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VanderWeele, Tyler

Unmeasured confounding may undermine the validity of causal inference with observational studies. Sensitivity analysis provides an attractive way to partially circumvent this issue by assessing the potential influence of unmeasured confounding on causal conclusions. However, previous sensitivity analysis approaches often make strong and untestable assumptions such as having an unmeasured confounder that is binary, or having no interaction between the effects of the exposure and the confounder on the outcome, or having only one unmeasured confounder. Without imposing any assumptions on the unmeasured confounder or confounders, we derive a bounding factor and a sharp inequality such that the sensitivity analysis parameters must satisfy the inequality if an unmeasured confounder is to explain away the observed effect estimate or reduce it to a particular level. Our approach is easy to implement and involves only two sensitivity parameters. Surprisingly, our bounding factor, which makes no simplifying assumptions, is no more conservative than a number of previous sensitivity analysis techniques that do make assumptions. Our new bounding factor implies not only the traditional Cornfield conditions that both the relative risk of the exposure on the confounder and that of the confounder on the outcome must satisfy but also a high threshold that the maximum of these relative risks must satisfy. Furthermore, this new bounding factor can be viewed as a measure of the strength of confounding between the exposure and the outcome induced by a confounder.

Sensitivity analysis can be useful in assessing how robust associations are to potential unmeasured or uncontrolled confounding. In this paper we introduce a new measure that we call the “E-value,” a measure related to the evidence for causality in observational studies, when they are potentially subject to confounding. 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 treatment and the outcome to fully explain away a specific treatment-outcome association, conditional on the measured covariates. A large E-value implies considerable unmeasured confounding would be needed to explain away an effect estimate. A small E-value implies little unmeasured confounding would be needed to explain away an effect estimate. We propose that in all observational studies intended to produce evidence for causality, the E-value be reported, or some other sensitivity analysis be used. We suggest calculating the E-value for both the observed association estimate (after adjustments for measured confounders) and for the limit of the confidence interval closest to the null. If this were to become standard practice, the ability of the scientific community to assess evidence from observational studies would be improved considerably, and ultimately, science would be strengthened.