Imputation without nightMARs: Graphical criteria for valid imputation of missing data

By revisiting imputation from the modern perspective of missing data graphs, we correct common guidance about which auxiliary variables should be included in an imputation model. We propose a generalized definition of missingness at random (MAR), called “z-MAR'', which makes explicit the set of variables to be analyzed and a separate set of auxiliary variables that are included in the imputation model, but not analyzed. We provide a graphical equivalent of z-MAR that we call the “m-backdoor criterion”. In a sense we formalize, a standard imputation model trained on complete cases is valid for a given analysis if and only if the m-backdoor criterion holds. As the criterion indicates, the standard recommendations to use all available auxiliary variables, or all that are associated with missingness status, can lead to invalid imputation models and biased estimates. This bias arises from collider stratification and can occur even with non-causal estimands. Instead, the set of auxiliary variables should be restricted to those that affect incomplete variables or missingness indicators. These auxiliary variables will always suffice to have a valid imputation model, if such a set does exist among the candidate auxiliary variables. Applying this result does not require full knowledge of the graph.