Calculating E-Values From Reported Results

Why Recover the Numbers Before Calculating an E-Value?

Published results do not always present an effect estimate in the form needed for an E-value calculation. A paper may report a risk ratio with a confidence interval, a regression coefficient and standard error, a p-value, or an odds ratio. These summaries can often be converted to a risk ratio and its uncertainty, but only if their scale and statistical construction are understood. A reproducible calculation therefore begins by identifying exactly what was reported, recovering the needed quantities, and checking that the conversion is appropriate.

The E-value is calculated on the risk-ratio scale. For an estimate above the null, it summarizes the minimum equal strength of association an unmeasured confounder would need to have with treatment and outcome, conditional on measured covariates, to explain the estimate; equivalently, it is the minimum possible value of the stronger association when the two strengths may differ. For an estimate below the null, first orient the estimate as an association greater than one by taking its reciprocal. The E-value for a confidence interval uses the limit closest to the null, not necessarily the point estimate.

Definition: For a risk ratio $RR$, define $Q=\max(RR,1/RR)$, so $Q\geq 1$. The point-estimate E-value is $$ E(Q)=Q+\sqrt{Q(Q-1)}. $$ If a confidence interval excludes 1, its E-value is calculated from the confidence limit closest to 1, reoriented in the same way. If the interval includes 1, its confidence-limit E-value is 1.

Recovering Estimates and Uncertainty

For a risk ratio estimated with a log-scale Wald confidence interval, the interval is symmetric around the estimated log risk ratio, even though the reported interval on the ratio scale is generally asymmetric. If the reported estimate is $RR$ and its two-sided $100(1-\alpha)\%$ interval is $(L,U)$, then an approximate standard error is

SE\{\log(RR)\}=\frac{\log(U)-\log(L)}{2z_{1-\alpha/2}}.

For a 95% interval, $z_{1-\alpha/2}$ is approximately 1.96. Conversely, given an estimated log risk ratio $b=\log(RR)$ and its standard error, the corresponding Wald interval is

\left(\exp\{b-z_{1-\alpha/2}SE\},\ \exp\{b+z_{1-\alpha/2}SE\}\right).

These calculations are appropriate when the reported interval is a log-scale Wald interval, or a sufficiently close approximation to one. Profile-likelihood, bootstrap, or other intervals need not be symmetric on the log scale; an SE reconstructed from their endpoints is then only a rough summary, not a recovery of the original method.

Worked Example: Reconstructing a Risk-Ratio Standard Error

Suppose a hypothetical report gives an adjusted risk ratio of 1.60 with a 95% confidence interval from 1.20 to 2.13. Assume the interval is a rounded log-scale Wald interval. Recover the approximate standard error and calculate the point and confidence-limit E-values.

Recover the standard error. The log endpoints are $\log(1.20)=0.1823$ and $\log(2.13)=0.7561$. Thus

SE\{\log(RR)\}=\frac{0.7561-0.1823}{2(1.96)} =\frac{0.5738}{3.92}=0.1464.

As a check, $\log(1.60)=0.4700$. The rounded interval is nearly centered at this value: its log midpoint is $(0.1823+0.7561)/2=0.4692$. Using the recovered SE gives $\exp(0.4700\pm1.96(0.1464))$, or approximately 1.201 to 2.132. The small difference from the published lower limit reflects rounding.

Point-estimate E-value. Since the estimate is above 1, $Q=1.60$:

E=1.60+\sqrt{1.60(1.60-1)} =1.60+\sqrt{0.96} =2.580.

Confidence-limit E-value. The confidence limit closest to 1 is 1.20, so $E=1.20+\sqrt{1.20(0.20)}=1.20+\sqrt{0.24}=1.690$. The point-estimate E-value is the minimum equal strength of association with treatment and outcome that could explain the estimate; with unequal strengths, one association may be stronger and the other weaker than the E-value. The confidence-limit value is smaller: associations of that strength could move the confidence interval to include the null, subject to the framework’s assumptions.

A reported regression coefficient may already be on the log-risk-ratio scale. If so, exponentiate it to obtain the risk ratio, or use it directly as $b=\log(RR)$ when constructing a log-scale interval. Do not exponentiate a coefficient whose model uses a different link or whose interpretation is not a risk ratio. When a two-sided p-value comes from a Wald test of the same coefficient against zero, an approximate standard error can also be recovered: find $z=\Phi^{-1}(1-p/2)$, then divide the absolute coefficient by $z$. This is an approximation, especially when the p-value is rounded.

