Reporting E-Value Analyses

What an E-Value Report Needs to Convey

An E-value is a sensitivity measure for unmeasured confounding. It expresses how strongly an unmeasured confounder would need to be associated with the exposure and outcome, conditional on the measured covariates, to account for an observed association on a risk-ratio scale. A useful report does more than give a single number: it identifies the effect measure and estimate, reports the point-estimate and confidence-limit E-values, explains the confounder-strength interpretation, and states important assumptions and scale conversions.

Let \(A\) denote an exposure or treatment, \(Y\) an outcome, \(L\) the measured covariates used in the analysis, and \(U\) an unmeasured confounder. For a risk ratio (RR) above 1, the E-value is the smallest common strength of association, on the risk-ratio scale, that \(U\) would need to have with both \(A\) and \(Y\), conditional on \(L\), to explain away the estimate. More generally, it is the minimum value of the larger of those two associations that could suffice: if one association is weaker, the other would need to be stronger. For an RR below 1, reverse the direction by taking its reciprocal before calculating.

The point-estimate E-value addresses how strong confounding would need to be to move the point estimate to the null. The confidence-limit E-value addresses how strong confounding would need to be to move the confidence interval to include the null. The latter is often the more conservative summary of statistical robustness: it uses the endpoint nearest the null, not the endpoint farthest away.

Why the Formula Has This Form

The formula can be understood through a standard upper bound on the bias from an unmeasured confounder. Suppose the confounder’s associations with exposure and outcome, conditional on measured covariates, are each bounded by \(\Gamma\), where \(\Gamma\geq 1\). The maximum bias factor on the risk-ratio scale is bounded by

To find the weakest equal-strength associations that could produce a bias factor as large as an observed risk ratio \(R^*\geq1\), set \(B(\Gamma)=R^*\). Multiplying through by the positive denominator gives a quadratic equation:

Solving the quadratic gives \(\Gamma=R^*\pm\sqrt{R^*(R^*-1)}\). The smaller root is below 1 when \(R^*>1\), so the relevant root satisfying \(\Gamma\geq1\) is \(R^*+\sqrt{R^*(R^*-1)}\), the E-value. This derivation explains the sensitivity benchmark; it does not establish that such a confounder exists or that confounding is the only possible source of bias.

Protective Estimates and the Confidence-Limit Endpoint

When an estimate is below 1, the formula is applied to its reciprocal. This puts the effect in the direction of an increase in risk and makes the calculation symmetric around the null on the ratio scale. For a confidence interval entirely below 1, use the upper confidence limit because it is closer to 1; take that limit’s reciprocal before applying the formula. Using the lower limit instead would describe how far the interval extends from the null, not how much confounding could bring it to the null.

If the confidence interval includes 1, report a confidence-limit E-value of 1. This convention indicates that no additional confounding is needed to make the interval include the null: it already does. It does not mean the point estimate has no sensitivity to confounding; the point-estimate E-value can still be greater than 1.

Reporting Estimates on Other Scales

The standard formula is defined for risk ratios. If the analysis reports a risk difference, odds ratio, or another effect measure, do not silently apply the risk-ratio formula to that number. Either report an appropriate E-value for that measure using a justified method, or clearly state and justify a conversion to a risk ratio. The conversion may depend on baseline risk, and uncertainty in that baseline risk can affect the result.

For a risk difference, a conversion is possible when the unexposed risk \(p_0\) and the treated or exposed risk are available on the same scale. If the risk difference is \(RD=p_1-p_0\), then the corresponding risk ratio is \(p_1/p_0=(p_0+RD)/p_0\), provided \(p_0>0\). Convert the estimate and relevant confidence limits consistently, and explain which baseline risk was used. A risk difference alone is not enough to determine a risk ratio.

For an odds ratio, one possible conversion uses the unexposed outcome risk \(p_0\). Given an odds ratio \(OR\), the corresponding risk ratio is

For example, if \(OR=2.00\) and \(p_0=0.20\), then \(RR=2.00/(0.80+0.40)=1.667\), giving an E-value of approximately \(1.667+\sqrt{1.667(0.667)}=2.721\). This differs from applying the E-value formula directly to 2.00. When an odds ratio is used as an approximation to a risk ratio because the outcome is rare, say that this is an approximation and report its basis; do not imply the measures are identical.

Assumptions, Diagnostics, and Limitations

E-values are not a substitute for sound design or analysis. Their causal interpretation presumes that the reported effect is a suitable estimate of the causal contrast of interest under the study’s design and analysis. The target population, exposure and outcome definitions, measured covariates, and effect measure should be clear. For a causal interpretation, standard requirements such as consistency, positivity, and appropriate control of measured confounding remain relevant. An E-value does not repair errors in those requirements or correct bias from selection, measurement error, model misspecification, or other mechanisms.

Before reporting, check that the effect measure is on the scale used by the formula, the null value is identified correctly, and the confidence-limit endpoint is the one nearest the null. Confirm that a protective estimate and its interval were reciprocated in the correct direction. If a conversion was used, verify that baseline risks and confidence limits are compatible with the study population and that the conversion is stated. A sensitivity benchmark calibrated using a poorly chosen baseline risk can be misleading even when its arithmetic is correct.

The interpretation also depends on what is meant by confounder strength. The E-value is a threshold based on conditional associations of \(U\) with exposure and outcome after accounting for \(L\), expressed on the risk-ratio scale. It does not say that a single confounder actually has those associations, that all unmeasured confounding has been ruled out, or that a confounder with one association below the E-value cannot produce bias. One association can be weaker if the other is correspondingly stronger.

An E-value is not a probability that the estimate is biased, a probability that the causal effect is true, or a measure of the prevalence of unmeasured confounding. It also does not summarize every possible source of uncertainty. Compare the benchmark with associations of plausible measured confounders when useful, while recognizing that measured confounders are not guaranteed to calibrate the strength of omitted ones.

A Clear Reporting Sequence

A compact report can be built in a consistent order. Include the effect measure and its estimate with confidence interval; give the point-estimate and confidence-limit E-values; identify any scale conversion and its inputs; then interpret the threshold conditionally and state the main limitations. For example: “The adjusted risk ratio was 1.80 (95% CI 1.20 to 2.40). The point-estimate E-value was 3.00 and the confidence-limit E-value was 1.69. These values describe the minimum confounder-strength benchmarks on the risk-ratio scale, conditional on measured covariates, needed to explain away the estimate or move the interval to include 1. The interpretation assumes the reported risk-ratio analysis is appropriate; the E-values do not address selection or measurement bias.” If the estimate was converted from a risk difference or odds ratio, add the conversion details rather than leaving the scale ambiguous.

E-values and Rosenbaum sensitivity analyses answer related but distinct questions. An E-value summarizes confounder strength on a risk-ratio scale; Rosenbaum bounds describe sensitivity to hidden bias in matched designs through a different parameterization. Report the method actually used, with its own assumptions and sensitivity parameter, rather than treating the metrics as interchangeable.