I found this source https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/1471-2288-9-57 and would appreciate any comments on the followed procedure:

1. I calculated the mean difference in proportions $diff_{mean}$ over all $m = 50$ imputed data sets, it's approximately $0.169$.

2. I calculated the mean within-variance as $$\overline{U} = \frac{1}{m}\sum _{i=1}^{m} U_i = 0.093208$$ where $U_i$ are the 50 variances of the 50 single data sets, calculated as

$$U_i = \sqrt{p_{i} * (1 - p_{i}) * (\frac{1}{n_1} + \frac{1}{n_2})}$$

($p_i$ being the 50 proportions of successes in the 50 data sets; $n_1$ and $n_2$ are the group sizes)

3. I calculated the between-variance
$$B = \frac{1}{m - 1}\sum _{i=1}^{m} (diff_i - diff_{mean})^2 \\
= \frac{1}{50 - 1}\sum _{i=1}^{50} (diff_i - 0.169)^2 \\
= 0.001570529$$
where $diff_i$ are the 50 differences of the proportions of the 50 single data sets.

4. I calculated the total variance as $$T = \overline{U} + (1 + \frac{1}{m}) * B = 0.09480994$$

5. The relative increase in variance was calculated as

$$r = \frac{(1 + \frac{1}{m}) * B}{\overline{U}} = 0.01718672$$

6. The degrees of freedom where calculated as

$$v = (m - 1)(1 + \frac{1}{r})^2 = 171637.2$$

7. Finally, the Wald-test-statistic was calculated as

$$Wald = \frac{(diff_{H_0} - diff_{mean})^2}{T}
= \frac{(0 - diff_{mean})^2}{T} = 0.3004891$$

$diff_{H_0} = 0$ as the null hypothesis states there is no difference in proportions.

Is the chosen procedure correct in my case? Or did I miss something? What would be the correct application of Rubin's rule?

Marshall, A., Altman, D. G., Holder, R. L., & Royston, P. (2009). Combining estimates of interest in prognostic modelling studies after multiple imputation: current practice and guidelines. BMC Medical Research Methodology, 9, 57.