Worked Example: Recovering an Interval From a Wald p-Value

A hypothetical paper reports a log-risk-ratio coefficient $b=0.4055$ and a two-sided Wald p-value of 0.040 for testing $H_0:b=0$. Recover an approximate standard error and 95% interval, then calculate the E-values.

The corresponding risk ratio is $\exp(0.4055)=1.500$. A two-sided p-value of 0.040 corresponds to $z\approx2.054$, so the approximate standard error is

SE=\frac{0.4055}{2.054}=0.1974.

The 95% log-scale interval is $0.4055\pm1.96(0.1974)$, which is approximately $0.0186$ to $0.7924$. Exponentiating gives a risk-ratio interval of approximately $(1.019,2.209)$. The point-estimate E-value is $1.500+\sqrt{1.500(0.500)}=2.366$. The closest confidence limit is 1.019, giving $1.019+\sqrt{1.019(0.019)}\approx1.158$. The interval just excludes 1, so its E-value is only slightly greater than 1. Because the p-value was rounded, both the reconstructed standard error and interval should be treated as approximate.

Protective Estimates and Confidence Limits

For a protective estimate, the numerical endpoint nearest 1 is still the endpoint closest to the null, but the association must be oriented above 1 before applying the formula. For a point estimate below 1, use its reciprocal. For a confidence interval entirely below 1, use the reciprocal of its upper endpoint: that endpoint is closest to 1. If an interval crosses 1, it does not exclude the null, so its confidence-limit E-value is 1. In particular, do not use the lower endpoint of a protective interval merely because it is the smaller number.

Worked Example: A Protective Risk Ratio

Suppose a hypothetical study reports a risk ratio of 0.70 with a 95% confidence interval from 0.55 to 0.89. Recover an approximate log-scale standard error and calculate the point and confidence-limit E-values.

The log endpoints are $\log(0.55)=-0.5978$ and $\log(0.89)=-0.1165$. The approximate standard error is

SE\{\log(RR)\} =\frac{-0.1165-(-0.5978)}{3.92} =\frac{0.4813}{3.92} =0.1228.

The log estimate is $\log(0.70)=-0.3567$, close to the interval’s log midpoint, $(-0.5978-0.1165)/2=-0.3572$. This supports (but does not prove) the assumed log-Wald construction; rounding accounts for the small difference.

For the point estimate, $Q=1/0.70=1.4286$, so $E=1.4286+\sqrt{1.4286(0.4286)}=1.4286+\sqrt{0.6122}=2.211$. For the confidence interval, the upper limit 0.89 is closest to 1. Reorient it as $Q=1/0.89=1.1236$, giving $E=1.1236+\sqrt{1.1236(0.1236)}\approx1.496$. The interval excludes 1, but its confidence-limit E-value is modest. These values summarize sensitivity to unmeasured confounding; they do not establish that such a confounder exists.

When the Report Gives an Odds Ratio

An odds ratio is not generally a risk ratio. If the outcome is not rare, treating an odds ratio as though it were a risk ratio can substantially overstate the risk-ratio effect and therefore distort the E-value. When an appropriate untreated or reference-group risk $p_0$ is available, an odds ratio $OR$ can be converted to a risk ratio using

RR=\frac{OR}{(1-p_0)+p_0 OR}.

This identity follows because the exposed risk implied by the odds ratio is $p_1=ORp_0/(1-p_0+ORp_0)$, and $RR=p_1/p_0$. For a confidence interval, convert both odds-ratio endpoints using the same $p_0$, then calculate the E-value from the converted risk ratio and the converted limit closest to 1. This is appropriate only when the reference risk corresponds to the population and estimand represented by the odds ratio. A single reference risk may not adequately convert an adjusted conditional odds ratio when baseline risks vary across covariate patterns; moreover, odds ratios are non-collapsible. In such cases, the resulting risk ratio and E-value are at best approximations. The earlier material on converting odds ratios before computing E-values develops this issue further.

Worked Example: Converting an Odds Ratio Before Calculation

In a hypothetical setting, an adjusted odds ratio is 2.00 with a 95% interval from 1.30 to 3.10. Suppose the relevant reference-group risk is $p_0=0.10$, and assume this risk is suitable for the conversion. Convert the estimate and interval to risk ratios before calculating E-values.

For the point estimate, $RR=2.00/\{(1-0.10)+0.10(2.00)\}=2.00/(0.90+0.20)=1.818$. Its E-value is $1.818+\sqrt{1.818(0.818)}\approx1.818+1.220=3.038$.

