Distinguishing “missing at random” and “missing completely at random”

Missing at random (MAR) and missing completely at random (MCAR) are ignorability conditions—when they hold, they guarantee that certain kinds of inferences may be made without recourse to complicated missing-data modeling. In this article we review the definitions of MAR, MCAR, and their recent generalizations. We apply the definitions in three common incomplete-data examples, demonstrating by simulation the consequences of departures from ignorability. We argue that practitioners who face potentially non-ignorable incomplete data must consider both the mode of inference and the nature of the conditioning when deciding which ignorability condition to invoke.