Introduces the “E-value”: the minimum risk-ratio-scale strength an unmeasured confounder would need with both exposure and outcome, given the measured covariates, to fully explain away an observed exposure-outcome association (or to shift its confidence interval to the null). Provides closed-form formulas and accessible software, proposed as a standard companion statistic for any observational effect estimate — including nutrition-cohort hazard ratios such as those reported for egg intake and CVD/T2D — to make residual/unmeasured confounding (e.g. healthy-user bias) quantitatively assessable rather than a purely qualitative worry.
relevance_note: The general quantitative-bias-analysis tool for gauging how much unmeasured confounding could plausibly explain away any given egg-cohort association.
Extracted (structured summary)
The E-value definition
O-57 - The E-value is the minimum RR-scale confounder strength needed to fully explain away an observed association
A general (mathematical) relationship between an observed effect estimate and the confounding that could account for it, applicable to any observational hazard/risk ratio — including the small egg–CVD and egg–T2D cohort estimates. It converts “residual confounding could explain this away” from a qualitative worry into a quantitative threshold: the E-value states how strong an unmeasured both-ways confounder (e.g. an unadjusted healthy-user factor) would have to be to reduce a given association to the null. The closed-form value and its application are given in the attached argument.
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Closed-form formula and application to cohort magnitudes
A-20 - The closed-form E-value formula shows small cohort hazard ratios need only modest unmeasured confounding to be explained away
Reasoning
Ding & VanderWeele’s confounding bound: for an unmeasured confounder U, let RR_EU be the maximum risk ratio relating exposure to U (across its levels) and RR_UD the risk ratio relating U to the outcome. The maximum factor by which such confounding can inflate an observed risk ratio is the bias factor B = (RR_EU · RR_UD) / (RR_EU + RR_UD − 1). Setting the confounding-adjusted risk ratio to 1 (the association fully explained away) and taking the symmetric worst case RR_EU = RR_UD = E — the case that minimises the required joint strength — and solving B = RR gives the closed form
E-value = RR + sqrt( RR · (RR − 1) ) for an observed RR ≥ 1.
For a protective estimate (RR < 1) the reciprocal 1/RR is used first. To ask what confounding would move the confidence interval to the null, the same formula is applied to the confidence limit nearer the null (the lower limit LL when RR > 1): E = LL + sqrt(LL·(LL − 1)); if that limit is already ≤ 1 the E-value is 1.
Worked application to nutrition-cohort magnitudes (numbers computed from the formula): an observed hazard ratio ≈ 1.2 — the order of magnitude reported for egg intake vs CVD/T2D — has E-value = 1.2 + sqrt(1.2 · 0.2) = 1.2 + sqrt(0.24) ≈ 1.2 + 0.49 = 1.69; a HR of 1.5 gives ≈ 2.37; a HR of 2 gives 2 + sqrt(2) ≈ 3.41. So an unmeasured confounder (e.g. an unadjusted healthy-user or dietary-pattern factor) associated with BOTH egg intake and the outcome by a risk ratio of roughly 1.7 each, beyond the measured covariates, would suffice to null a HR of 1.2, whereas a HR of 2 would need each of those two associations to be about 3.4. The inference is a valid algebraic consequence of the bounding factor B, which is itself a sharp (attainable) upper bound on confounding bias, so the E-value is the exact minimum joint strength required.
Validity (step 6): approved / checked. This is a mathematical claim and the algebra is elementary, so I traced it rather than trusting it. From the symmetric case RR_EU = RR_UD = E, the bias factor becomes B = E²/(2E − 1). Setting B = RR (association fully explained away) gives E² = RR(2E − 1) ⇒ E² − 2RR·E + RR = 0 ⇒ E = RR ± √(RR² − RR); the physical root is E = RR + √(RR(RR − 1)), matching the stated closed form (for RR ≥ 1; for RR < 1 apply it to 1/RR, and to the CI limit LL for the interval-null question — both as stated). The three worked values reproduce: RR = 1.2 → 1.2 + √0.24 = 1.69; RR = 1.5 → 1.5 + √0.75 = 2.37; RR = 2 → 2 + √2 = 3.41. The symmetric point is the correct minimiser: to null an association with B = RR you need both associations at least E, and max(RR_EU, RR_UD) is minimised on the diagonal, so E is the exact minimum joint strength — the sharpness/attainability of B is the Ding–VanderWeele result the interpretive sentence relies on. The qualitative reading in the statement (E increases monotonically in RR, so small HRs need only modest confounding, large HRs strong confounding) is a direct consequence of dE/dRR > 0. No undercutting defeater: given the bound B, the algebra is forced.
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