Convert each interval endpoint using the same reference risk. The lower endpoint gives $1.30/(0.90+0.10(1.30))=1.30/1.03=1.262$. The upper endpoint gives $3.10/(0.90+0.10(3.10))=3.10/1.21=2.562$. The converted interval is approximately $(1.262,2.562)$, and its closest limit to 1 is 1.262. Its E-value is $1.262+\sqrt{1.262(0.262)}\approx1.837$. These calculations use a fixed reference risk; uncertainty in that risk is not included in the interval or E-values.

Why the E-Value Formula Has This Form

A brief derivation helps check calculations and clarifies the boundary case. Let $Q\geq1$ be the risk-ratio-scale association to be explained. Under the standard E-value bounding relationship, the equal-strength threshold $E$ satisfies $Q=E^2/(2E-1)$. Rearranging gives

$$ E^2-2QE+Q=0. $$

The quadratic formula gives $E=Q\pm\sqrt{Q^2-Q}=Q\pm\sqrt{Q(Q-1)}$. The required root is the plus sign because the threshold must be at least 1. When $Q>1$, the other root is below 1: indeed, $\sqrt{Q(Q-1)}>Q-1$, since $Q(Q-1)>(Q-1)^2$, and hence $Q-\sqrt{Q(Q-1)}<1$. When $Q=1$, both roots equal 1. Thus the formula correctly gives an E-value of 1 at the null; there is no distinct smaller root in that boundary case.

Checks, Comparisons, and Limitations

Use a short diagnostic sequence before reporting a calculation:

Confirm that the effect measure is a risk ratio, or that any odds-ratio conversion has a defensible reference risk and target interpretation.
Check that an interval is on the stated scale and identify whether it is a log-Wald interval before treating its width as a standard error.
For a protective estimate, take the reciprocal; for a protective confidence interval, orient the upper endpoint nearest 1.
Check whether the interval excludes 1. If it crosses 1, the confidence-limit E-value is 1, even if the point-estimate E-value is larger.
Retain precision through intermediate steps and round only the final values. Rounded published limits or p-values make reconstructed quantities approximate.

A common mistake is to insert a protective risk ratio such as 0.70 directly into the formula, or to take the reciprocal of the wrong confidence limit. The formula is defined for $Q\geq1$, so inserting 0.70 does not calculate the E-value for that protective association. The correction is to use $Q=1/0.70$ for the point estimate and, for an interval wholly below 1, $Q=1/U$, where $U$ is its upper limit.

E-values and Rosenbaum bounds answer different sensitivity questions. An E-value expresses a confounder-strength threshold on a risk-ratio scale; Rosenbaum bounds describe sensitivity of a matched design to a specified degree of treatment-assignment bias. Neither replaces careful design assessment, and their numerical values should not be compared as though they shared a scale. E-values also do not measure bias from selection, outcome or exposure misclassification, model misspecification, or multiple sources of bias unless those are represented by the assumptions of the analysis. They are sensitivity summaries, not probabilities that confounding exists or proofs of causal validity.

Key takeaway: Reconstruct the effect and uncertainty on the risk-ratio scale first, verify the reported interval or test supports that reconstruction, orient protective effects correctly, and then calculate the point and confidence-limit E-values separately.

Check Your Understanding

Answer each question, then compare with the solutions below.

A risk ratio is 1.40 with a 95% confidence interval of 0.95 to 2.05. What is the confidence-limit E-value, and why?
A protective risk ratio is 0.80 with a confidence interval of 0.65 to 0.96. Which limit is used for the confidence-limit E-value, and what oriented $Q$ does it give?
A study reports a log-risk-ratio coefficient of 0.20 and standard error 0.10. Give the approximate 95% risk-ratio interval.
Why can an odds ratio of 2.00 not be converted to one unique risk ratio without further information?

Solutions. (1) The interval contains 1, so the confidence-limit E-value is 1; the data do not exclude the null at that confidence level. (2) Use the upper limit 0.96, which is closest to 1. Its oriented value is $Q=1/0.96=1.0417$. (3) The log interval is $0.20\pm1.96(0.10)=(0.004,0.396)$; exponentiating gives approximately $(1.004,1.486)$. (4) The conversion depends on the reference risk $p_0$: $RR=OR/\{(1-p_0)+p_0OR\}$. Different reference risks yield different risk ratios.