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 precise mechanism of missingness and making distributional assumptions. Bias arises when treatment effects differ between retained and nonretained participants or, among retained participants, the estimate is biased because conditioning on retention has induced a noncausal path between the treatment and outcome. We thus bound the overall treatment effect on the difference scale by specifying: 1) the unobserved treatment effect among nonretained participants; and 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 nonretained participants). As an analog to the E-value for confounding, we propose the M-value, which represents, for a specified treatment effect among nonretained 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.

Automated summary

We propose sensitivity analyses for complete-case estimates of treatment effects to address biases caused by non-random missing data. These analyses use simple summary data and avoid distributional assumptions. The proposed methods bound the overall treatment effect by considering unobserved treatment effects among nonretained participants and the strengths of confounding associations among retained participants. The introduction of the M-value as an analog to the E-value provides a measure of the strength of confounding associations required to mitigate the treatment effect. These methods can help evaluate the robustness of complete-case analyses to potential biases due to missing data.