The E-Value: Asking How Strong a Hidden Confounder Would Have to Be

The E-value is a number that summarizes how robust an observational finding is to unmeasured confounding. It states the minimum strength of association, on the risk-ratio scale, that some unmeasured factor would need to have with both the exposure and the outcome to fully explain away the observed result. A large E-value means only an implausibly strong hidden confounder could account for the finding; a small one means a fairly ordinary unmeasured factor could.

The worry the E-value addresses

Observational studies compare people who were not randomized, so the groups can differ in ways that affect the outcome. Researchers adjust for the confounders they measured, age, smoking, income, but they can never be sure they measured everything. The nagging question after any observational result is: could some factor nobody accounted for be producing this association?

Historically that worry was answered with hand-waving. The E-value was introduced to answer it with a number, giving readers a concrete sense of how much unmeasured confounding it would take to undo the finding.

What the number actually says

The E-value is the minimum strength of association that an unmeasured confounder would need to have, with both the exposure and the outcome, to fully explain away the observed effect, after the measured confounders are already accounted for. Strength is expressed as a risk ratio. So an E-value of 2 means that to erase the result, a hidden confounder would have to be associated with both the exposure and the outcome by a risk ratio of at least 2 each, beyond everything already adjusted for.

You can compute it for the main estimate and, separately, for the confidence-interval limit nearest the null. The second version asks how much confounding it would take to push the result to statistical non-significance, which is often the more relevant threshold.

Reading an E-value

Bigger is more reassuring. A large E-value means that only a strong, specific unmeasured confounder, one more strongly linked to both exposure and outcome than most known risk factors, could account for the association. That is hard to hide, because such a powerful factor would usually be recognized. A small E-value near 1 means that even a weak, common unmeasured factor could explain the result, so the finding is fragile.

The way to judge the number is to compare it against the confounders you already know about. If measured risk factors in the same study have associations of, say, 1.3, and the E-value is 3, an unmeasured confounder would have to be far stronger than anything measured, which strains belief. If the E-value is 1.3, a confounder no stronger than the ordinary ones could do it, and caution is warranted.

Why it uses two associations

The E-value refers to two associations, confounder with exposure and confounder with outcome, because a confounder only distorts a result if it is linked to both. A factor tied only to the outcome, or only to the exposure, does not create a spurious association between them. The E-value captures the joint requirement in a single worst-case number, assuming both associations are as large as the E-value on the risk-ratio scale.

This is why a hidden confounder has to clear a high bar to explain a strong result: it must be substantially connected to the exposure and to the outcome at the same time. Many candidate factors fail one half of that test, which is part of why sizable associations are harder to explain away than small ones.

What it does not do

The E-value is a sensitivity analysis, not a lie detector. It does not prove a result is causal, and it does not tell you that an unmeasured confounder of the required strength actually exists or does not. It only quantifies how strong such a confounder would have to be. Judging whether one that strong is plausible still takes subject knowledge.

It also addresses only unmeasured confounding, not other threats: selection bias, measurement error, reverse causation, and the like are outside its scope. And it assumes the measured confounders were handled correctly. A large E-value on top of a poorly adjusted analysis is not reassuring. Treat it as one useful input, not a verdict.

Putting it to work as a reader

When an observational study reports an E-value, read it as a stress test. Ask how it compares to the strength of the confounders the authors did measure; an E-value far above them is genuinely reassuring, one near them is not. Check whether they reported the E-value for the confidence-interval limit as well as the point estimate, since a result can be robust at its center yet fragile at its edge.

If no E-value is given, you can still apply the mindset: name a plausible unmeasured confounder and ask, honestly, whether it could be strongly linked to both the exposure and the outcome. Observational associations that survive that question deserve more weight than those that a single obvious omission could erase.