Worked Matched-Cohort E-Value Analysis

Why Work Through a Matched-Cohort E-Value?

Matching and covariate adjustment can improve comparability between treated and untreated participants on measured characteristics. They cannot, by themselves, rule out residual confounding by characteristics that were not measured or controlled. An E-value analysis puts the adjusted risk ratio into a sensitivity-analysis scale: it asks how strongly an unmeasured confounder would need to be associated with treatment and outcome, conditional on the measured covariates, to explain the estimate or move a confidence limit to the null.

The calculations below use a hypothetical matched-cohort analysis with an adjusted risk ratio (RR) of 1.60 and a 95% confidence interval of 1.174 to 2.180. The E-value is a benchmark for the strength of associations on the risk-ratio scale. It is not an estimate of the confounder's actual strength, and it does not give the probability that the result is biased.

Point-Estimate and Confidence-Limit Calculations

For a result above the null, the point-estimate E-value describes the minimum strength that an unmeasured confounder would need to have in its associations with treatment and outcome, conditional on the measured covariates, to explain the point estimate fully under the standard risk-ratio bound. The confidence-limit E-value uses the interval endpoint nearest 1. It therefore addresses a smaller challenge: the strength needed to move that endpoint to the null, rather than to move the point estimate itself.

These are joint-strength benchmarks. They do not mean that each of the two associations must equal the E-value. A confounder with a stronger association with one variable may require a weaker association with the other; the bound specifies combinations of strengths that could be sufficient. The usual interpretation also presumes that the adjusted RR is the effect measure of interest and that the measured-covariate adjustment and study design are otherwise adequate for that interpretation.

Why the E-Value Formula Has This Form

The formula comes from a worst-case bound for the confounding bias factor on the risk-ratio scale. Let \(a\) and \(b\), each at least 1, represent the relevant conditional treatment-confounder and outcome-confounder association strengths. Under the bound, the maximum bias factor is:

A bias factor \(B\) describes the maximum multiplicative distortion allowed by the bound. To explain an observed RR \(R>1\) completely, the required factor is at least \(R\). The E-value asks for the smallest common association strength \(r\) that makes the bound reach \(R\), so set \(a=b=r\) and solve:

The relevant root is \(r=R+\sqrt{R(R-1)}\), which is the E-value. The smaller root is not the association strength needed for the bound and is below 1 when \(R>1\). For the confidence-limit E-value, replace \(R\) with the confidence limit nearest 1. This algebra explains the formula; it does not establish that an unmeasured confounder with those associations exists.

Calibrating Against a Measured Confounder

A measured confounder can provide a scale for judging whether an E-value is large or modest in a particular application. The calibration requires associations defined on appropriate conditional scales. For a binary unmeasured confounder \(U\), one treatment-confounder association used in the standard risk-ratio bias-factor calibration is the ratio of the prevalence of \(U\) among treated participants to its prevalence among untreated participants, conditional on measured covariates \(C\). It is not interchangeable with the risk ratio of treatment comparing \(U=1\) with \(U=0\).

Calibration is a benchmark, not a reconstruction of the actual bias. It is most interpretable when the measured and hypothetical confounders are assessed on comparable conditional scales, and when relevant strata have adequate support. Large associations with both treatment and outcome can produce a large bias bound; a large association with only one does not automatically imply a large bias factor.

Intervals That Include the Null and Protective Estimates

The nearest-to-null rule matters. If a confidence interval for an RR includes 1, the data already allow the null value at that confidence level. The confidence-limit E-value is then 1 by convention: no additional confounding is needed to move the interval to include the null. This does not make the point-estimate E-value disappear; it means the confidence interval does not exclude the null before considering unmeasured confounding.

For a protective association, invert the RR before applying the formula, and use the confidence limit closest to 1 after orienting the effect in the protective direction. For example, if the estimate is 0.72, its reciprocal is \(1/0.72=1.389\), and the point-estimate E-value is \(1.389+\sqrt{1.389(1.389-1)}\approx2.124\). If the interval is 0.55 to 0.94, the limit closest to 1 is 0.94, whose reciprocal is 1.064; its confidence-limit E-value is approximately 1.324. Always state clearly whether the reported effect is harmful or protective, and which limit was used.

Assumptions, Diagnostics, and Common Pitfalls

An E-value does not repair defects in the matched-cohort analysis. A causal interpretation still depends on the design and analysis: treatment and outcome must be well-defined, the adjustment must address measured confounding appropriately, and the estimation and uncertainty calculations must respect the matching and cohort structure. Positivity, consistency, and an appropriate treatment of selection and missing data also matter. The E-value addresses only one sensitivity question: how strong residual confounding would have to be under a specified risk-ratio bound.

Before interpreting a calculation, check that the effect measure is an RR rather than an odds ratio or risk difference, that the direction is handled correctly, and that the confidence limit nearest 1 was used. In a matched cohort, check that the reported adjusted estimate targets a clearly defined population and that the confidence interval reflects the matched design. A mismatched estimand or an incorrectly computed interval makes a numerically correct E-value uninformative.

Common mistake: It is tempting to compare a measured-confounder bias factor with the RR itself, or to use \(P(A=1\mid U=1,C)/P(A=1\mid U=0,C)\) as though it were the required prevalence-ratio input. The correction is to calculate the conditional prevalence ratio \(P(U=1\mid A=1,C)/P(U=1\mid A=0,C)\) for the stated calibration, then compare the resulting \(B(a,b)\) with the relevant E-value. Also distinguish the two thresholds: moving a confidence limit to 1 is not the same task as explaining the point estimate completely.

E-values are only one sensitivity-analysis approach. Rosenbaum bounds address hidden bias in matched designs through a different parameterization of treatment assignment; they should not be converted into E-values by treating their sensitivity parameter as if it were an E-value. A measured-confounder calibration offers context, but it does not replace either sensitivity analysis or substantive knowledge about plausible confounders.