Clarifying missing at random and related definitions, and implications when coupled with exchangeability

We clarify the key concept of missingness at random in incomplete data analysis. We first distinguish between data being missing at random and the missingness mechanism being a missing-at-random one, which we call missing always at random and is more restrictive.We further discuss how, in general, neither of these conditions is a statement about conditional independence. We then consider the implication of the more restrictive missing-always-at-random assumption when coupled with full unit-exchangeability for the matrix of the variables of interest and the missingness indicators: the conditional distribution of the missingness indicators for any variable that can have a missing value can depend only on variables that are always fully observed. We discuss implications of this for modeling missingness mechanisms.