new parameters $\tilde{\boldsymbol{\theta}}$ from the large-sample normal approximation $N(\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{U}})$ of their posterior distribution, assuming non-informative priors;

3.  Draw a new probability of observing $x$, $\tilde{p}_{r}$, from the normal approximation $N\left(\hat{p}_{r}, \frac{\hat{p}_{r}\left(1-\hat{p}_{r}\right)}{n}\right)$, where $\hat{p}_{r}$ is the sample proportion of the response indicator of $x$, $\hat{p}_{r} = \frac{n_{++}^{\text{obs}}}{n_{++}}$;

4.  Draw a new probability of observed $x=1$, $\tilde{p}_{x}$, from the normal approximation $N\left(\hat{p}_{x}, \frac{\hat{p}_{x}\left(1-\hat{p}_{x}\right)}{n}\right)$, where $\hat{p}_{x}$ is the observed proportion of $x=1$, $\hat{p}_{x} = \frac{n_{1+}^{\text{obs}}}{n_{++}^{\text{obs}}}$;

5.  Derive the value of the calibrated-$\delta$ adjustment from the equation $$\frac{1}{n^{\text{mis}}}\sum_{i=1}^{n^{\text{mis}}}\text{expit}\left(\left(\tilde{\theta}_{0} + \delta\right) + \tilde{\theta}_{y}y_{i}\right) = \frac{p_{x}^{\text{pop}} - \tilde{p}_{x}}{\tilde{p}_{r}},$$ where $p_{x}^{\text{pop}}$ is the probability of $x=1$ in the population;

6.  Fit the logistic regression imputation model for $x$ conditional on $y$ (in step 1) to the complete records with the intercept adjustment fixed to $\delta$ to obtain the maximum likelihood estimates of the imputation models’ parameters $\hat{\boldsymbol{\theta}}$ and their asymptotic sampling variance $\widehat{\boldsymbol{U}}$;

7.  Draw new parameters $\dot{\boldsymbol{\theta}}$ from the large-sample normal approximation $N(\widehat{\boldsymbol{\theta}}, \widehat{\boldsymbol{U}})$ of their posterior distribution, assuming non-informative priors;

8.  Draw imputed values for $x$ from the above logistic regression imputation model, using the newly drawn parameters $\dot{\boldsymbol{\theta}}$ and calibrated-$\delta$ adjustment.

### Results {#subsubsec3.1.2}

Results of the simulation study are summarised graphically in 