I found this source https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/1471-2288-9-57 and would appreciate any comments on the followed procedure:
I calculated the mean difference in proportions $d_{mean}$ over all $m = 50$ imputed data sets, it's approximately $0.17$.
I calculated the mean within-variance as $$W = \frac{1}{m}\sum _{i=1}^{m} U_i = 0.008687747$$ where $U_i$ are the 50 variances of the 50 single data sets.
I calculated the between-variance $$B = \frac{1}{m - 1}\sum _{i=1}^{m} (d_i - d_{mean})^2 = 0.001570531$$ where $d_i$ are the 50 differences of the proportions of the 50 single data sets.
I calculated the total variance as $$T = W + (1 + \frac{1}{m}) * B = 0.01028969$$
The relative increase in variance was calculated as
$$r = \frac{(1 + \frac{1}{m}) * B}{W} = 0.1843909$$
- The degrees of freedom where calculated as
$$v = (m - 1)(1 + \frac{1}{r})^2 = 2021.65655$$
- Finally, the Wald-test-statistic was calculated as
$$Wald = \frac{(d_{H_0} - d_{mean})^2}{T} = \frac{(0 - d_{mean})^2}{T} = 2.768728$$
$d_{H_0} = 0$ 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.