Deriving the Risk-Ratio E-Value

From a Bias Bound to an E-Value

A risk-ratio E-value is obtained by asking a specific sensitivity question: how strong would residual confounding have to be, at minimum, to move an observed risk ratio to the null? The derivation begins not with a formula for the E-value, but with a bound on how much unmeasured confounding can change a risk ratio. Inverting that bound produces the familiar expression and shows why the two relevant confounder associations enter symmetrically.

Consider a binary treatment $A$, a binary outcome $Y$, measured covariates $C$ that have already been accounted for, and a possible unmeasured confounder $U$. Let $RR_{\text{obs}}$ be the observed or appropriately adjusted risk ratio, and let $RR_{\text{causal}}$ be the corresponding causal risk ratio for the target population. This tutorial derives the E-value for an observed harmful association, $RR_{\text{obs}}=r\geq 1$, when the null value is 1.

Definition: Let $R_{AU}$ be an upper bound on the largest risk ratio for treatment across levels of $U$, conditional on measured covariates. Let $R_{UY}$ be an upper bound on the largest risk ratio for outcome across levels of $U$, within treatment and measured-covariate strata. Both bounds are at least 1 and describe associations in the direction that could explain the observed treatment–outcome association. The maximum confounding bias factor is $$ BF(R_{AU},R_{UY})=\frac{R_{AU}R_{UY}}{R_{AU}+R_{UY}-1}. $$

These are risk-ratio-scale associations, not correlations and not statements that $U$ causes the outcome. In this sensitivity analysis, their role is to constrain how much the distribution of $U$ and the outcome risks may differ across relevant groups. If both association bounds equal 1, the bias factor is 1: these bounds allow no confounding bias.

What the Bias Factor Bounds

Under the confounding-bias bound, the observed risk ratio cannot exceed the causal risk ratio by more than the bias factor when confounding acts in the direction of the observed association. In the usual harmful-association orientation, this is expressed as

\frac{RR_{\text{obs}}}{RR_{\text{causal}}} \leq BF(R_{AU},R_{UY}).

Thus, if a proposed pair of association bounds makes $BF$ smaller than $RR_{\text{obs}}$, those bounds cannot, by themselves, account for the entire observed ratio under this result. Conversely, a bias factor at least as large as $RR_{\text{obs}}$ is sufficient in the bound to allow the causal ratio to be 1. This is a sensitivity bound, not a claim that the actual bias reaches its maximum.

The result is used for a causal risk-ratio estimand under a coherent treatment and outcome definition, consistency, and positivity for the treatment comparisons being made. The measured covariates must be handled in a way that defines the reported adjusted or standardized risk ratio. The bound concerns residual confounding by $U$; it does not repair selection bias, measurement error, or a misspecified estimand. Its association limits must be meaningful over the relevant covariate and treatment strata.

A central feature of the bound is that it depends on the two strengths through the same expression regardless of their order. A strong association of $U$ with treatment can compensate, in the bound, for a weaker association with outcome, and vice versa. The bound does not say the two associations are interchangeable as scientific descriptions; it says their maximum contribution to this risk-ratio bias bound is symmetric.

Deriving the E-Value

The E-value restricts attention to a common upper bound $E$ on each of the two associations: $R_{AU}\leq E$ and $R_{UY}\leq E$. Substituting the largest permitted pair into the bias factor gives

BF(E,E)=\frac{E^2}{2E-1}.

To explain away an observed ratio $r$, the bias factor must be at least $r$. The smallest common bound capable of reaching that threshold therefore solves $E^2/(2E-1)=r$. Multiplying by the positive denominator, for $E\geq1$, gives a quadratic equation:

E^2=r(2E-1) \quad\Longleftrightarrow\quad E^2-2rE+r=0.

The roots are $r\pm\sqrt{r^2-r}$. For $r>1$, the smaller root is below 1, so it cannot be a common association bound of the required size. The relevant root is the larger one. Since $r^2-r=r(r-1)$, the risk-ratio E-value is

Theorem: For an observed harmful risk ratio $r\geq1$, the point-estimate E-value is $$ E(r)=r+\sqrt{r(r-1)}. $$ It is the minimum common upper bound on the two risk-ratio associations that makes the maximum confounding bias factor as large as $r$. At $r=1$, the E-value is 1.

