E-Values for Confidence Limits

Why Calculate an E-Value for a Confidence Limit?

A point estimate describes the estimated association, but a confidence interval also reflects sampling uncertainty. For an observed risk ratio above 1, the confidence limit nearest the null is the lower limit; for a protective association below 1, it is the upper limit. The E-value for that limit summarizes how strong unmeasured confounding would have to be to move that boundary to the null, under a specified set of assumptions.

This is a different question from asking how strong confounding would need to be to explain away the point estimate. A confidence-limit E-value focuses on the boundary of the interval nearest no effect. It therefore describes the robustness of the interval’s incompatibility with the null, not just the size of the estimated association.

Definition: For a risk ratio \(r\) on the harmful side of the null, where \(r\geq 1\), its E-value is \(E(r)=r+\sqrt{r(r-1)}\). For a risk ratio below 1, first take its reciprocal: \(E(r)=E(1/r)\). For a confidence interval that excludes the null value 1, the confidence-limit E-value is calculated from the endpoint nearest 1, oriented to be at least 1.

The E-value is reported on the risk-ratio scale. For a harmful association, if the confidence interval is \([L,U]\) with \(L>1\), use \(L\). For a protective association with \(U<1\), use \(1/U\). This reciprocal places the limit on the same side of the null as a risk ratio greater than 1. In both cases the calculation is \(E(q)=q+\sqrt{q(q-1)}\), where \(q\) is the selected, oriented limit.

Deriving the Formula and the Confidence-Limit Rule

A short derivation shows what the formula encodes. A standard upper bound on the confounding bias factor, for a risk-ratio analysis conditional on measured covariates, is

B=\frac{RR_{AU}RR_{UY}}{RR_{AU}+RR_{UY}-1}.

Here \(RR_{AU}\) describes the strength of association between exposure \(A\) and an unmeasured confounder \(U\), conditional on measured covariates; \(RR_{UY}\) describes the association between \(U\) and outcome \(Y\), also conditional on measured covariates. These strengths are expressed on a risk-ratio scale. The bound is often used as a conservative summary: a bias factor no larger than \(B\) can be produced when the two associations have the specified strengths.

For an equal-strength benchmark, set \(RR_{AU}=RR_{UY}=x\), with \(x\geq 1\). Then \(B=x^2/(2x-1)\). To determine the common strength that could produce a bias factor \(r\), set \(B=r\) and solve:

\frac{x^2}{2x-1}=r \quad\Longrightarrow\quad x^2-2rx+r=0 \quad\Longrightarrow\quad x=r\pm\sqrt{r(r-1)}.

Because the relevant solution must satisfy \(x\geq 1\), the solution is \(x=r+\sqrt{r(r-1)}\). This gives the E-value formula. It is also the minimum strength of the larger of the two associations that could attain the required bias factor when the associations are allowed to differ. This does not mean every possible confounder must have equal-strength associations with exposure and outcome.

For a confidence interval excluding 1, use its nearest-null endpoint in place of \(r\). For example, when the entire interval is above 1, the lower limit is the smallest effect size still represented at the interval boundary. When the entire interval is below 1, the upper limit is the boundary closest to 1, so its reciprocal gives the corresponding oriented ratio. A larger E-value means stronger confounding associations would be needed, under the framework, to move that confidence limit to the null.

Calculating the Confidence-Limit E-Value

Check the scale and the null.
Confirm that the estimate and interval are risk ratios, with null value 1, and determine whether the interval excludes 1.

Select the nearest-null endpoint.
For an interval entirely above 1, select its lower limit. For an interval entirely below 1, select its upper limit and take its reciprocal.

Apply the formula.
For the oriented endpoint \(q\geq 1\), calculate \(q+\sqrt{q(q-1)}\). Keep adequate precision during the arithmetic, then round the reported value.

Interpret the result in context.
Describe the confounding strength the limit would require to reach the null, and state that this is a sensitivity summary, not a probability or proof of no confounding.

Worked Example: Harmful Association With a Lower Limit Near the Null

Suppose a hypothetical observational analysis estimates a risk ratio of 2.4 for an adverse outcome, with a 95% confidence interval from 1.3 to 4.1. The whole interval is above 1, so its nearest-null endpoint is \(L=1.3\). The point estimate is not used for this confidence-limit calculation.

Substitute the lower limit into the formula:

E(1.3)=1.3+\sqrt{1.3(1.3-1)} =1.3+\sqrt{0.39} \approx 1.3+0.6245 \approx 1.924.

The confidence-limit E-value is approximately 1.924. Under the stated risk-ratio sensitivity framework, an unmeasured confounder would need associations of roughly this magnitude, in the relevant conditional sense, with exposure and outcome to account for enough bias to move the lower confidence limit to 1. This does not establish that such a confounder exists or that the estimated causal effect is valid.

Worked Example: Protective Association

In a second hypothetical analysis, a risk ratio for an outcome is estimated as 0.55, with a 95% confidence interval from 0.38 to 0.82. The interval lies below 1. Its nearest-null endpoint is the upper limit, \(U=0.82\), not the lower limit. Orient that endpoint by taking its reciprocal:

q=\frac{1}{0.82}\approx 1.2195.

Now calculate:

E(q)=1.2195+\sqrt{1.2195(1.2195-1)} \approx 1.2195+\sqrt{0.2677} \approx 1.2195+0.5174 \approx 1.737.

