## The E value method

In a study with a specific exposure and outcome, the E value is the minimum strength of association that the unmeasured confounders should have with both the exposure and the outcome to explain away the observed association.8 The exposure refers to any characteristic that may explain or predict the presence of a study outcome.9 For a particular observed exposure-outcome risk ratio (RR), the E value can be calculated based on formula (1), involving only the estimate itself.10

The same approach applies to odds ratios and hazard ratios if the outcome is rare. If the outcome is common, the odds ratio and hazard ratio must be modified (eg, by replacing RR in formula (1) with the square root of the odds ratios).8

A strong association has a large E value, which suggests that the unmeasured confounding must be strongly associated with both the exposure and outcome to fully explain the association. In contrast, a small E value suggests that weak unmeasured confounding would be enough to explain the association. A lower E value might indicate that confounding rather than causality is a more plausible explanation than a higher E value. The E value does not have a specific range, and whether its value is considered large or small depends on the particular exposure and outcome and the amount of controlled confounding.11 For example, when examining the association between the use of glucocorticoids and the risk of suicide among patients with cancer, the incidence rate ratio was high (7.2) and the dose-response pattern showed that the highest cumulative dose was associated with a 20-fold increase in risk compared with non-use. In this example, the calculated E value indicated that to explain away the association, a hypothetical confounder would need be associated with a 14-fold higher use of glucocorticoids and a threefold greater risk of suicide. The authors judged that such a confounder is not likely to exist given the confounding already adjusted for in the analysis.12 Thus labelling the E value large or small depends on knowledge of the subject matter, the strength of the observed association, and amount of confounding removed.

The E value can be calculated without any assumptions regarding unmeasured confounders, such as being binary or consisting of only one unmeasured confounding factor.10 The E value estimates the overall strength of potential unmeasured confounding rather than the effect of individual confounding factors. With this information, investigators can assess whether one or several specific unmeasured confounders could plausibly explain away the observed association in a particular study.

In addition to the E value, Ding and VanderWeele introduced the joint bounding factor, B (formula 2).10

RR_{EU} denotes the strength of association between the unmeasured confounder and the exposure. B does not require assumptions about the structure of the unmeasured confounding.10 RR_{UD} denotes the strength of association between the unmeasured confounder and the outcome, as illustrated in a directed acyclic graph representing a simplified form of the confounding structure (figure 1).

Figure 1Causal directed acyclic graph showing the direction of hypothesised causal effects.23 Left: Directed acyclic graph for a generic association between unmeasured confounding, exposure, and outcome. Right: Directed acyclic graph for tobacco or alcohol as possible unmeasured confounders when examining the association between the use of antidepressants and risk of miscarriage. exRR_{EU}=strength of association between the unmeasured confounder and exposure; RR_{UD}=strength of association between the unmeasured confounder and outcome

The joint bounding factor, B, could take an infinite number of values depending on RR_{UD} and RR_{EU}. If the joint bounding factor B is set to equal the observed risk ratio, the joint bounding factor describes the different combinations of RR_{UD} and RR_{EU} that would have the joint minimum strength to explain away the association. The E value=B when RR_{UD}=RR_{EU}, which is one possible combination of RR_{UD} and RR_{EU}. A range of different possible combinations exist, which are often illustrative for the setting studied (figure 2).

Figure 2Illustration of different combinations of RR_{UD} and RR_{EU} for a possible joint bounding factor, B, of 1.41. RR_{EU}=strength of association between the unmeasured confounder and exposure; RR_{UD}=strength of association between the unmeasured confounder and outcome

For example, if RR_{EU} is known, the joint bounding factor can be used to estimate the minimum strength of association between the unmeasured confounder and the outcome (RR_{UD}) needed to explain away the association.8 In other settings, information might be available on the magnitude of the association between the strongest unobserved confounder and the outcome (RR_{UD}). The joint bounding factor can similarly be used to estimate RR_{EU}.

### Example: meta-analysis of the association between use of antidepressants in pregnancy and risk of miscarriage

One in five women have depression during pregnancy, and about 13% of pregnant women take antidepressants.13 14 Because both untreated depression and drug treatments might adversely affect pregnancy, treatment of depression in pregnant women involves balancing the benefits of treated depression against the potential treatment related risks to the mother and unborn child. Information about the safety of the use of antidepressants is therefore important for the healthcare professional when making these decisions.

We conducted a meta-analysis combining the evidence for the association between the use of antidepressants in the first trimester of pregnancy and the risk of miscarriage. Eligible studies were identified by a search in the PubMed and Embase databases from 2000 to February 2021, based on the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines (online supplemental appendix).15 The combined effect was estimated with the random effects model and Episheet software (figure 3).16

Figure 3Forest plot showing risk ratio and 95% confidence interval (CI) for each study included in the meta-analysis. A random effects model was used. Computing measures of heterogeneity (τ^{2}=0.039 and Q=223.28) showed low heterogeneity apart from the study by Johansen, but removing this study did not change the overall estimate of risk ratio substantially

The weighted estimate for the risk ratio of miscarriage in those who received treatment with antidepressants was 1.41 (95% confidence interval 1.22 to 1.63). To evaluate if this estimate is potentially biased by unmeasured confounding, we applied the E value, which we estimated at 2.17. Thus the unmeasured confounders must be associated with both use of antidepressants and risk of miscarriage by a risk ratio of at least 2.17 to fully explain away the observed risk ratio of 1.41. Tobacco and alcohol are prevalent and plausible confounders associated with both miscarriage and depression and hence with the use of antidepressants (figure 1).17–20 We used these confounders in our application of the E value method. Petersen et al estimated that the risk of substantial (>35 units/week) alcohol use and use of antidepressants was RR_{EU}=10.25.18 Sundermann et al found that alcohol use (>35 units/week) increased the risk of miscarriage (RR_{UD}=3.1).17 Because both estimates are stronger than the calculated E value, substantial alcohol use could explain the observed association between the use of antidepressants and risk of miscarriage.

Based on data from Johansen et al, we estimated that the association between smoking and use of antidepressants was RR_{EU}=2.06.21 Because the risk ratio was <2.17 (the E value), smoking could not fully explain the association when applying the E value method. By using the joint bounding factor, however, smoking could still explain the observed association if the association between smoking and miscarriage is sufficient. Figure 4 shows the different combinations of RR_{EU} and RR_{UD} that jointly would have the minimum strength required to explain the association.

Figure 4Different combinations of RR_{EU} and RR_{UD} that jointly would have the minimum strength required to explain the observed association in the meta-analysis. RR_{EU}=strength of association between the unmeasured confounder and exposure; RR_{UD}=strength of association between the unmeasured confounder and outcome

Smoking had an RR_{EU} value of 2.06, which requires RR_{UD}=2.3 to fully explain the association. Pineles et al found an increased risk of miscarriage associated with smoking of 1.32.19 Because this value is below the required RR_{UD} of 2.3, smoking is unlikely to explain the observed risk ratio of 1.41 on its own. The E value can similarly be applied to the results of any individual study.