E-Values for Protective Associations

Why Protective Risk Ratios Need Careful Orientation

A risk ratio below one indicates that the risk in the group named in the numerator is lower than the risk in the denominator group. For example, if the numerator group is treated and the denominator group is untreated, $RR=0.60$ means the observed risk in the treated group is 60% of the observed risk in the untreated group. This is a protective association in the chosen orientation.

The standard risk-ratio E-value formula is written for a ratio of at least one. For a protective estimate, the calculation therefore uses the reciprocal, which is greater than one. This is a computational step: it puts the magnitude of the contrast into the formula’s required orientation. It does not change which group had lower risk, nor does it turn a protective association into a harmful one.

Definition: Let $RR$ be a risk ratio with the risk in the group of interest in the numerator and the comparison-group risk in the denominator. When $0<RR<1$, the point-estimate E-value is calculated by applying the usual risk-ratio formula to $1/RR$: $$ E=\frac{1}{RR}+\sqrt{\frac{1}{RR}\left(\frac{1}{RR}-1\right)}. $$ The reported association remains protective in its original orientation. The reciprocal is used only to calculate the E-value.

The formula applies directly to a finite, positive risk ratio below one. If $RR=1$, the estimate is null and the formula gives $E=1$. If $RR=0$, its reciprocal is undefined; the E-value can only be described as tending to infinity in the limit as $RR$ approaches zero from above. If both risks are zero, the risk ratio itself is undefined, so this formula does not apply.

Deriving the Protective-Estimate Formula

For a risk ratio $R\geq 1$, the standard point-estimate formula is $R+\sqrt{R(R-1)}$. A protective estimate $RR$, with $0<RR<1$, can be put in that orientation by setting $R=1/RR$. Substitution gives the protective-estimate formula:

E=\frac{1}{RR}+\sqrt{\frac{1}{RR}\left(\frac{1}{RR}-1\right)}.

The same result can be written in a form that makes the dependence on the original protective ratio explicit. Since $RR$ is positive,

\sqrt{\frac{1}{RR}\left(\frac{1}{RR}-1\right)} =\sqrt{\frac{1-RR}{RR^2}} =\frac{\sqrt{1-RR}}{RR}, \qquad E=\frac{1+\sqrt{1-RR}}{RR}.

This is an algebraic check on the reciprocal calculation. Both forms give the same E-value, so they can help identify an error in inversion or arithmetic. As $RR$ approaches one from below, $E$ approaches one. As $RR$ approaches zero from above, the denominator shrinks while the numerator remains positive, and $E$ grows without bound.

Worked Example: A Risk Ratio of 0.60

Suppose a hypothetical study reports 12 outcome events among 200 treated participants and 20 among 200 untreated participants. Use treated risk divided by untreated risk as the orientation. The observed risks are $12/200=0.060$ and $20/200=0.100$, so

RR=\frac{0.060}{0.100}=0.60.

The point estimate is protective in this orientation: the treated group’s observed risk is 0.60 times the untreated group’s risk. For the E-value calculation, take the reciprocal without rounding it first:

\frac{1}{RR}=\frac{1}{0.60}=\frac{5}{3}, \qquad E=\frac{5}{3}+\sqrt{\frac{5}{3}\left(\frac{5}{3}-1\right)} =\frac{5}{3}+\sqrt{\frac{10}{9}} \approx 2.7208.

Thus the point-estimate E-value is approximately 2.721. The substantive estimate remains protective; the reciprocal is not a claim that treatment raised risk. Keep unrounded values during calculation: rounding the reciprocal to 1.67 and the difference to 0.67 first gives $1.67+\sqrt{1.67(0.67)}\approx2.728$. That is the result of the rounded intermediate inputs, not the more accurate calculation from $RR=0.60$.

Worked Calculations and Direction Checks

A reliable calculation starts by stating the numerator and denominator groups, then calculating the risk ratio, then taking its reciprocal only if it is below one. Reporting those steps makes it possible to see whether the direction of the observed association has been preserved.

Worked Example: A Stronger Protective Estimate

In a hypothetical cohort, 8 of 100 people in the exposed group experience an outcome, compared with 20 of 100 in the comparison group. Define $RR$ as exposed risk divided by comparison risk. Then

RR=\frac{8/100}{20/100}=\frac{0.08}{0.20}=0.40.

The exposed group has the lower observed risk. The reciprocal for the calculation is $1/0.40=2.5$, and the E-value is

E=2.5+\sqrt{2.5(2.5-1)} =2.5+\sqrt{3.75} \approx2.5+1.9365 =4.4365.

The E-value is approximately 4.437. This calculation summarizes the point estimate on the E-value scale; it does not change the observed risk ratio from 0.40 or say that exposure was harmful. The group labels and the original ratio should accompany the E-value whenever direction could otherwise be ambiguous.

Worked Example: Checking the Reciprocal With an Equivalent Form

Suppose a hypothetical study compares 30 events among 200 people in a treatment group with 40 among 200 in a comparison group. With treatment risk in the numerator,

RR=\frac{30/200}{40/200}=\frac{0.15}{0.20}=0.75.

