risk ratio scale, that an unmeasured confounder would need to have with both the treatment/exposure and the outcome to fully explain away a specific treatment–outcome association, conditional on the measured covariates.

Mathematical Formula

RR > 1: E-value = RR + √(RR × (RR - 1))
RR < 1 (Protective Effect): RR* = 1 / RR, E-value = RR* + √(RR* × (RR* - 1))
OR / HR (Common outcome ≥ 15%): RR ≈ √OR