Why Effect Size Matters
An E-value is a sensitivity measure: it describes how strong an unmeasured confounder’s associations with treatment and outcome would need to be, on the risk-ratio scale, to explain away an estimated association. Its numerical value depends directly on the estimated risk ratio. This tutorial examines that dependence. In particular, it explains why an association only slightly different from the null has an E-value close to one, and why E-values grow at approximately twice the risk ratio when the risk ratio is very large.
The calculations below concern a point estimate expressed as a risk ratio. They describe how the E-value changes as that estimate changes; they do not say whether an estimate is statistically precise, whether the study is free from bias, or whether a causal interpretation is justified. The earlier material on E-values for risk-ratio estimates establishes the basic interpretation; here the emphasis is on the function’s mathematical behavior and its practical consequences.
Deriving the E-Value Function
The behavior of the formula is easier to interpret after seeing why it has this form. Let \(\Gamma\) be an upper bound on each of two associations: the association between an unmeasured confounder and treatment, and the association between that confounder and outcome. The standard risk-ratio bias-factor bound is \(\Gamma^2/(2\Gamma-1)\), for \(\Gamma\geq1\). To ask what equal strength in both associations could suffice to account for an observed risk ratio \(R\), set the bound equal to \(R\) and solve.
The relevant solution is the larger root, because the minimum confounder-strength bound must be at least one and the smaller root does not give the required threshold when \(R>1\). Since \(R^2-R=R(R-1)\), this larger root is \(R+\sqrt{R(R-1)}\), the E-value. This derivation also makes an important interpretation clear: the value is a worst-case threshold under the bound, not a measured confounder association.
Behavior Near the Null
The null risk ratio is one. Write a risk ratio just above the null as \(R=1+\delta\), where \(\delta>0\) is small. Substitution gives
Thus, as \(\delta\) approaches zero, the leading change in \(E-1\) is \(\sqrt{\delta}\), not \(\delta\). The E-value still approaches one continuously, but it does so with a square-root pattern. Very small changes in the risk ratio near one can therefore produce a more noticeable change in \(E-1\) than a linear approximation would suggest. The slope of the function also becomes very large near the null: for \(R>1\),
As \(R\) decreases toward one, the denominator approaches zero, so this derivative grows without bound. This is a mathematical feature of the transformation, not evidence that a small observed association is especially robust. In fact, when \(R\) is close to one, the E-value is close to one as well: only weak confounder associations may be sufficient, under the framework, to move the point estimate to the null.
Worked Example: A Risk Ratio Just Above One
Suppose a hypothetical study reports an estimated risk ratio of \(1.01\). Calculate its E-value and compare it with the null.
The estimate is above one, so use \(R=1.01\) directly:
The point estimate is only 1% above the null, while its E-value is about 1.11. Under the standard interpretation, an unmeasured confounder associated with both treatment and outcome by risk ratios of roughly 1.11 each could be sufficient, in the worst-case bound, to explain away this point estimate. That is a modest threshold, not a statement that the estimate is false or that a confounder of that strength is present.
Behavior for Large Effects
For large \(R\), factor \(R^2\) out of the square root:
Using the expansion \(\sqrt{1-x}=1-x/2-x^2/8+O(x^3)\) for small \(x\), with \(x=1/R\), yields
So for a very large risk ratio, the E-value is approximately \(2R-0.5\), and its ratio to \(R\) approaches two. This is an approximation for large effects, not an alternative definition. For moderate effects, the exact formula is simple to evaluate and should be preferred.
Worked Example: Comparing Small and Moderate Risk Ratios
Consider hypothetical point estimates \(R=1.05\), \(R=1.10\), and \(R=1.25\). Calculate each E-value to see how a modest change in the estimated effect maps to the sensitivity scale.
For \(R=1.05\):
For \(R=1.10\):
For \(R=1.25\):
All three estimates are relatively close to one, and their E-values remain correspondingly modest. The first two values, in particular, illustrate why small associations yield E-values near one. The E-value is not a rescaled effect estimate with a fixed additive increment: the square-root term makes the transformation nonlinear.
