set $x_{i}$ to missing if $r_{i} = 0$;

5.  Impute missing values in $x$ $M=10$ times using standard MI and calibrated-$\delta$ adjustment MI in turn. For cases 2 and 3, calibrated-$\delta$ adjustment MI is performed as follows.

    1.  Draw a value $\tilde{p}_{x}^{\text{pop}}$ from the normal approximation $N\left(\hat{p}_{x}^{\text{pop}}, \frac{\hat{p}_{x}^{\text{pop}}\left(1-\hat{p}_{x}^{\text{pop}}\right)}{n^{\text{ex}}}\right)$, with values of $n^{\text{ex}} = 10\,000$ (case 2) and $1\,000$ (case 3). This is done by first taking a draw $\tilde{z}$ from the standard normal distribution, $z \sim N\left(0,1\right)$, followed by drawing $\tilde{p}_{x}^{\text{pop}} = \hat{p}_{x}^{\text{pop}} + \tilde{z}\sqrt{\frac{\hat{p}_{x}^{\text{pop}}\left(\hat{p}_{x}^{\text{pop}}\right)}{n^{\text{ex}}}}$;

    2.  Derive the calibrated-$\delta$ adjustment and perform MI according to the algorithm set out in \[subsubsec3.1.1\], using $\tilde{p}_{x}^{\text{pop}}$ as the reference proportion;

6.  For each MI method, fit the analysis model to each completed dataset and combine the results using Rubin’s rules.[@Rubin1987; @Barnard1999]

Step 5 is designed to mimic the full Bayesian sampling process, which is always the aim in proper (or Rubin’s) MI. Again, steps 1–6 are repeated $S=2\,000$ times under each of the four selection models M1–M4, so the same set of simulated independent datasets is used to compare the two MI methods under the same missingness scenario, but a different set of datasets is generated for each missingness scenario.[@Burton2006] The parameters of 