Interpreting E-Values as Confounder Strength

From an E-Value to a Confounder-Strength Threshold

An E-value is often reported as a single number, but its meaning is about a pair of associations. It asks how strongly an unmeasured confounder would have to be associated with both treatment and outcome, on a risk-ratio scale, to account for an observed treatment–outcome association. Interpreting that number requires distinguishing an equal-strength benchmark from the full range of possible combinations.

Suppose a hypothetical observational analysis estimates a risk ratio $R>1$ for an outcome comparing treated with untreated people. Let $A$ denote treatment, $Y$ the outcome, $C$ the measured covariates used in the analysis, and $U$ an unmeasured confounder. The relevant associations are considered conditional on $C$: how strongly $U$ predicts treatment, and how strongly $U$ predicts the outcome within treatment levels. The E-value concerns residual confounding after accounting for $C$, not confounding already removed by the analysis.

Definition: For an observed risk ratio $R>1$, the E-value is the smallest value $E\geq 1$ such that an unmeasured confounder with associations of strength $E$ with treatment and outcome, on the risk-ratio scale and conditional on measured covariates, could suffice to explain away the observed association. The equal-strength E-value is $$ E=R+\sqrt{R(R-1)}. $$ More generally, if the confounder–treatment and confounder–outcome association strengths are $a$ and $b$, the standard risk-ratio bias factor is $$ B(a,b)=\frac{ab}{a+b-1}. $$ This pair can explain away $R$ when $B(a,b)\geq R$.

The bias factor gives a useful way to interpret the E-value. A larger value of $B(a,b)$ allows more attenuation of the observed risk ratio when considering residual confounding. To explain away an observed ratio $R$, the associations must be strong enough for the bias factor to reach at least $R$. If the bias factor is below $R$, that pair of association strengths is not sufficient under this bound.

Why the Equal-Strength Formula Works

The E-value uses the threshold case in which the two association strengths are equal: $a=b=x$. Setting the bias factor equal to $R$ gives

\frac{x^2}{2x-1}=R.

Multiplying by $2x-1$ and collecting terms yields a quadratic equation. Its solutions are $R\pm\sqrt{R^2-R}$; the relevant solution is the larger one, which is at least 1 and meets the confounding-strength threshold.

x^2-2Rx+R=0 \qquad\Longrightarrow\qquad x=R+\sqrt{R^2-R} =R+\sqrt{R(R-1)}.

This derivation shows why the E-value is an equal-strength benchmark. It is one point on the boundary $B(a,b)=R$, not a claim that both associations must always equal that number. A stronger association with treatment can compensate for a weaker one with outcome, and vice versa. To interpret unequal strengths, solve the bias-factor inequality using the association that is specified.

Worked Examples

Worked Example: Equal-Strength Associations for a Risk Ratio of 1.8

In a hypothetical analysis, the estimated risk of an outcome is 1.8 times as high among treated people as among untreated people. What is the E-value, and what does it say about equal-strength confounding associations?

Here $R=1.8$. Substitution into the formula gives

E=1.8+\sqrt{1.8(1.8-1)} =1.8+\sqrt{1.44} =1.8+1.2 =3.000.

The equal-strength threshold is 3.000. To check it, substitute $a=b=3$ into the bias factor:

B(3,3)=\frac{3\times 3}{3+3-1} =\frac{9}{5} =1.8.

Thus, under the risk-ratio bias bound, equal-strength associations of 3.000 with treatment and outcome could suffice to move the observed ratio to the null. This is a sensitivity threshold, not evidence that such a confounder exists or that either association has been measured.

Worked Example: Checking the Arithmetic for a Risk Ratio of 5/3

Suppose hypothetical outcome risks are 40% in the treated group and 24% in the untreated group. Their ratio is $R=0.40/0.24=5/3$. Calculate and verify the equal-strength E-value.

Using $R=5/3$,

E=\frac{5}{3}+\sqrt{\frac{5}{3}\left(\frac{5}{3}-1\right)} =\frac{5}{3}+\sqrt{\frac{10}{9}} =\frac{5+\sqrt{10}}{3} \approx 2.721.

The square root is $\sqrt{10}/3$, approximately 1.054; adding it to $5/3$, approximately 1.667, gives approximately 2.721. To verify, use the unrounded value $E=(5+\sqrt{10})/3$. The equal-strength bias factor is

B(E,E)=\frac{E^2}{2E-1}=\frac{5}{3}.

For an explicit check of the identity, $E^2=(35+10\sqrt{10})/9$ and $2E-1=(7+2\sqrt{10})/3$; their ratio simplifies to $5/3$. The interpretation is that equal-strength associations of approximately 2.721 could suffice under the bound. Reporting 3.000 would be incorrect: it would give $B(3,3)=9/5=1.8$, which is larger than the observed ratio $5/3$ and is not the minimum equal-strength threshold.

Worked Example: Finding the Required Outcome Association

For a hypothetical analysis, let the observed risk ratio be $R=2$. Suppose substantive knowledge suggests a possible unmeasured confounder could be associated with treatment with strength $a=3$. What outcome-association strength $b$ would be needed to reach the explain-away threshold?

Start with $B(a,b)\geq R$, and substitute $a=3$, $R=2$:

\frac{3b}{3+b-1}\geq 2 \qquad\Longrightarrow\qquad 3b\geq 2(b+2) \qquad\Longrightarrow\qquad b\geq 4.

