E-value analogs for bias due to missing data in treatment effect estimates

Background: Complete-case analyses can be biased if missing data are not missing completely at random. Methods: We propose simple sensitivity analyses that apply to complete-case estimates of treatment effects; these analyses use only simple summary data and obviate specifying the mechanism of missingness and making distributional assumptions. Bias arises when: (1) treatment effects differ between retained and non-retained participants; or (2) among non-retained participants, the estimate is biased because conditioning on retention has induced a backdoor path. We thus bound the overall treatment effect on the difference scale by specifying: (1) the unobserved treatment effect among non-retained participants; (2) the strengths of association that unobserved variables have with the exposure and with the outcome among retained participants (``induced confounding associations''). Working with the former sensitivity parameter subsumes certain existing methods of worst-case imputation, while also accommodating less conservative assumptions (e.g., that the treatment is not detrimental even among non-retained participants). We propose analogs to the E-value for confounding that represent, for a specified treatment effect among non-retained participants, the strength of induced confounding associations required to reduce the treatment effect to the null or to any other value. Results: We apply the methods to a published randomized trial on financial incentives for smoking cessation. Conclusion: These methods could help characterize the robustness of complete-case analyses to potential bias due to missing data. The methods can also be used for general selection bias when the probability of selection is known.