What E-Values Do Not Measure

What an E-Value Summarizes

An E-value compresses a sensitivity question into one number: how strong would unmeasured confounding have to be, on a risk-ratio scale, to explain away an observed association? It is useful as a benchmark, but it is not a probability that confounding exists and it does not tell us how common an unmeasured confounder is. Those distinctions matter because the number alone cannot describe the actual distribution of a confounder or reconstruct the bias in a particular study.

Let \(A\) denote an exposure, \(Y\) an outcome, \(U\) an unmeasured confounder, and \(C\) measured covariates. For a risk ratio \(R\) greater than 1, the point-estimate E-value is \(R+\sqrt{R(R-1)}\). For an estimate below 1, the reciprocal is used as the risk ratio before applying the formula. Here we focus on risk-ratio estimates; the interpretation depends on the measure and its scale.

Definition: For an observed risk ratio \(R\geq 1\), the E-value is \(E=R+\sqrt{R(R-1)}\). It is the minimum common strength, on the risk-ratio scale, that the associations of an unmeasured confounder with exposure and outcome would each need to reach to explain away the observed association, under the standard bounding framework and conditional on measured covariates. It is a strength benchmark, not a prevalence, probability, or estimate of a particular confounder’s parameters.

The phrase “minimum common strength” is important. It refers to an equal-strength benchmark for the confounder–exposure and confounder–outcome associations. Unequal strengths can also produce enough bias, and the E-value does not tell us which pair of strengths is plausible. Nor does it identify a particular confounder: \(U\) may represent one factor or a collection of unmeasured factors summarized for sensitivity analysis.

Why the E-Value Is Not a Prevalence or Probability

The E-value formula takes the observed risk ratio as its input. No prevalence of \(U\), probability of an outcome given \(U\), or probability that confounding is present appears in that calculation. Consequently, two analyses with the same risk ratio have the same point-estimate E-value, even if the underlying populations differ in how common a candidate confounder is. The number cannot be read as “the confounder is present in this fraction of people” or “there is this chance that the result is confounded.”

Prevalence can matter greatly in a fuller bias analysis. The amount of bias generated by a confounder depends not only on how it relates to exposure and outcome, but also on its distribution and on other features of the data-generating process. The E-value instead uses a bound based on the strengths of associations. It does not solve for a prevalence, and an E-value alone cannot supply the inputs needed for a prevalence-based calculation.

Worked Example: Computing a Strength Benchmark, Not a Probability

Suppose a hypothetical study reports an adjusted risk ratio of \(2.00\) for an adverse outcome comparing exposed with unexposed participants. Calculate the point-estimate E-value and interpret it.

Substitute \(R=2.00\) into the definition:

E=R+\sqrt{R(R-1)} =2.00+\sqrt{2.00(2.00-1)} =2.00+\sqrt{2.00} \approx 3.414.

Under the standard risk-ratio bounding interpretation, an unmeasured confounder associated with exposure and outcome by risk ratios of about \(3.414\) each, conditional on measured covariates, would be at the equal-strength benchmark for explaining away the observed risk ratio of \(2.00\). This is not a claim that \(3.414\) percent of participants have the confounder, nor that there is a 3.414-fold probability of confounding. The E-value says nothing by itself about how many participants have \(U\).

The Bound Behind the Benchmark

A common risk-ratio bias bound makes the distinction between association strength and prevalence more explicit. Write \(R_{AU}\) for a measure of the association between exposure and the unmeasured confounder, conditional on \(C\), and \(R_{UY}\) for the association between the confounder and outcome, conditional on exposure and \(C\). In the standard formulation these are upper-bound measures of association strength, taken to be at least 1. The maximum bias factor is bounded by:

B=\frac{R_{AU}R_{UY}}{R_{AU}+R_{UY}-1}.

This result is used under the assumptions that the effect and the confounder associations are appropriately represented on the risk-ratio scale, the measured covariates are conditioned on, and the associations used in the bound apply to the target population and contrast. It is a bound, not a promise that the observed bias equals \(B\). A particular confounder’s actual bias can be smaller. To explain away an observed risk ratio \(R\), the bound must be large enough to reach \(R\).

If both association strengths are set to the same value \(k\), then \(B=k^2/(2k-1)\). Setting this bound equal to \(R\) gives the equal-strength threshold that defines the E-value:

\frac{k^2}{2k-1}=R \quad\Longrightarrow\quad k^2-2Rk+R=0 \quad\Longrightarrow\quad k=R\pm\sqrt{R(R-1)}.

For \(R\geq1\), the relevant threshold is the larger root, \(k=R+\sqrt{R(R-1)}\). The smaller root does not provide the required association strength. This derivation shows why the E-value is an association-strength threshold: the inputs are the two association measures, not the prevalence of \(U\).

Worked Example: Unequal Association Strengths

For a hypothetical risk ratio of \(2.00\), suppose a candidate unmeasured confounder has exposure-association strength \(R_{AU}=4.00\). Find the outcome-association strength \(R_{UY}\) that makes the bias bound equal to \(2.00\).

Set \(B=2.00\) in the bound and let \(R_{UY}=x\):

\frac{4.00x}{4.00+x-1}=2.00 \quad\Longrightarrow\quad 4.00x=2.00(x+3.00) \quad\Longrightarrow\quad 2.00x=6.00 \quad\Longrightarrow\quad x=3.00.

Thus, for this pair of strengths, \(R_{AU}=4.00\) and \(R_{UY}=3.00\) make the bound reach the observed risk ratio. This unequal pair differs from the equal-strength E-value benchmark of about \(3.414\) for each association. Neither calculation tells us how prevalent \(U\) is.