The confidence-limit E-value is approximately 1.737. It summarizes the confounding strength needed to move the interval’s upper limit, on the protective side, to the null. Taking the reciprocal of the endpoint first is essential: applying the formula directly to 0.82 would not implement the defined calculation.

Worked Example: Point Estimate and Confidence Limit Tell Different Stories

Suppose a hypothetical study reports a risk ratio of 1.8 with a 95% confidence interval from 1.05 to 3.0. The point-estimate E-value is \(E(1.8)\), while the confidence-limit E-value uses \(1.05\), the lower endpoint nearest 1.

For the point estimate:

E(1.8)=1.8+\sqrt{1.8(0.8)} =1.8+\sqrt{1.44} =3.000.

For the lower confidence limit:

E(1.05)=1.05+\sqrt{1.05(0.05)} =1.05+\sqrt{0.0525} \approx 1.279.

The point estimate has an E-value of 3.000, but the confidence-limit E-value is about 1.279. These are not contradictory. The point estimate is farther from the null, while the interval’s lower boundary is close to the null. The latter can be shifted to 1 by substantially weaker confounding associations than would be needed to explain away the point estimate.

Assumptions, Diagnostics, and Comparisons

The calculation itself is simple, but its interpretation depends on the analysis it summarizes. The effect measure must be a risk ratio, or a measure that has been appropriately represented on the risk-ratio scale. The confidence interval must be computed for the intended estimand using a defensible sampling model and design. The causal interpretation also relies on the usual assumptions of the underlying study, including a well-defined exposure and outcome, consistency, and no uncontrolled bias other than the unmeasured confounding being considered. An E-value does not verify these assumptions.

Before reporting the result, check the following:

Confirm the measure. The formula here is for risk ratios. Do not apply it directly to an odds ratio or hazard ratio without an appropriate conversion or method.
Choose the correct endpoint. For an interval above 1, use the lower limit; for one below 1, use the upper limit and its reciprocal.
Check the null and interval. The interpretation of a confidence-limit E-value as the confounding needed to move the interval boundary to no effect is most useful when the interval excludes 1. If it already includes 1, it is already statistically compatible with the null.
Check rounding. Retain several digits through the square root and round only the final E-value. Small differences in the last digit may result from rounding.

The confidence-limit E-value complements, rather than replaces, a confidence interval. The interval conveys sampling uncertainty under its model; the E-value adds a sensitivity summary for unmeasured confounding. It does not modify the interval, provide a new p-value, or assign a probability to the null hypothesis. In contrast, Rosenbaum bounds use a sensitivity parameter for departures from randomized assignment within matched sets. That parameter and an E-value are different quantities, so their numerical values should not be compared as though they shared a scale.

There are also limits to what this summary can show. It does not identify a confounder, prove that no confounder is strong enough, or measure bias from selection, measurement error, model misspecification, or other sources unless those are explicitly incorporated into an analysis. It reduces a complex sensitivity question to a particular risk-ratio summary, and its meaning depends on the chosen estimand and confidence interval. A useful report therefore gives the estimate, interval, endpoint used, E-value, and a contextual explanation of the assumptions.

Common Mistakes

Using the point estimate for the interval E-value. The point-estimate E-value answers how strong confounding would need to be to explain away that estimate. The confidence-limit E-value uses the endpoint nearest the null.
Using the farther endpoint. For an interval above 1, the upper limit is not the endpoint closest to 1. For a protective interval, use the upper limit and take its reciprocal.
Calling the E-value a probability. An E-value of 1.9 is not a 90% probability of robustness, nor is it a probability that the null is true. It is a sensitivity measure on the risk-ratio scale.
Treating the E-value as proof of causality. A large value does not rule out confounding or other biases; it describes the strength required under the stated framework.
Rounding intermediate values too early. For instance, \(1.3+\sqrt{0.39}\approx1.924\), not 1.925. Keep intermediate precision and round the final result consistently.

Key takeaway: For an interval excluding the risk-ratio null of 1, orient the confidence limit nearest the null to be at least 1, then calculate \(q+\sqrt{q(q-1)}\). This E-value summarizes the confounding strength needed to move that interval boundary to no effect, under the assumptions of the framework.

Check Your Understanding

For each problem, identify the nearest-null limit, show the calculation, and interpret the result. Solutions follow the questions.

A hypothetical harmful risk ratio has a 95% confidence interval from 1.2 to 2.4. Calculate the confidence-limit E-value.
A hypothetical protective risk ratio has a 95% confidence interval from 0.50 to 0.80. Which endpoint should be used, and what is its E-value?
Why is the confidence-limit E-value different from the point-estimate E-value when the lower confidence limit is close to 1?
What does a confidence-limit E-value of 1.8 not tell you?

Solutions. (1) Use the lower endpoint, \(q=1.2\): \(E(1.2)=1.2+\sqrt{1.2(0.2)}=1.2+\sqrt{0.24}\approx1.690\). (2) Use the upper endpoint \(0.80\) and take its reciprocal, so \(q=1/0.80=1.25\). Then \(E(1.25)=1.25+\sqrt{1.25(0.25)}=1.25+\sqrt{0.3125}\approx1.809\). (3) The point estimate measures distance from the null at the estimate, while the confidence-limit calculation uses the interval boundary nearest the null, which may be much closer. (4) It does not give the probability that the result is robust, prove that no confounder exists, or rule out other biases; it is a conditional sensitivity summary under the risk-ratio framework.

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