Missing data: a unified taxonomy guided by conditional independence

Recent work (Seaman et al., 2013; Mealli & Rubin, 2015) attempts to clarify the not always well-understood difference between realised and everywhere definitions of missing at random (MAR) and missing completely at random. Another branch of the literature (Mohan et al., 2013; Pearl & Mohan, 2013) exploits always-observed covariates to give variable-based definitions of MAR and missing completely at random. In this paper, we develop a unified taxonomy encompassing all approaches. In this taxonomy, the new concept of ‘complementary MAR’ is introduced, and its relationship with the concept of data observed at random is discussed. All relationships among these definitions are analysed and represented graphically. Conditional independence, both at the random variable and at the event level, is the formal language we adopt to connect all these definitions. Our paper covers both the univariate and the multivariate case, where attention is paid to monotone missingness and to the concept of sequential MAR. Specifically, for monotone missingness, we propose a sequential MAR definition that might be more appropriate than both everywhere and variable-based MAR to model dropout in certain contexts.