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When performing mulitple imputations with the MICE package in R, I found some resources (http://www.statisticalhorizons.com/more-imputations) that recommend around 10 imputations. I was wondering if there is any drawback of e.g. performing 1000 or 10000 imputations as it should lead to more precise results and is computationally easily feasible?

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Depending on your needs it may or may not be a problem regarding computation ressources, but speaking in general it does not help much too impute 10000 times. Note that depending on the implementation, a large number might be a problem considering both computing time and memory needed. Rubin (who invented) the method investigated the question of how many imputations are needed and found that usually a small number suffices.

The governing equation is $$ T_m = (1 + \frac{\gamma}{m})T_{\infty} $$ where $T_m$ is the variance estimate for the imputation with m imputed datasets, $T_\infty$ is the theoretical optimal case of an infinite number of different imputations and $\gamma$ is the rate of missing information. In the case of a single variable without covariates $\gamma$ is actually equal to the fraction of missing values. With covariates and missing values in more than one variable it is a bit more complicated.

Now, for a $\gamma$ of say 30% you get $$T_{10}=1.03T_{\infty}$$ which is already quite close. Since this is the variance your confidence interval for your estimates would then be larger than the theoretical optimal case by a factor of $\sqrt{1.03}$. This additional error is very small and shows that it is not often worth imputing many more times.

You can get additional information here: http://sites.stat.psu.edu/~jls/mifaq.html and in the book Flexible Imputation of Missing Data by van Buuren, which was my primary source for this. Rubin's book Multiple Imputation for Nonresponse in Survey should also contain a (in fact, the original) discussion on this, but I haven't read it.

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