Quantify the joint unmeasured-confounder strength needed to explain away a risk-ratio association, with protective and confidence-interval conventions.
E-value for unmeasured confounding
The E-value for unmeasured confounding summarises how strong an omitted confounder's associations with both an exposure and an outcome would need to be, on the risk-ratio scale and beyond measured covariates, to explain away an observed association. In quantitative research, it helps put a number on an omitted-variable challenge to an observational policy comparison. It does not identify the policy's causal effect. VanderWeele and Ding (2017)
This causal sensitivity measure is distinct from e-values used for always-valid inference. It is neither a p-value nor a probability that confounding is absent.
A policy comparison with an unrecorded reason for assignment
Consider a fictional execution study. The exposure A is use of one routing policy rather than another; the binary outcome Y is exceeding a prespecified shortfall threshold over the same evaluation horizon. Within a fixed stratum of recorded market and order characteristics C=c, suppose the observed risk ratio is R=2. Unrecorded urgency U could influence both policy assignment and the outcome: that is confounding.
For a risk ratio R≥1, the null-target E-value is
E(R)=R+R(R−1).
Here,
E(2)=2+2≈3.414.
The number describes a joint strength threshold. It does not say that an omitted factor is present, nor that a factor called “urgency” actually has associations of that size.
Why two associations matter
Let a be the largest ratio of confounder-category probabilities between the exposed and unexposed groups, conditional on C=c:
a=umaxP(U=u∣A=0,C=c)P(U=u∣A=1,C=c).
Let b be the largest outcome-risk ratio across categories of U, taking the maximum over both exposure groups within that stratum. These sensitivity parameters are at least 1; U can represent several omitted variables jointly. They are not correlations or regression coefficients.
The joint bounding factor is
B(a,b)=a+b−1ab.
If C and U suffice to control confounding, the bound for this positively associated exposure gives Rcausal≥R/B(a,b). Explaining away R=2 through confounding alone therefore requires B(a,b)≥2. Ding and VanderWeele (2016), Result 1
Setting a=b=s gives s2/(2s−1)=2, whose admissible solution is s=2+2. 3.414 is the minimum equal-strength benchmark. With unequal strengths, at least one must reach that benchmark; both need not. For example,
B(3,4)=612=2,B(2,2)=34.
Thus the pair (3,4) reaches the null bound even though one strength is below 3.414. The pair (2,2) only lowers the bound to 2/(4/3)=1.5.
Original sensitivity plot for observed risk ratio 2. The equal-strength boundary is at 3.414 on both axes; unequal pairs (3, 4) and (4, 3) also reach the null bound, while (2, 2) does not.
The shaded region meets the joint bounding-factor condition B ≥ 2. It shows mathematical compatibility with explaining away the fictional association, not evidence that such confounding exists.
Report the estimate and its nearest-null interval limit
Input on the risk-ratio scale
Calculation
E-value
Observed ratio 2
2+2
3.414
Protective ratio 0.5
Invert first: 1/0.5=2
3.414
Lower 95% confidence limit 1.2, with the interval entirely above 1
1.2+1.2(0.2)
1.690
Confidence interval includes 1
Null already included
1
For a protective association whose entire interval is below 1, use the upper limit nearest 1 and invert it before applying the formula. The confidence interval calculation asks how much confounding could move its nearest boundary to the null; it is different from moving the point estimate to the null. VanderWeele and Ding (2017), Table 1
The example assumes an actual risk ratio for a binary outcome at a specified horizon. An odds ratio or hazard ratio cannot automatically be inserted as if it were that risk ratio; extensions require their own scale assumptions. Author manuscript, Table 2
A large E-value does not establish causal robustness. It leaves selection bias, measurement error, model misspecification and the plausibility of the proposed confounder strengths unresolved. The bound also needs a causal interpretation supported by the study design. Here the analysis is explicitly within C=c; an averaged effect over heterogeneous covariate strata needs the corresponding aggregation argument. Ding and VanderWeele (2016), qualifications and covariate-stratum discussion
For a practical setting in which assignment and outcomes can share an unrecorded cause, see Post-Trade Analysis (public preview; full lesson requires membership).
