the logistic regression [@Nemes2009];

2.  Simulate a binary indicator of response $r$ of $x$ from each of the selection models M1–M4 (Table \[tab:analytic\_study\_selections\]). Values of $1.5$ and $-1.5$ are chosen for $\alpha_{y}$ and $\alpha_{x}$ in M2 and M3, respectively, to reflect strong odds ratios (OR) of observing $x$ (OR $= 4.5$ and $0.2$, respectively). For M4, $\alpha_{y} = 1.5$ and $\alpha_{x} = -1.5$ are chosen as bias in the three MI methods under evaluation is likely to be apparent with these coefficients predicting missingness in $x$. For all selection models, $\alpha_{0}$ is altered to achieve approximately $45\%$ missing $x$. For M1, $\alpha_{0}$ is calculated directly as $\text{ln}\left(\frac{0.55}{0.45}\right)$; for M2–M4, $\alpha_{0} = -0.2; 1.35$; and $0.75$ appear to work well;

3.  For $i=1 \ldots 5\,000$, set $x_{i}$ to missing if $r_{i}=0$;

4.  Impute missing values in $x$ $M=50$ times using standard MI and calibrated-$\delta$ adjustment MI in turn;

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

Steps 1–5 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 three MI methods under the 