Reply to Sjölander and VanderWeele on ‘Bias factor, maximum bias and the E-value’

Sjölander and VanderWeele do not report faults in our demonstration of additional formulas for the E-value given the stated assumptions and the conventional relative risk definitions adopted,1 and likewise, for our extension of the E-value interpretation on the odds ratio scale. In their letter, Sjölander and VanderWeele say that ‘under the original definitions (…), RRXU and RRUY are always ≥1’ and provide definitions of the RR equations for the E-value (their equation 5) that make it valid regardless of the direction of the association between U and (X, Y).1 These RR equations are also found in Ding and VanderWeele,2 but not in the paper introducing the E-value.3 Ideally, we would have worked out and reported the general applicability of the unique E-value formula using the more complex definitions of the RR as found in a paper previous to the one formally introducing the E-value.2 However we had a different goal, which was to facilitate interpretation by generating additional E-values directly interpretable as RRUY and RRXU <1 and using different, more familiar definitions of the RR formulas. As a result, under the stated assumptions, our equations simplify the sensitivity analysis, as E-values (RRUY and RRXU) are immediately interpretable in the direction postulated by the investigator. Sjölander and VanderWeele suggest that it is incorrect to assume our conclusions hold for a different set of definitions. However, nowhere in the paper did we make a statement to that effect. On the contrary, we say that ‘An alternative way to introduce the B equation is to derive it from the BF equation’,4 which is not the approach used by Ding and VanderWeele.2 Moreover, the BF uses simpler but different RR formulas than the ones used by Ding and VanderWeele.2 We conventionally assumed that our conclusions applied with the proposed methods, without having to specify that this may not be the case with other methods. Finally, Sjölander and VanderWeele believe our assumptions are ‘fairly strong’ and would often be violated in real scenarios. Most methods have limitations and, for ours, these limitations remain to be shown. The E-value has its own limitations, one requiring the same magnitude of effect of the unmeasured confounder on exposure and outcome, but it remains an attractive method to use. Our equations are valid for a binary U, with the risk ratio RRXU defined as P(U|X = 1)/P(U|X = 0), and the RRUY defined as common to X = 0 and X = 1. These choices were made explicit in our paper and were used by many others,5–9 including Schlesselman.7 Ding and VanderWeele2 compare their results with Schlesselman’s, concluding that Schlesselman’s results are valid but not as general as their own, since the former require extra assumptions. We repeatedly mention that our approach uses more assumptions, thereby implying more simplicity but less flexibility. We have no qualms in recognizing that the cost of simplicity is less applicability because of a greater likelihood of violated assumptions; however, such an observation is not specific to our work. It remains to be seen if simple methods for more restricted situations are less useful than more general but complex ones.

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