Letters

# E-Values for Mendelian Randomization

Swanson, Sonja A.; VanderWeele, Tyler J.

Author Information
doi: 10.1097/EDE.0000000000001164
• Free

## To the Editor:

The Mendelian randomization (MR) approach uses genetic variants to attempt to estimate causal effects of an exposure on an outcome with instrumental variable (IV) techniques.1 The traditional application of this approach requires several assumptions, including that the genetic variants do not affect the outcome except through the exposure (i.e., exclusion restriction) and share no cause with the outcome. These assumptions are often not evaluated and are often even not stated. One concern with the latter assumption is that there may be confounding of the effect of the variant on the outcome by ancestry (i.e., population stratification). Such confounding can also arise from assortative mating or dynastic effects. Although the assumption cannot be verified, the sensitivity or robustness of estimates to violations of this assumption can be evaluated using traditional bias analysis techniques.

A particularly simple, easy-to-use approach to evaluate the sensitivity of estimates to confounding is the E-value. The E-value is defined as the minimum strength of association on the risk ratio scale that an unmeasured confounder would need to have with both the exposure and the outcome, conditional on the measured covariates, to completely explain away an observed exposureâ€“outcome association.2 The E-value can be reported for both the estimate and the limit of the confidence interval closest to the null. It is straightforward to calculate, and for an observed risk ratio of magnitude RRobs, it can be obtained by

If the initial risk ratio is less than 1, then it is inverted first before applying the formula. The E-value is nothing more than a simple transformation of the risk ratio to the confounding scale of the minimum association required between the unmeasured confounder and the exposure and the outcome, assuming both confounding associations are equal. It would be straightforward to report in MR analyses. To compute the E-value, all that is needed is the variantâ€“outcome association. This is all that is needed because, when using the standard IV ratio, for example, a numerator equal to zero, of course, implies an effect estimate of zero. Similar logic pertains to any consistent IV-based estimator such that a zero causal effect of the exposure on the outcome implies a null variantâ€“outcome association.3

To demonstrate the ease of computation, consider a recent MR study that suggested 3.6 additional years of education decreased the risk of coronary heart disease (CHD) by about one-third.4 The analyses used 162 genetic variants that have been previously associated with education, with the strongest variantâ€“CHD association to be an odds ratio of 1.07. The E-value for this particular variant is then 1.34; for other variants, the E-value would be less than 1.34. Modest confounding could explain these away. In contrast, Chen et al.5 used a single variant in the ALDH2 gene to study the effects of alcohol intake on risk of hypertension. Among males, the variantâ€“hypertension association was an odds ratio of 2.42. The E-value then is 4.27. The E-value for the lower limit of the confidence interval (1.66) is 2.71. As the analysis was conducted in an ethnically homogeneous Asian population, this E-value may be large enough to reasonably conclude that any residual confounding by ancestry is unlikely to explain away the effect.

Unfortunately, the variantâ€“outcome association is rarely reported, as many MR analyses report only IV estimates of the causal effect. We believe this should change. Ultimately, the statistical evidence for a non-null causal effect for the exposure on the outcome derives from the variantâ€“outcome association and it is therefore important to evaluate the robustness of this association to both confounding and exclusion restriction violations. The reporting of the variantâ€“outcome association itself is therefore of central importance. It allows for evaluating robustness to confounding using the E-value or similar techniques. It also directly allows for evaluating robustness to violations of the exclusion restriction, because the magnitude of the variantâ€“outcome association itself is precisely equal to the magnitude of the exclusion restriction violation that would be required to explain away the IV estimate.6,7 The reporting of the variantâ€“outcome association directly is further advantageous insofar as it is all that is needed to test a causal null hypothesis and is subject to far fewer assumptions and thus biases than IV estimation of the effect of the exposure on the outcome.7,8 Indeed, many of the issues regarding exposure definitions and well-defined interventions are rendered moot when viewing the variantâ€“outcome association as a test of a causal null hypothesis.3,7

The reporting of the E-value and the variantâ€“outcome association certainly does not resolve all biases concerning MR analyses. Moreover, this reporting also focuses on understanding the magnitude of bias that may explain away a non-null effect, whereas investigators may also be interested in understanding the plausible magnitude of bias or the relative magnitude compared with a non-MR approach.9 Bias analytic techniques other than the E-value may be preferable if some background knowledge exists about the unmeasured confounder.10 Nonetheless, we believe reporting the E-value and the variantâ€“outcome association should be routine practice in MR analyses and would be an important step forward in more accurately evaluating the robustness of the purported evidence.

Sonja A. Swanson

Department of Epidemiology

Erasmus MC

Rotterdam, The Netherlands

s.swanson@erasmusmc.nl

Tyler J. VanderWeele

Department of Epidemiology

Harvard T. H. Chan School of Public Health

Boston, MA

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