Why Summarize Sensitivity With an E-Value?
An adjusted risk ratio can remain vulnerable to confounding by variables that were not measured or adequately controlled. A sensitivity analysis asks how strong such confounding would need to be to account for the observed association. The E-value gives a compact benchmark on the risk-ratio scale: for an estimate above 1, it is the smallest common strength of association that an unmeasured confounder would need to have with treatment and outcome, conditional on measured covariates, to move the estimate to the null under a standard confounding bound.
This is a benchmark, not an estimate of the confounder’s actual strength. A large E-value does not prove that the result is free from bias, and a small one does not prove that bias exists. The introductory material on Rosenbaum bounds concerns a different sensitivity framework; the distinction matters because the parameters and assumptions are not interchangeable.
Here, “treatment association” means the strength of the relationship between the unmeasured confounder and treatment, and “outcome association” means its strength of association with the outcome, conditional on treatment and measured covariates. The strengths are measured on risk-ratio scales; they are not regression coefficients, odds ratios, or probabilities of confounding. In an application, these are understood as bounds on the relevant associations across covariate strata and confounder levels, not necessarily as two single observed ratios.
Where the Formula Comes From
The standard risk-ratio confounding bound summarizes the largest bias factor allowed by two association-strength bounds. Let \(R_T\geq1\) bound the unmeasured confounder’s association with treatment, and let \(R_Y\geq1\) bound its association with the outcome. Under the bound, the confounding bias factor cannot exceed
If the observed risk ratio is \(R\), explaining it fully down to 1 requires a bias factor of at least \(R\). The E-value asks for the smallest common bound: set \(R_T=R_Y=e\) and solve \(B(e,e)=R\). This equal-strength criterion supplies one interpretable threshold; it does not say that the two true associations must actually be equal.
The relevant solution is \(e=R+\sqrt{R(R-1)}\). The other root is at most 1 for \(R\geq1\), so it cannot be the required confounding-strength bound \(e\geq1\) (except at the null, where both roots equal 1). This algebra explains the formula; the confounding bound itself is a sensitivity result, not a claim that a particular unmeasured confounder exists.
Worked Example: Calculating an E-Value for a Risk Ratio of 2.4
Suppose, hypothetically, an adjusted analysis of a binary health outcome reports an estimated risk ratio of \(R=2.4\) comparing treatment with no treatment. The risk ratio is above the null value of 1, so use the formula for \(R\geq1\):
The E-value is approximately 4.23. Under the standard bound, an unmeasured confounder would need treatment-association and outcome-association strengths of about 4.23 each, conditional on measured covariates, in the equal-strength worst-case benchmark to explain the point estimate fully to the null. This does not mean either association has been measured to be 4.23, nor does it give the probability that unmeasured confounding is present.
Unequal Associations and the Meaning of “Minimum”
The E-value’s equal-strength interpretation is useful, but actual confounder associations need not be equal. For an unequal pair of strengths, use the bias bound directly. If \(B(R_T,R_Y)\) is below the observed \(R\), that pair is not strong enough under this bound to explain the entire estimate away. If it is at least \(R\), it is strong enough in principle under the bound; this does not establish that the confounder has those associations or that the resulting bias has the required direction.
Worked Example: Checking an Unequal Pair of Strengths
Continue with the hypothetical estimate \(R=2.4\). Suppose a proposed sensitivity scenario gives an unmeasured confounder a treatment-association bound of \(R_T=8\) and an outcome-association bound of \(R_Y=3\), both conditional on measured covariates.
This pair reaches the bias factor needed to explain the estimate down to 1 under the bound. Notice that one strength, 3, is below the E-value 4.23 while the other, 8, is above it. The E-value is the minimum common ceiling, not a rule that every unequal pair must have both components at least 4.23. Conversely, a proposed pair of 5 and 3 gives \(B(5,3)=15/(5+3-1)=2.143\), which is below 2.4 and is insufficient to explain the full estimate under this bound.
