A risk ratio of 3.9 with an interval from 1.8 to 8.7. To explain this away entirely, an unmeasured confounder would have to be associated with both the exposure and the outcome by a risk ratio of 7.2630 each, above and beyond every covariate already adjusted for. To move the interval to the null it would still need 3.0 on both — exactly 3, since 1.8 plus the square root of 1.8 times 0.8 is 1.8 plus 1.2. Confounders that strong are rare and usually known about, which is what makes a finding of this size hard to dismiss.
risk ratio 3.9000 · interval 1.8000 to 8.7000
An unmeasured confounder would need 7.2630 on both sides to explain this away
Shifting the interval limit of 1.8000 to the null takes a weaker confounder: 3.0000 on both sides. That second number is the one that matters for whether the finding survives, since a result whose interval can be moved to 1 is no longer significant however far the point estimate travels.
E-value, point estimate
7.2630
needed on BOTH associations
E-value, interval limit
3.0000
to move 1.800 to 1
On the risk-ratio scale
3.9000
as entered
Identity check
3.900000
returns the observed 3.900000
What a confounder of each strength would have to do
For each association with the exposure, the association with the outcome needed to explain the result away
Association with the exposure
Association with the outcome needed
Verdict
1.50
no strength suffices
cannot explain it away
2.00
no strength suffices
cannot explain it away
2.50
no strength suffices
cannot explain it away
3.00
no strength suffices
cannot explain it away
4.00
117.000
implausibly strong
5.00
14.182
implausibly strong
8.00
6.659
implausibly strong
The E-value is the point where the two columns meet — the weakest confounder that works if it is equally associated with both. Rows above it trade: a confounder more strongly tied to the exposure needs less on the outcome. Rows whose exposure association is at or below the observed ratio cannot explain it away at any outcome strength, which is a hard bound rather than a judgement.
Substituting the E-value back into the bias formula returns 3.9000000000 against the observed 3.9000000000. That is exact rather than close: E squared over twice E minus one equals the observed ratio algebraically, which is what defines the E-value in the first place.
An E-value is a bound, not a defence. It says how strong confounding would need to be, and says nothing about whether a confounder that strong exists. It also assumes the measured covariates were adjusted for properly, which is what a balance table shows and weighting diagnostics price.
Point and interval Trade-off grid Exact closed form Odds and hazard ratios Identity printed as a check
What this tool shows
A risk ratio of 3.9 would take an unmeasured confounder of 7.2630 on both the exposure and the outcome to explain away. A risk ratio of 1.15 takes only 1.5653. That second number is the uncomfortable one, because associations of 1.6 with both an exposure and an outcome are everywhere — diet, income, education, physical activity all clear it routinely. The E-value turns “there might be unmeasured confounding” from a reviewer’s objection into a number the reader can weigh against what is actually known about the field.
The E-value for the point estimate and separately for the interval limit
A grid showing what each exposure-side strength requires on the outcome side
The hard bound: confounders no stronger than the observed ratio cannot explain it away at all
Odds ratios and hazard ratios converted before the bound is applied, with a rare-outcome switch
Protective effects inverted, so a ratio below 1 is not misread as a weak one
The exact identity that checks the closed form, printed on every run
Point and interval Trade-off grid Exact closed form Odds and hazard ratios
A bound on what confounding would have to be, not evidence about what it is.
Updated 13 September 2026 · Works in any browser, no installation
The E-value is the minimum strength of association an unmeasured confounder would need with both the exposure and the outcome, above and beyond every covariate already adjusted for, to fully account for an observed association. It is computed from the observed ratio alone by a closed form, needs no assumption about the confounder’s distribution, and gives a number a reader can compare against the associations actually known in the field — which is what turns an unanswerable objection into a judgement someone can make.
At a glance
Formula shown
E = RR + √(RR(RR − 1)) for a risk ratio above 1; for one below 1, invert first and apply the same formula. It is the value at which a confounder equally associated with exposure and outcome exactly reproduces the observed association through the bias formula RR_obs = (RR_EU·RR_UD)/(RR_EU + RR_UD − 1). Substituting E for both gives E²/(2E − 1), which equals the observed ratio exactly. Odds ratios are square-rooted first when the outcome is common; hazard ratios are converted by the corresponding transformation.
Scenario support
Reporting an observational association alongside how much confounding would undo it, answering a reviewer who objects that a study cannot rule out confounding, comparing the robustness of two findings of different sizes, and deciding whether a small but significant result is worth acting on.
Educational estimate
Planning support from the values you enter — not professional advice.
When a significant result needs almost nothing to undo it
The second preset is a risk ratio of 1.15 with an interval from 1.02 to 1.30. Statistically significant, publishable, and of exactly the size that fills observational epidemiology and much of applied social science.
Its E-value is 1.5653, and the interval limit needs only 1.1628. So an unmeasured factor associated with both the exposure and the outcome by about 1.6 would account for the entire finding, and one associated at 1.16 with both would be enough to render it non-significant.
Associations of that size are not exotic. Income relates to almost every health exposure and almost every health outcome at something like that strength. So do education, diet quality, and physical activity. A study that adjusted for three of them and not the fourth has plausibly left enough on the table to produce the whole result.
The contrast with the first preset is the point. A risk ratio of 3.9 needs 7.2630 on both sides, and an unmeasured confounder of that strength would be remarkable — strong enough that anyone in the field would probably already know about it. Two findings, both significant, and one is far harder to dismiss than the other. The p-values do not distinguish them at all.