This is a minimum in the common-bound sense: if both associations are no greater than a value below $E(r)$, the bias-factor bound is below $r$, so it does not permit those associations alone to move the causal risk ratio all the way to 1. At the threshold, the bound uses $R_{AU}=R_{UY}=E(r)$. The equality describes the extremal boundary of the sensitivity calculation; it does not establish that a real confounder has equal associations or achieves the bound.

Worked Example: Deriving the E-Value for a Risk Ratio of 3.6

Suppose a hypothetical analysis reports an adjusted risk ratio of $r=3.6$. Calculate the point-estimate E-value and interpret it.

Use the derived expression, substituting the reported ratio:

E(3.6)=3.6+\sqrt{3.6(3.6-1)} =3.6+\sqrt{3.6(2.6)} =3.6+\sqrt{9.36} \approx 3.6+3.059 =6.659.

The result is unitless. Under the stated risk-ratio bias bound, an unmeasured confounder would need associations of at least about 6.659-fold with both treatment and outcome, conditional on the measured covariates and in the relevant directions, to move the point estimate to the null if the two bounds were equal. Unequal associations can also reach the threshold; the E-value is not a requirement that both actual associations have this exact value.

Why the Formula Is Symmetric

The symmetry is visible first in the bias factor itself: exchanging $R_{AU}$ and $R_{UY}$ leaves its value unchanged. For example, association bounds of 3 and 4 give the same factor as bounds of 4 and 3. When a single common upper bound is sought, both are set to $E$; the resulting equation depends only on $E^2$ and $2E-1$. Neither association is privileged in the algebra.

A short check confirms that the common-bound factor increases as $E$ increases for $E\geq1$. Differentiating gives

\frac{d}{dE}\left(\frac{E^2}{2E-1}\right) =\frac{2E(2E-1)-2E^2}{(2E-1)^2} =\frac{2E(E-1)}{(2E-1)^2}\geq0.

Thus, the larger root of the quadratic is the threshold: smaller common bounds produce a smaller maximum bias factor, while larger bounds can produce a factor at least as large as $r$. This also clarifies why the E-value uses the larger quadratic root rather than simply reporting both roots.

Worked Example: Unequal Associations and the Same Bias Bound

Suppose a hypothetical sensitivity assessment allows $R_{AU}=3$ and $R_{UY}=4$. Calculate the maximum bias factor and compare it with an observed risk ratio of 2.

BF(3,4)=\frac{3(4)}{3+4-1} =\frac{12}{6} =2.

The bias factor reaches 2, so this pair of association bounds is sufficient, at the bound, to allow an observed ratio of 2 to be reduced to a causal ratio of 1. Reversing the two inputs gives $BF(4,3)=12/6=2$, illustrating the symmetry.

For comparison, the common-bound E-value for $r=2$ is

E(2)=2+\sqrt{2(2-1)} =2+\sqrt{2} \approx3.414.

The unequal pair $3,4$ is not the E-value itself. Rather, it is one pair whose bias factor reaches the threshold. The E-value reports the smallest equal upper bound that does so; it is not a statement that each real association must equal 3.414.

Using Unequal Bounds and Interpreting Robustness

The E-value compresses a two-dimensional sensitivity question into one number. If credible information about the two associations is available separately, it is often more informative to calculate their bias factor directly. When both association bounds are no greater than proposed values $a$ and $b$, respectively, the bound is $ab/(a+b-1)$. Dividing the observed ratio by that factor gives a lower bound on the causal ratio in the direction of the observed association.

Worked Example: Applying Plausible Unequal Bounds

In a hypothetical study, let $RR_{\text{obs}}=2.5$. Suppose subject-matter information motivates upper bounds $R_{AU}=2$ and $R_{UY}=5$, conditional on the measured covariates. Assess what the bias bound implies.

