What does an E-value reported alongside an observational risk ratio actually mean?

## The quantity being reported An E-value answers a single question: how strongly would an unmeasured variable have to be related to both the exposure and the outcome, over and above everything already adjusted for, in order to explain away the association we observed? It is a minimum. Anything weaker on either association cannot fully account for the finding, whatever its prevalence. The scale is the risk ratio. If the E-value is 2.4, the hidden variable must be associated with the exposure by a risk ratio of at least 2.4 **and** with the outcome by a risk ratio of at least 2.4. Those are joint requirements. A variable that predicts the outcome tenfold but is evenly distributed across treated and untreated groups is not a confounder at all, and one that is wildly unbalanced across groups but unrelated to the outcome is equally harmless. ## Computing it For an observed risk ratio RR of 1 or more, the E-value is `RR + sqrt(RR * (RR - 1))`. - RR = 1.0 gives 1.0 - no confounding at all is needed, since there is nothing to explain. - RR = 1.2 gives about 1.69. - RR = 1.5 gives about 2.37. - RR = 3.0 gives about 5.45. For a protective association with RR below 1, take the reciprocal first and apply the same formula. Effect measures that are not risk ratios - odds ratios for common outcomes, hazard ratios, standardised mean differences - are converted to an approximate risk ratio scale before the formula is applied, and that conversion is itself an approximation worth flagging. ## Two E-values, not one Good practice reports the E-value for the point estimate and for the confidence limit nearest the null. Suppose a study reports a risk ratio of 1.5 with an interval from 1.2 to 1.9. The point-estimate E-value is about 2.4; the E-value for the limit of 1.2 is about 1.7. The interpretation splits cleanly: confounding at strength 2.4 on both arms would move the estimate to the null, while confounding at only 1.7 would be enough to make the interval include the null. The second number is usually the decision-relevant one, because it is what it takes to lose the claim of a non-zero effect. The limit E-value is always the smaller of the two for a positive association. ## Reading the size There is no universal threshold. Judgment comes from comparison. Compute the observed associations of the measured covariates with the exposure and the outcome; if the strongest measured covariate reaches only 1.4 on each side and the E-value is 2.4, then a confounder would have to be substantially stronger than anything you have already seen in this domain, which is a real argument. If the E-value is 1.3 and several measured covariates comfortably exceed that, the finding is fragile and should be described that way. The scale of the outcome matters too. In a well-studied domain where the causes of the outcome are largely known, a moderate E-value can be persuasive because the space of candidate confounders is nearly exhausted. In a domain full of unmeasured behavioural variables - health-seeking behaviour, motivation, socio-economic position - even a large E-value may not be comforting, because those variables plausibly reach large associations on both sides. ## Common errors The most frequent mistake is treating the E-value as a probability, or as a claim that a confounder of that strength exists. It is neither: it is a threshold on a hypothetical. The second is applying it to only one association - saying the confounder must predict the outcome by 2.4 while forgetting that it must also be that unbalanced across exposure groups. The third is confusing it with prevalence: an E-value of 2.4 says nothing about how common the confounder is. The classical bias formulas do involve prevalence, and the E-value is deliberately the worst case over prevalence, which is why it is a conservative bound and why a confounder meeting the E-value threshold only *could* explain the result, at the most damaging prevalence. A fourth error is over-claiming robustness. A large E-value does not establish causation. It rules out weak confounding only. Selection into the sample, measurement error in the exposure, and outright model misspecification are separate threats that the E-value does not address at all. ## Why it caught on The E-value's appeal is that it needs no assumptions about the unmeasured variable - not its distribution, not its prevalence, not whether it is binary or continuous - and it is a single number computable from an already-published estimate. That makes it cheap enough that it can be reported routinely, which is exactly what turns sensitivity analysis from a specialist exercise into a default part of how an observational claim is presented.