The minimum is $b=4$. Verification gives $B(3,4)=12/(3+4-1)=12/6=2$. Therefore, given the specified treatment association of 3, an outcome association of at least 4 is required to reach the bound. The equal-strength E-value for $R=2$ is $2+\sqrt{2}\approx3.414$, which is a different threshold: it describes the equal-strength case, not the specified $a=3$ case.

Worked Example: When a Specified Association Is Not Strong Enough

Keep the observed risk ratio at $R=2$, but suppose the treatment association is specified as $a=1.5$. Can any finite outcome association strength $b$ reach the explain-away threshold?

For $a>R$, rearranging the bias-factor inequality gives

b\geq \frac{R(a-1)}{a-R}.

That rearrangement requires $a-R>0$. Here $a=1.5$ is below $R=2$, so there is no finite $b$ that reaches the threshold. In fact, as $b$ grows without limit, the bias factor approaches $a$:

\lim_{b\to\infty} B(1.5,b) =\lim_{b\to\infty}\frac{1.5b}{b+0.5} =1.5<2.

Thus even an arbitrarily strong outcome association cannot make this specified treatment association sufficient under the bias bound. If $a=R$, the limit would equal $R$, but no finite $b$ would attain it. These edge cases are important: the formula for a required $b$ is not applicable when its denominator is zero or negative.

Using a Specified Association and Checking the Interpretation

For a specified treatment-association strength $a$, begin with

\frac{ab}{a+b-1}\geq R.

When $a>R$, multiply through by the positive denominator and rearrange:

ab\geq R(a+b-1) \qquad\Longrightarrow\qquad b(a-R)\geq R(a-1) \qquad\Longrightarrow\qquad b\geq\frac{R(a-1)}{a-R}.

The boundary value is the minimum $b$ that reaches the threshold; any larger value also reaches it. The expression is symmetric in $a$ and $b$, so the same calculation applies if the outcome association is specified and the required treatment association is sought.

In practice, treat $a$ or $b$ as a plausible sensitivity value only when it has a defensible interpretation on the risk-ratio scale and refers to associations conditional on the measured covariates. The calculation does not estimate these associations from the study data when $U$ is unmeasured. It asks what would be required, given an assumed value for one of them.

Assumptions, Diagnostics, and Common Pitfalls

This interpretation is for a risk-ratio estimate and the standard risk-ratio bias bound. The treatment and outcome associations must be understood on compatible risk-ratio scales, conditional on the measured covariates in the analysis. The E-value does not establish that the causal assumptions hold: the treatment and outcome must still be well-defined, the analysis must address measured confounding appropriately, and other sources of bias may remain. If the reported effect measure is not a risk ratio, it should not be inserted as $R$ without a justified conversion or an appropriate method for that measure.

A useful diagnostic is to check the bias factor directly for any proposed pair $(a,b)$. If $B(a,b)<R$, that pair does not meet the explain-away threshold. If $B(a,b)=R$, it is exactly on the threshold. This arithmetic check can catch mistaken use of the equal-strength E-value, as well as errors from rounding intermediate values.

Mistake: “Both associations must equal the E-value.” The E-value gives the equal-strength point on the threshold, not the only possible combination. Use $B(a,b)$ or the rearranged inequality when one strength differs.
Mistake: “An E-value of 3 means there is a 3-in-1 chance of confounding.” It is not a probability, a p-value, or a measure of the chance that a confounder exists. It is a threshold for association strengths on a specified scale.
Mistake: “An E-value proves the result is robust.” Whether a threshold is large or small is a substantive judgment. Compare it with plausible confounder associations, and remember that the E-value does not cover every possible source of bias.

E-values and Rosenbaum bounds summarize sensitivity in different ways. E-values express the strength of confounder–treatment and confounder–outcome associations on a risk-ratio scale. Rosenbaum bounds express sensitivity through a parameter governing treatment-assignment odds within matched sets. Neither scale is automatically interchangeable with the other, so a numerical value from one should not be read as a numerical value on the other.

The main limitation for this tutorial’s interpretation is that a single E-value compresses a two-association tradeoff into one equal-strength summary. It does not identify the confounder, reveal its actual association with treatment or outcome, or guarantee that unmeasured confounding is the only concern. Its usefulness is as a transparent threshold: specify plausible association strengths, calculate the bias factor, and judge whether they could be sufficient in context.

Key takeaway: An E-value is an equal-strength confounder threshold. For unequal strengths, compare $B(a,b)=ab/(a+b-1)$ with the observed risk ratio $R$. A specified association $a>R$ requires $b\geq R(a-1)/(a-R)$; if $a\leq R$, no finite $b$ reaches the threshold.

Check Your Understanding

Use the bias factor to answer each question. The solutions follow the questions.

For an observed risk ratio $R=3$, calculate the equal-strength E-value.
For $R=2$ and a specified treatment-association strength $a=4$, calculate the minimum outcome-association strength $b$.
For $R=1.8$, does the pair $a=2$, $b=5$ reach the explain-away threshold?
Explain why an E-value of 2.5 should not be described as a 2.5-fold probability that confounding is present.

Solutions. (1) $E=3+\sqrt{3(2)}=3+\sqrt{6}\approx5.449$. (2) Since $a>R$, $b\geq2(4-1)/(4-2)=3$; indeed, $B(4,3)=12/6=2$. (3) $B(2,5)=10/(2+5-1)=10/6\approx1.667<1.8$, so this pair does not reach the threshold. (4) An E-value is a risk-ratio-scale sensitivity threshold for associations with treatment and outcome, not a probability, p-value, or estimate of whether a confounder exists.