The direction and meaning of an association also matter. The bounding argument concerns confounding that could bias the treatment-outcome comparison away from the null. A variable can be associated with treatment and outcome yet produce little bias, or bias in the opposite direction, depending on how those associations align. The E-value does not determine those directions or reconstruct a corrected risk ratio.
Applying the Benchmark to Adjusted Results
For an adjusted estimate, interpret the E-value relative to the covariates included in the analysis. It asks about additional unmeasured confounding after conditioning on those measured covariates, subject to the assumptions of the adjustment and sensitivity bound. It does not repair inadequate measurement of included covariates, selection bias, measurement error, model misspecification, or violations of consistency and positivity. Nor does it establish that the adjusted estimate has a causal interpretation in the first place.
Worked Example: A More Modest Adjusted Association
Imagine a hypothetical matched observational study reports an adjusted risk ratio of \(R=1.5\) for a treatment and a binary outcome. The adjustment includes measured age group and baseline disease severity. Calculate the point-estimate E-value:
The benchmark is about 2.37. In context, a single unmeasured confounder would need associations with treatment and outcome of roughly this strength, conditional on the measured age group and disease severity, under the equal-strength worst-case interpretation to explain the point estimate to the null. This conditional wording is important: the calculation does not assess confounding by variables already adjusted for, and the benchmark can differ if the covariate set or estimand changes.
The point estimate alone also does not describe sampling uncertainty. This calculation says nothing by itself about whether the estimate is precise or statistically distinguishable from 1. It is a sensitivity summary for the point estimate only.
Diagnostics, Comparisons, and Limits
Before reporting an E-value, check that the input is a risk ratio with the intended treatment and outcome orientation, that it is at least 1 for the formula used here, and that the estimate and adjustment set are stated. Check whether the reported statistic is truly a risk ratio rather than an odds ratio or hazard ratio; those measures cannot simply be substituted into this formula as though they were the same quantity. Also inspect whether the outcome is sufficiently well-defined and whether the conditional associations being benchmarked are scientifically interpretable.
Rosenbaum sensitivity analysis and E-values answer related but different questions. Rosenbaum’s \(\Gamma\) parameter bounds how much hidden bias can alter treatment-assignment odds within matched sets. A risk-ratio E-value summarizes a confounding-strength benchmark for a risk-ratio estimate. Neither parameter converts mechanically into the other: they use different scales and structures. Other sensitivity analyses can specify a particular confounder distribution or model and estimate bias under those explicit choices. Such analyses can be more tailored, but require stronger modeling choices; the E-value is compact precisely because it summarizes a bound rather than specifying a full data-generating model.
A common mistake is to say, “The E-value is 4.23, so an unmeasured confounder must have at least a 4.23-fold association with both treatment and outcome.” That overstates the result. The equal-strength threshold is the smallest common bound; an unequal pair, such as 8 and 3 in the example, can attain the same bias factor. Another mistake is to say, “The E-value is large, so unmeasured confounding is ruled out.” The E-value does not reveal whether a confounder of that strength is plausible, whether several confounders jointly satisfy a relevant bound, or whether other sources of bias are present. Those judgments require substantive knowledge and additional design or sensitivity analyses.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Calculate the E-value for a hypothetical risk-ratio estimate of 3.0.
- For an estimate of 2.0, calculate the bias bound for association strengths \(R_T=4\) and \(R_Y=2\). Is the pair sufficient to explain the estimate fully to the null?
- In words, what does an E-value of 2.5 mean under the equal-strength interpretation?
- Why is it incorrect to interpret a large E-value as proof that the estimate is causal?
Solutions. (1) \(E(3)=3+\sqrt{3(2)}=3+\sqrt{6}=5.449\), approximately 5.45. (2) \(B(4,2)=4(2)/(4+2-1)=8/5=1.6\). Since 1.6 is below the required factor 2.0, this pair is insufficient under the bound. (3) It is the smallest common upper bound on the unmeasured confounder’s treatment and outcome association strengths, conditional on measured covariates, that could explain the point estimate to the null under the standard bound. (4) It measures sensitivity to one form of unmeasured confounding for the point estimate; it does not rule out confounding, other biases, or failures of assumptions needed for causal interpretation.