the between-imputation variance of Rubin’s variance estimator.

An extension of the simulation study presented in section \[subsec3.1\] is conducted to explore this setting.

### Method {#subsubsec3.2.1}

This extended simulation study of a fully observed binary outcome y and a partially observed binary covariate $x$ follows the same method described in \[subsubsec3.1.1\], except that two variations of the population proportions of $x$ are evaluated in the imputation step of calibrated-$\delta$ adjustment MI. The reference distribution is assumed to either come from a census or equivalent (case 1), or be estimated in an external dataset of larger size (case 2) or smaller size (case 3) than the study sample.

Suppose that in an external dataset of size $n^{\text{ex}}$ which comes from the same population as the study sample, the sample proportion $\hat{p}_{x}^{\text{pop}}$ provides an unbiased estimate of the population proportion $p_{x}^{\text{pop}}$. Assuming that the sampling distribution of the sample proportions is approximately normal, its standard error is given by $$\text{SE}\left(\hat{p}_{x}^{\text{pop}}\right) = \sqrt{\frac{\hat{p}_{x}^{\text{pop}}\left(1-\hat{p}_{x}^{\text{pop}}\right)}{n^{\text{ex}}}}.$$ The data generating mechanism and analysis procedures are as follows.

1.  For cases 2 and 3, the following two steps are performed to incorporate the sampling behaviour of $\hat{p}_{x}^{\text{pop}}$, which is estimated in an external dataset of size $n^{\text{ex}}$, into 