The M-value: A simple sensitivity analysis for bias due to missing data in treatment effect estimates.

Complete-case analyses can be biased if missing data are not missing completely at random. 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 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 on average even among non-retained participants). As an analog to the E-value for confounding, we propose the M-value, which represents, 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. These methods could help characterize the robustness of complete-case analyses to potential bias due to missing data.