First calculate the maximum bias factor:

BF(2,5)=\frac{2(5)}{2+5-1} =\frac{10}{6} \approx1.667.

Then use the bound $RR_{\text{obs}}/RR_{\text{causal}}\leq BF$. Rearranging, with positive quantities, gives

RR_{\text{causal}} \geq\frac{RR_{\text{obs}}}{BF} =\frac{2.5}{1.667} \approx1.500.

Under the assumed association limits and the applicability conditions of the bound, the causal risk ratio is bounded below by about 1.500. This is not a confidence interval and does not account for sampling uncertainty. It is a sensitivity statement conditional on the proposed limits. The bounds are unequal, which is precisely why a direct bias-factor calculation can be more useful than a single E-value.

Diagnostics, Alternatives, and Limits

Before interpreting an E-value, check whether the calculation matches the estimand and scale. Confirm that the reported measure is a risk ratio, not an odds ratio or hazard ratio; those measures require separate treatment. Confirm that the association bounds refer to the residual confounder after measured-covariate adjustment and are conditional on the relevant variables. Check positivity and whether the target population and standardization are clear. An E-value computed from a number that does not represent the intended risk-ratio estimand can be numerically correct but substantively misleading.

The most important practical diagnostic is calibration. Compare the E-value, or proposed $R_{AU}$ and $R_{UY}$, with associations of measured covariates that are known to affect both treatment and outcome. This does not prove that unmeasured confounding is weaker or stronger; it gives a context for judging whether the required magnitudes are plausible. Also assess whether the confounder could act in the direction needed to explain the result. The bound describes a worst-case direction, not every possible causal structure.

The principal alternative is to report a range of bias-adjusted estimates over a grid of plausible $R_{AU}$ and $R_{UY}$ values. That retains the asymmetry of subject-matter beliefs and shows how conclusions change across combinations. Rosenbaum sensitivity bounds address a different framework, typically hidden bias in treatment assignment within matched designs and tests of hypotheses. They should not be treated as the same bound or converted into this risk-ratio formula without justification. Design-based sensitivity analyses, negative-control analyses, and quantitative bias analyses using more detailed information about $U$ may answer complementary questions.

A common mistake is to read a large E-value as proof that unmeasured confounding is absent or that the causal conclusion is secure. The calculation does not estimate the actual strength of a hidden confounder, prove that such a confounder exists, or account automatically for random error and other biases. The correction is to state the result conditionally: under the risk-ratio bias bound, association strengths at least this large could be required to explain away the point estimate, subject to the assumptions and direction specified. The formula quantifies one form of sensitivity; it does not replace study design, substantive knowledge, or uncertainty analysis.

Key takeaway: The E-value follows by setting both confounder-association bounds equal to $E$ in the maximum bias factor and solving $E^2/(2E-1)=r$. Its symmetry comes directly from the symmetric bias bound. Interpret it as a threshold for a specified risk-ratio sensitivity analysis, not as evidence that a result is free from bias.

Check Your Understanding

Answer each question, then compare with the solutions below.

For an observed risk ratio of 4, derive and calculate the point-estimate E-value.
Calculate the bias factor for association bounds of 2.5 and 3.5. Does it change if the bounds are exchanged?
If $R_{AU}\leq2$ and $R_{UY}\leq3$, what is the maximum bias factor under the formula?
Why is the smaller root of the quadratic not the E-value for an observed risk ratio greater than 1?
State one conclusion an E-value does not justify.

Solutions. (1) $E(4)=4+\sqrt{4(3)}=4+\sqrt{12}\approx7.464$. (2) $BF(2.5,3.5)=2.5(3.5)/(2.5+3.5-1)=8.75/5=1.750$; exchanging the inputs leaves the result unchanged. (3) $BF(2,3)=6/(2+3-1)=1.500$. (4) The smaller root $r-\sqrt{r(r-1)}$ is below 1 when $r>1$, whereas association bounds are defined to be at least 1; the larger root is the threshold on the relevant domain. (5) It does not prove that unmeasured confounding is absent or that the causal effect is established; it quantifies the confounding strength that the specified bound would require to explain the point estimate.