1\right)$
  ----------------------------

  : Note: $r$: response indicator of $x$; $j$ and $k$: index categories of $x$ and $y$, respectively; $j, k$ take values $0/1$.

&

  -------
   Label
  -------

  : Note: $r$: response indicator of $x$; $j$ and $k$: index categories of $x$ and $y$, respectively; $j, k$ take values $0/1$.

\
$\alpha_{0}$ & $p_{r}$ & M1\
$\alpha_{0} + \alpha_{y}y$ & $p_{r_{k}}$ & M2\
$\alpha_{0} + \alpha_{x}x$ & $p_{r_{j}}$ & M3\
$\alpha_{0} + \alpha_{x}x + \alpha_{y}y$ & $p_{r_{jk}}$ & M4\

\

In addition, suppose that $y$ is fully observed, while some data in $x$ are set to missing (i.e. the sample contains no individuals with missing $y$ and observed $x$, Table \[tab:analytic\_study\_r=1\]). Let $r$ be the response indicator taking values $1$ if $x$ is observed and $0$ if $x$ is missing. Four different missingness mechanisms considered for $x$ and the corresponding selection models are presented in Table \[tab:analytic\_study\_selections\]. Observed cell counts, $n_{jk}^{\text{obs}}$, can be written as a product of the full-data cell counts, $n_{jk}$, and the cell-wise probability of observing $x$, $p_{r_{jk}}$, such that $n_{jk}^{\text{obs}} = n_{jk}p_{r_{jk}}$.

To perform standard MI of missing values in $x$, an imputation model $$\label{eq:standard_imp_model} 
        \text{logit}\left[p\left(x=1\mid y\right)\right] = \theta_{0} + \theta_{y}y,$$ is fitted to the $n_{++}^{\text{obs}}$ 