E-value for Unmeasured Confounding calculator

Calculate the E-value for a risk ratio, hazard ratio or odds ratio and its confidence interval: how strong an unmeasured confounder would have to be to explain an observational association away.

What it does

Turns "but what about confounders you did not measure?" into a number. The E-value is the minimum strength of association — on the risk-ratio scale, with both the exposure and the outcome — that an unmeasured confounder would need in order to explain an observed association away entirely. A second E-value does the same for the confidence bound nearer the null: the least a confounder would need to move the interval to include 1.

When to use it

After any Cox, logistic or risk-ratio estimate from observational data, when a reader will reasonably ask whether something you did not measure could account for the result. It is a sensitivity analysis, not a test: a large E-value says the association is robust to moderately strong unmeasured confounding; a small one says a plausible confounder could remove it.

Worked example

A cohort finds business accounts regress to a managed plan with a hazard ratio of 1.73 (95% CI 1.19 to 2.49), and 41% of accounts had the event, so the HR is first converted to an approximate risk ratio of 1.46. The E-value is 2.28 for the estimate and 1.51 for the lower bound. An unmeasured confounder would need a risk ratio of at least 2.28 with both segment and the outcome to explain the estimate away — and only 1.51 to push the interval across 1, which is a much weaker ask.

Cautions

• An E-value is not evidence of causation. It says how strong an unmeasured confounder would have to be, not whether one exists — and confounders of that strength are common in some fields and rare in others, which only subject knowledge can judge.
• Hazard and odds ratios must be put on the risk-ratio scale first. For a common outcome (more than about 15% by the end of follow-up) the conversions are HR → RR ≈ (1 − 0.5^√HR)/(1 − 0.5^√(1/HR)) and OR → RR ≈ √OR; for a rare outcome the HR or OR is used directly. Applying the formula to a raw odds ratio for a common outcome overstates the E-value.
• The E-value for the confidence bound is usually the more honest number to report: it is what a sceptic needs to explain away the statistical significance, not the point estimate.
• It is 1 whenever the interval already includes the null — no unmeasured confounding is needed for the data to be consistent with no effect.

Alternatives

• Adjusted regression — the confounder was measured — then adjust for it rather than reasoning about its hypothetical strength
• A designed study — a causal answer is genuinely needed; no sensitivity analysis substitutes for randomisation or a quasi-experimental design
• Bias analysis with named confounders — you can put numbers on a specific confounder's prevalence and effect — the E-value is the agnostic bound, not the whole story

How to read the output

E-value (estimate)

The risk ratio an unmeasured confounder would need with both exposure and outcome to reduce the observed association to the null. Compare it with the strength of the measured confounders you did adjust for. If the strongest known confounder had a risk ratio of 1.5 with the outcome, an E-value of 4 is reassuring; an E-value of 1.4 is not.

E-value (CI bound)

The same quantity for the confidence bound nearer 1 — how strong a confounder would need to be to move the interval to include the null. This is the number that answers "could the significance be confounding?", and it is always smaller than the estimate's E-value. Report both.

Risk ratio used

The estimate after conversion to the risk-ratio scale. Equal to the input for a risk ratio, or for a hazard/odds ratio with a rare outcome. If the outcome is common and you told the calculator it was rare, the E-value is too large. The threshold is about 15% by the end of follow-up.

How this calculator is validated · Which statistical test should I use?