The reciprocal is $1/0.75=4/3$. Applying the definition gives

E=\frac{4}{3}+\sqrt{\frac{4}{3}\left(\frac{4}{3}-1\right)} =\frac{4}{3}+\sqrt{\frac{4}{9}} =\frac{4}{3}+\frac{2}{3} =2.

The equivalent expression gives the same result:

E=\frac{1+\sqrt{1-0.75}}{0.75} =\frac{1+0.5}{0.75} =2.

Both calculations yield an E-value of 2. The original risk ratio still indicates a protective association: the treatment-group risk is three-quarters of the comparison-group risk. The reciprocal does not reverse that conclusion.

Conditions, Diagnostics, and Common Errors

The arithmetic requires a risk ratio that is defined and on the risk-ratio scale. For a sensitivity interpretation about a causal comparison, the contrast must also be meaningful for the target population, outcome, and follow-up period. The E-value calculation alone does not establish that the study identifies a causal effect. In particular, it does not verify the design and measurement conditions required for a causal interpretation.

Before reporting an E-value for a protective estimate, check the following:

Confirm the ratio’s direction. Record which group’s risk is the numerator and which is the denominator. A ratio below one is protective only relative to that stated orientation.
Check that the ratio is positive. The reciprocal formula is finite only for $0<RR<1$. At $RR=0$, report that the finite formula is not defined and, if relevant, describe the limiting value as infinite. Do not calculate a finite reciprocal.
Keep the scale consistent. The formula here is for a risk ratio, not a risk difference, odds ratio, or hazard ratio. Those measures cannot simply be inserted as if they were risk ratios.
Retain precision until the end. Round the reported E-value, not intermediate reciprocals and square roots. As the 0.60 example shows, early rounding can produce a different displayed result.
Check the null boundary. At $RR=1$, the formula gives $E=1$. For values just below one, the E-value should be just above one. A much larger result near the null suggests a direction or arithmetic error.

A frequent mistake is to say, “The reciprocal is greater than one, so the association is harmful.” That reasoning confuses the calculation’s orientation with the study’s original comparison. The reciprocal is used to express the magnitude in the orientation required by the formula; the reported $RR=0.60$, for example, still means lower observed risk in the numerator group.

Another mistake is to calculate $1/RR$ correctly but then substitute the original $RR$ into the formula for ratios at least one. The formula requires a value of at least one. Check that the value entering that formula is $1/RR$ for a protective estimate, and check that the final result is at least one.

What This Calculation Does—and Does Not—Provide

An E-value is a sensitivity summary based on a point estimate. It is not a replacement for the estimated risk ratio, a measure of the observed absolute risk difference, or evidence that a particular unmeasured confounder exists. Nor does a large value establish that bias is absent. It quantifies the result of a defined transformation; conclusions about a study still depend on design, measurement, and the plausibility of alternative explanations.

The E-value also differs from Rosenbaum sensitivity bounds used in matched designs. Those bounds describe sensitivity through a parameter governing differences in treatment-assignment odds within matched sets and can be used to assess how inference changes under hidden bias. An E-value is instead computed from a risk-ratio estimate. The quantities have different definitions and are not interchangeable merely because both concern unmeasured confounding.

Likewise, matching or adjustment for measured covariates and an E-value answer different questions. Matching and adjustment address measured covariates under their respective assumptions; the point-estimate E-value summarizes sensitivity to unmeasured confounding on the risk-ratio scale. Neither calculation, by itself, resolves problems such as poorly defined groups, incompatible follow-up, or a risk ratio that cannot be estimated.

Key takeaway: For a protective estimate with $0<RR<1$, calculate the point-estimate E-value from $1/RR$, retain the original group orientation when interpreting the association, and do not treat the reciprocal as a reversal of the substantive result. An estimate of zero has no finite reciprocal E-value.

Check Your Understanding

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

A study reports $RR=0.50$, with the treated group’s risk in the numerator. Calculate the point-estimate E-value.
A study reports $RR=0.90$. Is its E-value expected to be close to one or much larger? Calculate it to verify.
What should be reported if the estimated risk ratio is exactly zero because the numerator-group risk is zero and the denominator-group risk is positive?
In a sentence, explain why using $1/RR$ does not turn a protective estimate into a harmful association.

Solutions. (1) $1/RR=2$, so $E=2+\sqrt{2(2-1)}=2+\sqrt{2}\approx3.414$. (2) It should be close to one. Here $1/0.90=1.1111$, so $E=1.1111+\sqrt{1.1111(0.1111)}\approx1.462$, using unrounded inputs for the calculation. (3) The reciprocal is undefined at zero; there is no finite E-value from the formula, though the limit as a positive $RR$ approaches zero is infinite. (4) The reciprocal is a computational orientation used in the formula; the original group risks and $RR$ still determine which group had lower observed risk.