$\theta_{y}^{\text{obs}} = \theta_{y}^{\text{mis}}$, since $$\begin{aligned}
    \theta_{y}^{\text{mis}} - \theta_{y}^{\text{obs}} &= \text{ln}\left(\frac{\left(1-p_{r_{00}}\right)\left(1-p_{r_{11}}\right)p_{r_{01}}p_{r_{10}}}{\left(1-p_{r_{01}}\right)\left(1-p_{r_{10}}\right)p_{r_{00}}p_{r_{11}}}\right)\\
    &= \text{ln}\left(\frac{\text{exp}\left(\alpha_{0} + \alpha_{x}\right)\text{exp}\left(\alpha_{0} + \alpha_{y}\right)}{\text{exp}\left(\alpha_{0}\right)\text{exp}\left(\alpha_{0} + \alpha_{x} + \alpha_{y}\right)}\right) \\
    &= 0.
    \end{aligned}$$ The log odds of $x=1$ for $y=0$ in the observed and missing data are given by $$\begin{aligned}
        \theta_{0}^{\text{obs}} = \text{ln}\left(\frac{n_{10}p_{r_{10}}}{n_{00}p_{r_{00}}}\right); \\
        \theta_{0}^{\text{mis}} = \text{ln}\left(\frac{n_{10}\left(1-p_{r_{10}}\right)}{n_{00}\left(1-p_{r_{00}}\right)}\right),
    \end{aligned}$$ which implies that the correct adjustment in the imputation model’s intercept should be $$\begin{aligned}
        \theta_{0}^{\text{mis}} - \theta_{0}^{\text{obs}} &= \text{ln}\left(\frac{\left(1-p_{r_{10}}\right)p_{r_{00}}}{\left(1-p_{r_{00}}\right)p_{r_{10}}}\right) \\
        &= \text{ln}\left(\frac{\text{exp}\left(\alpha_{0}\right)}{\text{exp}\left(\alpha_{0}+\alpha_{x}\right)}\right) \\
        &= -\alpha_{x},
    \end{aligned}$$ which is again minus the log odds ratio of observing $x$ in .

Derivation of the calibrated-$\delta$ adjustment
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The analytic calculations above confirm that in a $2\times 2$ contingency table setting, appropriately adjusting the intercept of the imputation model for the covariate $x$ can sufficiently correct bias introduced by MNAR mechanisms under which missingness in $x$ depends on either its values or both its values and the outcome (M3 and M4). The population distribution of $x$ can be used to calculate the correct adjustment in the imputation model’s intercept. This adjustment is referred to as the *calibrated-$\delta$ adjustment* to clarify its relationship to van Buuren et al.’s $\delta$ adjustment.[@VanBuuren1999]

The probability of $x=1$ can be written in terms of the conditional probabilities among subjects 