Worked Example: A Large Risk Ratio and Its Approximation
A hypothetical estimate has risk ratio \(R=10\). Find the exact E-value, then compare it with the large-effect approximation.
The exact calculation is
The first-order large-effect approximation gives
The approximation is close to the exact value, differing by about \(0.0132\). At this effect size the E-value is nearly twice the risk ratio, as the asymptotic expansion predicts. For smaller risk ratios, the approximation can be less accurate; the exact formula remains straightforward and avoids that approximation error.
Protective Associations and Comparisons
For an estimated risk ratio below one, the direction is protective, but the E-value calculation uses the distance from the null on the reciprocal scale. For instance, a risk ratio of \(0.90\) is converted to \(1/0.90\), not treated as a negative or subtracted from one. The E-value is therefore always at least one, and an estimate close to one in either direction produces an E-value close to one.
Worked Example: A Protective Estimate Near the Null
Suppose a hypothetical study reports \(RR=0.90\). Convert to the reciprocal scale and calculate the E-value, retaining the unrounded reciprocal until the final result.
Then
The E-value is about 1.4625. It is computed from the magnitude of the association on the reciprocal scale; the protective direction does not change the minimum-strength interpretation.
E-values can help compare sensitivity thresholds for risk-ratio point estimates, but they answer a different question from the effect estimate itself. A risk ratio describes the estimated relative outcome risks; an E-value summarizes a confounding-strength threshold under a particular bound. Neither the E-value nor the point estimate describes precision by itself. An E-value for a point estimate should not be substituted for an E-value based on a confidence limit when the question concerns uncertainty in the estimate.
| Quantity | What it describes | What it does not establish |
|---|---|---|
| Risk ratio | Estimated relative risk in the compared groups | That the association is causal or precisely estimated |
| Point-estimate E-value | Confounder-strength threshold for explaining away the point estimate under the standard bound | That a confounder exists or that no other bias matters |
| Large-effect approximation | A convenient approximation, \(2R-0.5\), when \(R\) is large | The exact E-value for every risk ratio |
Diagnostics, Limitations, and Common Mistakes
A useful check is to confirm the direction and scale before calculating. For \(R\geq1\), the E-value must be at least \(R\), since the square-root term is nonnegative, and it must equal one at \(R=1\). As \(R\) grows, the E-value increases. For a protective estimate, first take the reciprocal; a result below one signals a calculation or orientation error. Retain precision in intermediate calculations, particularly when the original estimate is close to one, and round only the final value.
A frequent mistake is to say that an E-value of, for example, 1.11 means the treatment effect is 11% more robust, or that a confounder with an association of 1.11 has been found. Neither statement follows. The value is a threshold expressed as a risk-ratio association strength under a specified worst-case framework. It also does not account for every possible source of bias, establish exchangeability, or settle whether an observational association is causal. The estimate’s scale must be appropriate for the risk-ratio formula; using an odds ratio as if it were a risk ratio can lead to a misleading result when the outcome is not rare.
The central pattern is useful but limited: near the null, E-values are close to one, while for very large risk ratios they are close to twice the risk ratio. These facts help explain the numbers; they do not replace study-specific assessment of confounding, design, measurement, or uncertainty.
Check Your Understanding
Use the formula and reasoning above. Each solution is included after the questions.
- Calculate the E-value for a hypothetical risk ratio of \(1.50\).
- A hypothetical study reports \(RR=0.80\). What risk ratio should be used in the E-value formula, and what is the resulting E-value?
- In one sentence, describe how \(E-1\) behaves when \(R=1+\delta\) and \(\delta\) is small and positive.
- For a very large \(R\), what simple approximation describes the E-value? Is it exact?
Solutions. (1) \(E=1.50+\sqrt{1.50(0.50)}=1.50+\sqrt{0.75}\approx2.366\). (2) Use \(R=1/0.80=1.25\), so \(E=1.25+\sqrt{1.25(0.25)}\approx1.809\). (3) Since \(E(1+\delta)-1=\sqrt{\delta}+\delta+O(\delta^{3/2})\), the leading behavior is \(\sqrt{\delta}\). (4) \(E\approx2R-0.5\); it is an asymptotic approximation, not the exact formula for every \(R\).