For another hypothetical pair, \(R_{AU}=1.702\) and \(R_{UY}=4.00\), the bias bound is:

B=\frac{1.702(4.00)}{1.702+4.00-1} =\frac{6.808}{4.702} \approx 1.448.

This bound is below \(2.00\), so these association strengths are not sufficient, by this bound, to explain away a risk ratio of \(2.00\). The value \(1.448\) is a maximum bias factor on the risk-ratio scale; it is not a probability or a percentage of participants with \(U\).

What a More Complete Bias Model Requires

A quantitative bias analysis can ask a more specific question by assigning values or distributions to quantities such as the prevalence of \(U\) in exposure groups, the association between \(U\) and \(Y\), and possibly the causal effect of \(A\). Those inputs allow a calculation tailored to a proposed data-generating process. An E-value does not choose those inputs, and it cannot replace them when the scientific question depends on prevalence or on a particular confounder’s structure.

Worked Example: Prevalence Enters a Specific Model

Consider a hypothetical binary confounder \(U\) and no causal effect of exposure on outcome. Suppose the outcome risk is \(0.30\) for \(U=1\) and \(0.10\) for \(U=0\), regardless of exposure. The prevalence of \(U=1\) is \(0.60\) among exposed people and \(0.20\) among unexposed people. Calculate the resulting crude risk ratio.

The outcome risk among exposed people is the prevalence-weighted average:

P(Y=1\mid A=1) =0.60(0.30)+0.40(0.10) =0.18+0.04 =0.22.

Among unexposed people:

P(Y=1\mid A=0) =0.20(0.30)+0.80(0.10) =0.06+0.08 =0.14.

Therefore the crude risk ratio is \(0.22/0.14\approx1.571\), despite the stipulated absence of a causal exposure effect. In this specified model, the prevalence of \(U\) is essential for obtaining the group risks. The point-estimate E-value based on \(1.571\) is approximately \(1.571+\sqrt{1.571(0.571)}\approx2.518\), but that number does not recover the prevalences \(0.60\) and \(0.20\). The prevalence values were inputs to the model, not outputs of the E-value calculation.

Assumptions, Diagnostics, and Limitations

An E-value is only as interpretable as the estimate and scale to which it is applied. For a risk-ratio interpretation, use a risk ratio for the relevant exposure contrast and outcome, and specify whether the estimate is crude or adjusted. For an adjusted estimate, the sensitivity associations are understood conditional on the measured covariates used in that analysis. If the reported measure is not a risk ratio, it should not be inserted uncritically into this formula; the measure and any conversion must be appropriate to the setting.

The calculation also does not verify that the study estimate is unbiased apart from unmeasured confounding. Selection bias, measurement error, model misspecification, and other threats are not ruled out by a large E-value. A large value means that stronger associations would be required under the bound to explain away the estimate through the specified unmeasured-confounding mechanism; it does not certify the causal interpretation. A small value indicates that comparatively modest confounder associations could suffice under that framework, not that such confounding actually occurred.

Useful checks include confirming the direction of the contrast, using the correct risk-ratio scale, identifying the measured covariates being conditioned on, and comparing the implied association strengths with substantive knowledge or measured covariates. If the scientific question is whether a particular confounder could generate a particular amount of bias, specify a model with its prevalence and outcome associations rather than treating the E-value as a complete answer.

The E-value also differs from other sensitivity approaches in what it summarizes. It gives a compact RR-scale benchmark; a quantitative bias analysis can encode specified prevalence and association parameters, while Rosenbaum bounds describe sensitivity to hidden bias in matched designs using a different framework. These approaches answer related but nonidentical questions. A numerical value from one framework should not be interpreted as if it were a parameter from another.

Common Mistake

Wrong reasoning: “The E-value is 3.414, so the unmeasured confounder must occur in 3.414 times as many exposed people,” or “there is a 3.414-fold probability that confounding explains the association.”

Correction: The E-value is a threshold for association strengths on the risk-ratio scale under a bounding argument. It supplies neither a confounder prevalence nor a probability that bias exists. To estimate the consequences of a specified prevalence, build a bias model that includes that prevalence and the relevant outcome associations.

Key takeaway: An E-value is a sensitivity benchmark for the strength of unmeasured confounding associations needed to explain away a risk-ratio estimate. It is not a prevalence estimate, a probability, a complete bias model, or evidence that the required confounder exists.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A hypothetical risk ratio is \(1.50\). What is its point-estimate E-value, and what does that value summarize?
Can an E-value of \(2.50\) be interpreted as a 2.50 probability that the result is confounded? Explain.
If \(R_{AU}=4.00\), what value of \(R_{UY}\) makes the bias bound equal to \(1.50\)?
Name two kinds of information a prevalence-based quantitative bias analysis may require that are not supplied by an E-value.

Solutions. (1) \(E=1.50+\sqrt{1.50(0.50)}=1.50+\sqrt{0.75}\approx2.366\). It summarizes the equal-strength RR-scale confounder-association threshold under the standard bound. (2) No. It is an association-strength benchmark, not a probability; it does not quantify whether confounding exists. (3) Solve \(4x/(4+x-1)=1.50\): \(4x=1.50(x+3)\), so \(2.50x=4.50\) and \(x=1.80\). (4) Examples include the prevalence of the confounder in exposure groups and its association with the outcome. The E-value alone specifies neither.

What You'll Learn