record analysis (CRA) under various missingness mechanisms for $x$.

When the population distribution is ‘known’ {#subsec3.1}
-------------------------------------------

Below we consider the setting where the population distribution of the incomplete variable is obtained from a population census or equivalent, i.e. it is ‘known’. The uncertainty associated with having to estimate the population distribution is explored in \[subsec3.2\].

### Method {#subsubsec3.1.1}

Similar to the analytic study presented in \[sec2\], the analysis model in this simulation study is a logistic regression model for a fully observed binary outcome $y$ on an incomplete binary covariate $x$. Calibrated-$\delta$ adjustment MI is compared to standard MI and CRA under four missingness mechanisms of increase complexity. The data generating mechanism and analysis procedures are as follows.

1.  Simulate $n=5\,000$ complete values of the binary $0/1$ covariate $x$ and binary $0/1$ outcome $y$ from the following models $$\begin{aligned}
            &x \sim \text{Bernoulli}\left(p_{x}^{\text{pop}} = 0.7\right); \nonumber\\
            & \text{logit}\left[p\left(y=1 \mid x\right)\right] = \beta_{0} + \beta_{x}x, \label{eq:moi_sim1}
        \end{aligned}$$ 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 $\beta$ parameters are used throughout to make bias comparable across all simulation settings. This sample size is chosen to minimise the issue of small-sample bias associated with 