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I would appreciate any help to understand the statistical difference between using random effects and Rubin's rule to obtain pooled parameter estimates from multiply imputed datasets.

For example, if .imput is a variable representing each imputation stage, chl the outcome variable, age the predictor, and data the multiply imputed dataset, what would be the rationale to choose results_Random over results_Rubinin the example below?

data <- mice(nhanes, seed = 23109)
results_Random <- lm(chl~ age + (age |.imput), data=data)
results_Rubin <- pool(with(data, lm(chl ~ age))))

While I understand the complexity of the law of total variance needed to pool the variance, it seems to me that that would not be an issue using Bayesian statistics because the uncertainty around the parameter estimate is not estimated using standard error anymore, it is instead based on quantiles of the posterior distribution of pooled estimates of the parameter.

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  • $\begingroup$ " the uncertainty around the parameter estimate is not estimated using standard error anymore" is not true, imagine that no data (or similarly hardly any) were missing, you performed multiple imputation. Obviously your uncertainty about parameters is not suddenly 0, just because the estimates don't vary across multiple imputations. If you look at the formula for Rubin's rule you'll see that the SEs in each imputation also enter. $\endgroup$ – Björn Feb 5 at 5:40

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