Both associations, not one
The most common misreading is treating the E-value as a single association. It is the strength required on both legs — the confounder has to be associated with the exposure by that much and with the outcome by that much. A factor strongly related to the exposure but weakly related to the outcome does not produce much confounding, and vice versa.
The grid on the tool makes the trade explicit. On the first preset a confounder associated with the exposure at 5 would need 14.18 on the outcome; at 8 it would need 6.66. The E-value of 7.2630 is simply the point where the two are equal, which is the weakest a confounder can be if it is equally tied to both.
The grid also shows a hard bound rather than a judgement: a confounder associated with the exposure at 1.5, 2, 2.5 or 3 cannot explain away a risk ratio of 3.9 at any outcome strength. The bias formula has no solution there. That is often the most useful line in the table, because it rules out whole classes of candidate confounder without anyone having to argue about plausibility.
The limit matters more than the point
Two E-values are reported and the second is usually the one that decides anything. Moving the point estimate to the null is a strong requirement; moving the interval limit to the null is enough to destroy the claim of significance, and it always takes less.
On the first preset the point needs 7.2630 and the limit needs exactly 3 — exactly, because 1.8 plus the square root of 1.8 times 0.8 is 1.8 plus 1.2. A confounder of 3 on both sides would leave a result no longer distinguishable from nothing, and 3 is a great deal more plausible than 7.26.
The fourth preset shows the degenerate case handled properly. Its interval already includes 1, so the limit E-value is exactly 1: no confounding at all is needed, because the data does not distinguish the result from the null to begin with. The point estimate still carries an E-value of 1.6899, and reporting that number alone — which is easy to do and occasionally done — would dress a null finding in the language of robustness.
Odds ratios, hazard ratios, and protective effects
The bound is defined on the risk ratio scale, so other measures are converted before it is applied. An odds ratio approximates a risk ratio when the outcome is rare, and diverges from it sharply when the outcome is common — in which case the square root is the standard approximation, and the tool applies it when the outcome is marked common. An odds ratio of 4 with a common outcome is read as a risk ratio of 2, which lowers the E-value substantially and honestly.
Hazard ratios get the corresponding transformation. The distinction matters because quoting an E-value computed from a common-outcome odds ratio as though it were a risk ratio overstates robustness by a wide margin, and the conversion is easy to skip.
Protective effects are inverted first. A risk ratio of 0.45 is read as 2.2222, giving 3.8703. The bound is symmetric in that sense — a halving and a doubling require equally strong confounding — and inverting avoids the natural misreading of a number below 1 as a weak association.
What an E-value does not do
It is a bound, not a defence. It says how strong confounding would have to be. It says nothing whatever about whether a confounder that strong exists, and a large E-value is not evidence that the association is causal. The judgement about plausibility belongs to whoever knows the field, and the number exists to give them something specific to judge.
It also assumes the measured covariates were handled properly. The bound is about confounding beyond what was adjusted for, so if the adjustment itself failed — unbalanced covariates, a misspecified model, weights resting on three observations — the E-value is answering the wrong question. Check the balance table and the weighting diagnostics first.
And it addresses confounding only. Selection bias, measurement error, differential loss to follow-up and reverse causation are all untouched by it, and any of them can produce an association without a confounder anywhere. Designs that sidestep confounding structurally — an instrument, a policy change, a cutoff — trade this problem for a different one rather than removing it.
Reporting it
Give both numbers: the E-value for the point estimate and for the interval limit. The second is what a sceptical reader will use, and reporting only the first looks like selection.
Then do the part the arithmetic cannot. Name the confounders that were not measured, and say what is known about how strongly they relate to this exposure and this outcome. An E-value of 1.57 next to a known unadjusted risk factor of 1.8 is a finding in serious trouble; the same 1.57 in a setting where everything plausible has been measured and nothing exceeds 1.2 is reasonably secure. The number is an input to that argument, not a substitute for it.
Sources and methodology
References for the bound, its derivation and the caveats on its use.
Method. The closed form is checked against six independently computed values and against the quadratic it satisfies, E² − 2(RR)E + RR = 0. The defining identity is verified directly: substituting the E-value back into the bias formula returns the observed ratio, at 0.0e+0 for most values tested and 2.2e-16 at worst. The grid is confirmed to meet the closed form exactly at the E-value itself, and protective effects are confirmed to give the same bound as their reciprocals. That engine is verified on every change against 556 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
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Inverse Probability WeightingWeight units by the inverse of their treatment probability, with the effective sample size and the heaviest weight's share of the total on every run.
Difference-in-DifferencesEstimate a policy effect from treated and control groups before and after, with both naive comparisons, all four cell means and a pre-trend placebo test.
Instrumental VariableTwo-stage least squares with the first-stage F, the partial R-squared, the reduced form and the ordinary least squares estimate reported beside it.
Relative RiskRisk ratio and odds ratio from one table with the divergence between them plotted: they agree to half a percent at a 1% baseline, and at an 80% baseline the odds ratio is exactly half the risk ratio.
Odds RatioOdds ratio, relative risk, risk difference and number needed to treat from one 2x2 table — because an odds ratio of 6.00 can describe a relative risk of 1.50.
An educational tool. The E-value bounds how strong unmeasured confounding would have to be and provides no evidence about whether such a confounder exists; a large value is not evidence of causation. It assumes measured covariates were adjusted for adequately, and addresses confounding only — selection bias, measurement error and reverse causation are not covered. Odds-ratio and hazard-ratio conversions for common outcomes are standard approximations.