MISSING AT RANDOM AND IGNORABILITY FOR INFERENCES ABOUT INDIVIDUAL PARAMETERS WITH MISSING DATA

In a landmark paper, Rubin (1976 Biometrika) showed that the missing data mechanism can be ignored for likelihood-based inference about parameters when (a) the missing data are missing at random (MAR), in the sense that missingness does not depend on the missing values after conditioning on the observed data, and (b) distinctness of the parameters of the data model and the missing-data mechanism, that is, there are no a priori ties, via parameter space restrictions or prior distributions, between the parameters of the data model and the parameters of the model for the mechanism. Rubin (1976) described (a) and (b) as the “weakest simple and general conditions under which it is always appropriate to ignore the process that causes missing data”. However, it is important to note that these conditions are not necessary for ignoring the mechanism in all situations. We propose conditions for ignoring the missing-data mechanism for 2 likelihood inferences about subsets of the parameters of the data model. We present examples where the missing data are ignorable for some parameters, but the missing data mechanism is missing not at random (MNAR), thus extending the range of circumstances where the missing data mechanism can be ignored.