A Worked E-Value Calculation From a Risk Ratio

From a Reported Risk Ratio to an E-Value

A reported risk ratio is often easier to interpret than a complicated model, but it does not show how sensitive the estimate is to unmeasured confounding. An E-value expresses that sensitivity on the risk-ratio scale: it gives a benchmark for how strongly an unmeasured confounder would need to be associated with treatment and outcome, conditional on measured covariates, to account for an observed association. This tutorial works through the arithmetic from a reported risk ratio and its confidence interval, including estimates below one and intervals that include the null.

The calculations are summaries of sensitivity, not tests for the presence or absence of confounding. They do not establish that an unmeasured confounder exists, nor that a causal interpretation is justified. The estimate must still be interpreted in light of the study design, measurement, and assumptions underlying the causal analysis.

An E-value of, for example, 2.5 is a benchmark for the strength of associations on a risk-ratio scale: in the equal-strength interpretation, an unmeasured confounder would need associations of at least 2.5-fold with both treatment and outcome, conditional on measured covariates, to explain away the point estimate. More generally, unequal associations could do so if one were stronger and compensated for the other being weaker. The E-value does not say that the confounder actually has those associations.

A Calculation Sequence

For an estimate above one, use the reported risk ratio directly. For an estimate below one, taking the reciprocal reverses the comparison so that the effect is expressed in the direction above one. For a confidence interval, this reversal also reverses endpoint order: an interval \((L,U)\) becomes \((1/U,1/L)\). The endpoint nearest 1 on the oriented scale is the one used for the confidence-limit E-value.

These steps give a reliable sequence: check whether the interval includes 1; orient a non-null estimate above one; calculate the point E-value; then calculate the E-value for the confidence limit nearest one. Keep full precision through the square root and round only the final result.

Protective Estimates and Reversed Intervals

A protective association is represented by a risk ratio below one when the outcome risk is lower in the exposed group. The formula is not applied directly to that number: the reciprocal expresses the association in the reverse direction, above one. For an interval wholly below one, the upper endpoint is closest to one before reversal; after reversal, that endpoint becomes the lower limit of the oriented interval.

Why the Confidence Limit Nearest One Matters

For \(r>1\), write \(g(r)=r+\sqrt{r(r-1)}\). If \(r\) increases, then both \(r\) and \(r(r-1)\) increase, so \(g(r)\) increases. Therefore, among confidence limits above one, the limit closest to one has the smallest E-value and is the relevant limit for describing how much confounding could move the interval to include the null. For an interval below one, the reciprocal transformation gives the same logic: use the endpoint closest to one after orienting the interval.

If the reported interval includes 1, it already includes the null value for a risk ratio. By convention, the confidence-limit E-value is then 1: no additional unmeasured confounding is needed to move that interval to include the null. The point-estimate E-value can still be greater than 1, because the point estimate itself may be away from the null even when the interval is wide.

Diagnostics, Comparisons, and Limitations

Several checks catch common calculation errors. Confirm that the reported measure really is a risk ratio; an odds ratio or hazard ratio should not be inserted into this formula without the appropriate treatment for that measure. Check that the estimate’s direction is clear, that a protective ratio has been reciprocated, and that the interval endpoints have been reordered after taking reciprocals. Finally, check whether the confidence interval crosses or touches 1 before reporting a confidence-limit E-value.

A common mistake is to use the farther confidence limit from the null. For an estimate of 1.80 with limits 1.20 and 2.70, the 2.70 limit is farther from 1, but 1.20 is the relevant limit because it is the first endpoint that could be moved to the null as confounding increases. Another mistake is to round an intermediate square root and then round the sum. For example, \(1.30+\sqrt{0.39}\approx1.9245\), which rounds to 1.924 at three decimal places; rounding the square root too early can change the final digit.

E-values and Rosenbaum sensitivity parameters answer different sensitivity questions. An E-value uses associations on a risk-ratio scale to summarize how strong unmeasured-confounder associations would need to be under its bounding framework. A Rosenbaum sensitivity parameter concerns departures from random treatment assignment within matched sets. Neither number can be substituted for the other, and their magnitudes are not directly comparable. Standard confidence intervals, meanwhile, summarize sampling uncertainty under a statistical model; an E-value does not replace that interval or incorporate all sources of uncertainty.

An E-value is a compact benchmark, not a full sensitivity analysis. It does not identify the confounder, show whether the required associations are plausible, or account automatically for multiple sources of bias, selection, measurement error, or model misspecification. Its interpretation also depends on the effect measure and causal contrast being meaningful. A large E-value alone is not proof that a result is robust in practice; compare the benchmark with substantive knowledge about plausible confounders and the study’s design.