Bias factor, maximum bias and the E-value: insight and extended applications

Abstract

Unmeasured confounding can bias the relationship between exposure and outcome. Sensitivity analyses generate bias-adjusted measures but these are not much used; this may change with the availability of the E-value (for evidence for causality in observational studies), appealing for its ease of calculation. However, as currently proposed, the E-value has some practical limitations that may reduce its use. We first provide some insight into the relationship between two established measures for unmeasured confounding: 'the bias factor' and the maximum value this bias factor can take ('the B bias'). These measures are the statistical foundation for the E-value. We use them to develop new E-value formulas for situations when it is not currently applicable such as e.g. when, not unusually, a negative relation between unmeasured confounder and outcome and a positive one with exposure are postulated. We also provide E-values on the odds ratio scale because, currently, even when using the odds ratio as the study measure in the calculation of E-value, the result is to be interpreted as a relative risk, which is somewhat inconvenient. The additional formulas for the E-value measure make it applicable in all possible scenarios defined by the combined directions between unmeasured confounder and both the exposure and outcome. In addition, E-value measures can now be interpreted as odds ratios if the observed results are reported on the same scale. The E-value is part of newer sensitivity analyses methods for unmeasured confounding. We provide insight into its structure, underscoring its advantages and limitations, and expand its applications.