complete records (Table \[tab:analytic\_study\_r=1\]) to obtain the $\theta$ parameter estimates, where $$\theta_{0}^{\text{obs}} = \text{ln}\left(\frac{n_{10}^{\text{obs}}}{n_{00}^{\text{obs}}}\right); \quad \theta_{y}^{\text{obs}} = \text{ln}\left(\frac{n_{11}^{\text{obs}}n_{00}^{\text{obs}}}{n_{01}^{\text{obs}}n_{10}^{\text{obs}}}\right).$$ When $x$ is MCAR or MAR conditional on $y$, we can obtain an unbiased estimate of the association between $x$ and $y$ in the missing data by fitting the above logistic regression imputation model to the complete records. No adjustment is needed in the intercept of the imputation model, and standard MI provides unbiased estimates of the marginal distribution of $x$ as well as the association between $x$ and $y$. We focus on two general MNAR mechanisms described below.

$x$ is MNAR dependent on $x$
----------------------------

Under this missingness mechanism, the posited model for the response indicator $r$ of $x$ is given by $$\label{eq:selection_mnar_x}
        \text{logit}\left[p\left(r=1\mid x\right)\right] = \alpha_{0} + \alpha_{x}x,$$ and the corresponding probabilities of observing $x$ are $$p\left(r=1 \mid x=j\right) = p_{r_{j}} = \text{expit}\left(\alpha_{0} + \alpha_{x}x\right); \quad j=0,1.$$ For imputation model , the log odds ratios of $x=1$ for $y=1$ compared to $y=0$ in the observed and missing data are $$\begin{aligned}
        \left[\theta_{y} \mid r=1\right] &= \theta_{y}^{\text{obs}} = \text{ln}\left(\frac{n_{00}p_{r_{0}}n_{11}p_{r_{1}}}{n_{01}p_{r_{0}}n_{10}p_{r_{1}}}\right) = \text{ln}\left(\frac{n_{00}n_{11}}{n_{01}n_{10}}\right); \\
                \left[\theta_{y} \mid r=0\right] &= \theta_{y}^{\text{mis}} = \text{ln}\left(\frac{n_{00}\left(1-p_{r_{0}}\right)n_{11}\left(1-p_{r_{1}}\right)}{n_{01}\left(1-p_{r_{0}}\right)n_{10}\left(1-p_{r_{1}}\right)}\right) = \text{ln}\left(\frac{n_{00}n_{11}}{n_{01}n_{10}}\right),
    \end{aligned}$$ respectively. Hence, $\theta_{y}^{\text{obs}} = \theta_{y}^{\text{mis}}$, which are also the same 