E-Values for Multiple Unmeasured Confounders

Why Multiple Unmeasured Factors Require a Joint View

An unmeasured confounder is not necessarily one hidden variable. The unmeasured part of a study might include health status, access to care, and health-seeking behavior together. These factors can be related to one another, can jointly predict treatment, and can jointly predict the outcome. An E-value for an estimated risk ratio summarizes how strong the relevant unmeasured confounding would need to be to explain the association, but it does not assign a separate E-value to each hidden factor and then combine them.

The key move is to regard all unmeasured factors together as a joint profile, denoted \(U\). This profile could be a vector such as \(U=(U_1,U_2)\), with categories for each combination of the factors. The joint profile’s association with exposure and outcome—not a sum of standalone component summaries—is what enters the risk-ratio bias bound. The resulting E-value is a threshold for that joint confounding structure.

Definition: For a risk-ratio estimate \(R>1\), the E-value is the smallest common upper bound \(\Gamma\) on the joint unmeasured-confounder associations with exposure and outcome that could produce a bias factor as large as \(R\), under the standard risk-ratio confounding bound. Here, \(U\) denotes the full joint profile of unmeasured factors. The E-value is a sensitivity threshold; it is not a probability that confounding exists.

The Joint Risk-Ratio Bias Bound

Let \(A\) denote exposure, \(Y\) the outcome, and \(U\) the joint profile of unmeasured confounders. On the risk-ratio scale, summarize the strength of the joint profile’s association with exposure by \(\Gamma_{AU}\), and its association with the outcome by \(\Gamma_{UY}\). Each is at least 1. For a categorical joint profile, these quantities can be understood as upper bounds on the largest relevant risk-ratio contrasts across its categories: exposure risks across \(U\) categories for \(\Gamma_{AU}\), and outcome risks across \(U\) categories at a given exposure level for \(\Gamma_{UY}\). They must describe the joint profile, not just one component at a time.

Under the standard risk-ratio bias bound, the maximum factor by which this unmeasured confounding can move the observed risk ratio away from the causal risk ratio is bounded by:

B \leq \frac{\Gamma_{AU}\Gamma_{UY}}{\Gamma_{AU}+\Gamma_{UY}-1}.

This is a worst-case bound, not a prediction of the actual bias. It concerns the risk-ratio scale and requires that the association bounds adequately describe the joint profile and its outcome relationship. The study must also be interpreted under the usual causal framework: the exposure and outcome are well-defined, the relevant causal contrast is meaningful, and the bias being assessed is unmeasured confounding rather than a mixture of unrelated errors. If other biases are plausible, an E-value for confounding alone does not quantify them.

For the E-value, suppose both association dimensions are bounded by the same number \(\Gamma\). Set the bias factor equal to the observed risk ratio \(R\), the threshold at which confounding could move that estimate to the null:

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

Solving gives the familiar expression. Multiplying both sides by \(2\Gamma-1\), then rearranging, produces a quadratic equation:

R(2\Gamma-1)=\Gamma^2 \quad\Longrightarrow\quad \Gamma^2-2R\Gamma+R=0.

The larger root is the relevant strength threshold, since \(\Gamma\geq 1\):

\Gamma = R+\sqrt{R(R-1)}.

Thus, the E-value asks how strong both associations would have to be if they were equally strong. Unequal strengths are also possible: the bias factor can be evaluated directly using distinct values of \(\Gamma_{AU}\) and \(\Gamma_{UY}\).

Worked Example: The E-Value for a Joint Confounder Profile

Suppose a hypothetical study estimates a risk ratio of \(R=3.00\). The unmeasured profile \(U\) jointly represents two factors, such as baseline frailty and access to care. Find the E-value and interpret its meaning.

Use the risk-ratio formula:

E = R+\sqrt{R(R-1)} =3.00+\sqrt{3.00(3.00-1)} =3.00+\sqrt{6.00} =5.449.

The joint unmeasured profile would need associations of at least about \(5.449\) with both exposure and outcome, on the specified risk-ratio scales, if the two strengths were equal, to be sufficient under the bound to explain an observed risk ratio of 3.00 completely. This does not mean each factor separately needs an association of 5.449, nor that the two factors actually have that strength. It is a threshold for the joint profile.

Why Component E-Values Cannot Be Added

An E-value is calculated from an effect estimate. It is not a unit of confounding strength that can be assigned to each candidate variable and added. Even when investigators describe two factors separately, each factor’s association with exposure and outcome does not determine the joint profile’s associations by simple addition. Dependence between factors, overlap in the people they identify, and interactions can all change the joint pattern.

A useful way to see the issue is to compare the two association dimensions for the joint profile. If factor \(U_1\) is strongly associated with exposure and \(U_2\) is strongly associated with outcome, their joint profile may or may not be strongly associated with both. Their relationships could overlap, offset one another, or become stronger in particular combinations. Therefore, separate one-factor summaries do not, by themselves, specify \(\Gamma_{AU}\) and \(\Gamma_{UY}\).

Worked Example: Two Factors Do Not Produce an Additive E-Value

Suppose an observed risk ratio is \(3.00\), with the E-value \(5.449\) calculated above. For illustration, assess two factors one at a time. Assume each factor, considered alone, has an exposure association bound of 2 and an outcome association bound of 2.

For either factor considered alone, the maximum bias factor under the bound is:

B_{\text{single}} \leq \frac{2(2)}{2+2-1} =\frac{4}{3} =1.333.

