The full-data distribution is assumed to be identical to the population distribution, such that the population marginal distribution of $x$ is given by $p_{j}^{\text{pop}} = \frac{n_{j+}}{n_{++}}$. The data generating model is $$\text{logit}\left[p\left(y=1\mid x\right)\right] = \beta_{0} + \beta_{x}x,$$ whose parameters can be written in terms of cell counts, $\beta_{0} = \text{ln}\left(\frac{n_{01}}{n_{00}}\right)$ and $\beta_{x} = \text{ln}\left(\frac{n_{11}n_{00}}{n_{01}n_{10}}\right)$.

\[tab:analytic\_study\_full\_data\]

                         $y=0$      $y=1$     $\sum_{j=0}^{1}x$
  -------------------- ---------- ---------- -------------------
  $x=0$                 $n_{00}$   $n_{01}$       $n_{0+}$
  $x=1$                 $n_{10}$   $n_{11}$       $n_{1+}$
  $\sum_{k=0}^{1} y$    $n_{+0}$   $n_{+1}$       $n_{++}$

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

\[tab:analytic\_study\_r=1\]

                                  $y=0 \mid r=1$          $y=1 \mid r=1$       $\sum_{j=0}^{1} x\mid r=1$   Population
  --------------------------- ----------------------- ----------------------- ---------------------------- ------------
  $x=0 \mid r=1$               $n_{00}^{\text{obs}}$   $n_{01}^{\text{obs}}$     $n_{0+}^{\text{obs}}$       $n_{0+}$
  $x=1 \mid r=1$               $n_{10}^{\text{obs}}$   $n_{11}^{\text{obs}}$     $n_{1+}^{\text{obs}}$       $n_{1+}$
  $\sum_{k=0}^{1}y\mid r=1$    $n_{+0}^{\text{obs}}$   $n_{+1}^{\text{obs}}$     $n_{++}^{\text{obs}}$     
  $\sum_{k=0}^{1}y\mid r=0$    $n_{+0}^{\text{mis}}$   $n_{+1}^{\text{mis}}$     $n_{++}^{\text{mis}}$     

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

\[tab:analytic\_study\_selections\]

[lcc]{}

  ---------------------------------------------------------
  Linear predictor of selection model
  $\text{logit}\left[p\left[(r=1 \mid x, y\right)\right]$
  ---------------------------------------------------------

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

&

  ----------------------------
     Selection probability
   $p\left(r_{jk} = 