Missing data mechanisms

Definition

Let $Y$ be the complete data, a random variable with values in a product space, split as $Y = (Y_{\mathrm{obs}}, Y_{\mathrm{mis}})$ into its observed and its missing parts. Let $R$ be the missingness indicator: $R_j = 1$ if entry $j$ of $Y$ is observed and $R_j = 0$ otherwise. The missing data mechanism is the conditional distribution of $R$ given $Y$, written $p(r \mid y; \phi)$ with a parameter $\phi$.

1. The data are missing completely at random (MCAR) if $p(r \mid y; \phi) = p(r; \phi)$ for all $y$, that is, $R$ does not depend on $Y$.
2. The data are missing at random (MAR) if $p(r \mid y_{\mathrm{obs}}, y_{\mathrm{mis}}; \phi) = p(r \mid y_{\mathrm{obs}}; \phi)$ for all $y_{\mathrm{mis}}$, that is, $R$ depends on $Y$ only through the observed part.
3. The data are missing not at random (MNAR) if they are not MAR.