It would be incorrect to add these bias factors, or to add two E-values, and treat the result as a joint assessment. The factors are not independent units of bias, and the single-factor calculations do not state the joint association bounds. To illustrate what additional information is needed, suppose a defensible joint model instead establishes \(\Gamma_{AU}=4\) and \(\Gamma_{UY}=4\) for the combined profile. Then:

B_{\text{joint}} \leq \frac{4(4)}{4+4-1} =\frac{16}{7} =2.286.

That joint bound is below 3.00, so under these specified bounds the joint profile is not strong enough to explain the entire estimated risk ratio. The value \(4\) in this illustration is an assumed joint bound, not something established by adding the single-factor calculations. A different dependence pattern or joint model could yield different bounds.

Assessing Joint Strength in Practice

When the number of unmeasured factors is larger than one, a useful sensitivity analysis makes its joint-profile assumptions explicit. For example, investigators might define categories for combinations of binary factors, or define a composite profile with clinically interpretable levels. They can then specify plausible upper bounds on the profile’s association with exposure and on its association with outcome. If the two dimensions are unequal, calculate the bias bound directly rather than forcing them to be equal.

The following hypothetical calculation shows how joint categories can be used. The numbers are population risks chosen for illustration, not data from an actual study.

Worked Example: Applying Unequal Bounds to Joint Categories

Suppose two unmeasured binary factors form four joint categories, \(U=(0,0),(1,0),(0,1),(1,1)\). Assume the exposure risks across these categories are 0.10, 0.15, 0.20, and 0.30. Suppose the outcome risks, evaluated within either exposure level, are 0.04, 0.08, 0.12, and 0.16. For this illustration, treat the largest-to-smallest risk ratios across categories as the association bounds.

The exposure association bound for the joint profile is:

\Gamma_{AU}=\frac{0.30}{0.10}=3.00.

The outcome association bound is:

\Gamma_{UY}=\frac{0.16}{0.04}=4.00.

The joint bias factor is therefore bounded by:

B \leq \frac{3(4)}{3+4-1} =\frac{12}{6} =2.000.

If the hypothetical observed risk ratio were 2.50, this specified joint profile would not reach the bias factor needed to explain it completely, since \(2.000<2.50\). The conclusion is conditional on the assumed joint categories, the association bounds, and the risk-ratio bias bound; it does not establish that no unmeasured confounding exists.

Diagnostics, Assumptions, and Limits

A joint E-value analysis is only as informative as the joint profile and association bounds it represents. Several checks help prevent an apparently precise number from concealing weak assumptions:

Define the joint profile. State which factors are included and what the joint categories mean. A profile that omits an important factor does not bound confounding from that omitted factor.
Check the scale and population. The bias formula here is for risk ratios. The exposure and outcome association bounds must refer to the same analysis population and the intended joint profile.
Look for sparse or empty categories. If some combinations of factors are rare or absent, their joint associations may be poorly estimated or not estimable from the available data. A numerical bound should not suggest certainty where there is little information.
Consider interactions and dependence. The effect of one factor may differ by the level of another, and factors may cluster together. A product or sum of separate associations is justified only by additional modeling assumptions; it is not a general rule.
Separate sensitivity from diagnosis. A large E-value does not show that strong confounding is present, while a smaller one does not prove that it is. The value describes a threshold, not the probability or existence of a particular confounder.

This approach also has limits. The E-value compresses a potentially complicated joint distribution into two association bounds and one threshold. Different joint distributions can share the same bounds but imply different actual bias. The standard bound is deliberately conservative and may not describe the likely bias in a particular setting. It also does not measure selection bias, measurement error, or other departures from the study’s assumptions.

Common Mistake: Adding Separate E-Values

Common mistake: “The first unmeasured factor has an E-value of 2 and the second has an E-value of 3, so their combined E-value is 5.” This reasoning is wrong because E-values are not additive measures assigned to individual confounders. The correction is to define the joint profile and assess its exposure and outcome association bounds, accounting for dependence and the chosen model, then apply the joint bias bound.

The joint calculation can still be informed by knowledge about separate factors, but translating that knowledge into joint bounds requires explicit assumptions. For example, a multiplicative model might imply that certain conditional association ratios combine multiplicatively. That conclusion depends on the model and on how the factors are distributed; it cannot be inferred from standalone E-values alone.

Key takeaway: An E-value summarizes the strength required of the joint unmeasured-confounder profile. For unequal joint associations, use \(B \leq \Gamma_{AU}\Gamma_{UY}/(\Gamma_{AU}+\Gamma_{UY}-1)\). Do not add component E-values; first justify the joint association bounds.

Check Your Understanding

Answer each question using the joint-profile interpretation and the risk-ratio bias bound.

A risk-ratio estimate is 2.00. What is its E-value, to three significant figures?
A joint profile has \(\Gamma_{AU}=2\) and \(\Gamma_{UY}=5\). Calculate its maximum bias factor under the bound.
Why can’t two separately calculated E-values be added to obtain the E-value for the two factors together?
A joint profile produces a bias factor bound of 1.80, while the observed risk ratio is 2.20. What does this comparison show, and what does it not show?

Solutions. (1) \(E=2+\sqrt{2(2-1)}=2+\sqrt{2}=3.414\), or 3.41. (2) \(B \leq 2(5)/(2+5-1)=10/6=1.667\). (3) E-values are thresholds calculated from an effect estimate, not additive units of confounding strength; the joint associations depend on the joint distribution and any modeling assumptions. (4) Under the specified bounds, the profile’s maximum bias factor is below the 2.20 estimate and cannot explain it completely by this bound. This does not prove that confounding is absent or address other sources of bias.