the data generating mechanism in repeated simulations.

    1.  Simulate $n^{\text{ex}} = 10\,000$ (case 2) or $1\,000$ (case 3) complete values of the binary $0/1$ covariate $x$ from the model $$x \sim \text{Bernoulli}\left(p_{x}^{\text{pop}} = 0.7\right);$$

    2.  Obtain the sample proportion $\hat{p}_{x}^{\text{pop}}$ of $x$, which is an unbiased estimate of the population proportion $p_{x}^{\text{pop}}$;

2.  Simulate $n=5\,000$ complete values of the binary $0/1$ covariate $x$ and binary $0/1$ covariate $y$ from the models $$\begin{aligned}
            &x \sim \text{Bernoulli}\left(p_{x}^{\text{pop}} = 0.7\right); \nonumber \\
            &\text{logit}\left[p\left(y=1 \mid x\right)\right] = \beta_{0} + \beta_{x}x \label{eq:moi_sim2},
        \end{aligned}$$ where $\beta_{0}$ and $\beta_{x}$ are arbitrarily set to $\text{ln}\left(0.5\right)$ and $\text{ln}\left(1.5\right)$, respectively. The same values of the $\beta$ coefficients are used throughout to make bias comparable across all simulation settings;

3.  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. For M4, $\alpha_{y} = 1.5$ and $\alpha_{x} = -1.5$ are used. In 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 $0.75$ are used;

4.  For $i=1, \ldots, 5\,000$, 