with observed and missing $x$ $$    p\left(x=1\right) = p\left(x=1 \mid r=1\right)p\left(r=1\right) + p\left(x=1 \mid r=0\right)p\left(r=0\right),$$ where $p\left(x=1\right)$ is the population proportion; $p\left(x=1 \mid r=1\right)$ , $p\left(r=1\right)$, and $p\left(r=0\right)$ can be obtained from the observed data. Thus, $p\left(x=1 \mid r=0\right)$ can be solved for as $$\label{eq:p_x=1_partitioned2}
    p\left(x=1 \mid r=0\right) = \frac{p\left(x=1\right) - p\left(x=1 \mid r=1\right)p\left(r=1\right)}{p\left(r=0\right)}.$$ Note that $p\left(x=1 \mid r=0\right)$ can be further written as $$\begin{aligned}
    p\left(x=1 \mid r=0\right) &= \sum_{k=0}^{1}p\left(x=1 \mid y=k, r=0\right)p\left(y=k \mid r=0\right) \nonumber\\
    &= \sum_{k=0}^{1}\text{expit}\left(\theta_{0}^{\text{mis}} + \theta_{y}^{\text{mis}} I\left[y=k\right]\right)\frac{n_{+k}^{\text{mis}}}{n_{++}^{\text{mis}}} \nonumber\\
    &= \frac{1}{n_{++}^{\text{mis}}}\text{expit}\left(\theta_{0}^{\text{mis}} + \theta_{y}^{\text{mis}} I\left[y=k\right]\right)n_{+k}^{\text{mis}}, \label{eq:p_x=1_r=0}\end{aligned}$$ where $I\left[A\right]$ is an indicator function taking values 1 if $A$ is true and 0 otherwise. It is shown earlier that when $x$ is MNAR dependent on either the values of $x$ or both $x$ and $y$, $\theta_{y}^{\text{obs}}=\theta_{y}^{\text{mis}}$; is therefore equal to $$\begin{aligned}
    p\left(x=1 \mid r=0\right) &= \frac{1}{n_{++}^{\text{mis}}}\text{expit}\left(\theta_{0}^{\text{mis}} + \theta_{y}^{\text{obs}} I\left[y=k\right]\right)n_{+k}^{\text{mis}} \\
    &= \frac{1}{n_{++}^{\text{mis}}}\text{expit}\left(\left(\theta_{0}^{\text{obs}} + \delta\right) + \theta_{y}^{\text{obs}} I\left[y=k\right]\right)n_{+k}^{\text{mis}} \\
    &= \frac{1}{n^{\text{mis}}}\sum_{i=1}^{n^{\text{mis}}}\text{expit}\left(\left(\theta_{0}^{\text{obs}} + \delta \right) + \theta_{y}^{\text{obs}}y_{i}\right),\end{aligned}$$ where $\delta$ is the adjustment factor in the intercept of the imputation model for $x$. The value of the calibrated-$\delta$ adjustment can be obtained numerically from and using interval bisection [@Russ1980; @Burden2011] (or any other root-finding method).

When the population marginal distribution of the incomplete 