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  "body": "\n## Definition\n\nLet $Y$ be the complete data, a [[random variable|0DKRR22AEQ]] with values in a product space, split as $Y = (Y_{\\mathrm{obs}}, Y_{\\mathrm{mis}})$ into its observed and its missing parts.\nLet $R$ be the missingness indicator: $R_j = 1$ if entry $j$ of $Y$ is observed and $R_j = 0$ otherwise.\nThe *missing data mechanism* is the conditional [[distribution|0QB913WM5E]] of $R$ given $Y$, written $p(r \\mid y; \\phi)$ with a parameter $\\phi$.\n\n1. 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 $Y$.\n2. 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.\n3. The data are *missing not at random* (MNAR) if they are not MAR.\n\n## Remarks\n\nMCAR implies MAR.\n\nExample: let $Y = (Y_1, Y_2)$, where $Y_1$ (age) is always observed and $Y_2$ (income) is missing with probability $h(Y_1)$ for a function $h$ that is not constant.\nThen $p(r \\mid y)$ depends only on $y_1 = y_{\\mathrm{obs}}$, so the data are MAR, but not MCAR.\n\nNon-example: let $Y_2$ be missing with probability $h(Y_2)$ for a function $h$ that is not constant.\nThen $p(r \\mid y)$ depends on the missing value itself, so clause 2 fails and the data are MNAR.\n\nAssume the data are MAR and the parameters $\\theta$ of the data model and $\\phi$ of the mechanism are distinct, that is, they vary in a product set.\nThen the mechanism is called *ignorable*.\nIn that case, the [[likelihood|05FSTXM92W]] of $\\theta$ based on $(y_{\\mathrm{obs}}, r)$ is proportional to the likelihood based on $y_{\\mathrm{obs}}$ alone.\n",
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