Choosing $m$ value when using multiple imputation (MI) Multiple imputation creates $m$ new imputed datasets by taking each missing value and replacing it by analyzing the $m$ imputed values (for example: using the mean). Is there a rule of thumb or a method (and I don't mean to Grid Search) to choose the ideal $m$? I am aware of the trade-off regarding the running time.
I have been searching in the literature, but in the papers that I read, they mention the $m$ value they used and do not explain why they chose it. For example:

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*Bennet (2001) recommends choosing $m$ between 5 and 10

*Hawthorne and  Elliott (2005) claim $m$ value should be  usually
less than 10

*Janesn et al. (2010) chose $m = 10$

*Royston et al. recommend choosing $m = 3$ or $m = 5$

*Schafer and Graham (2002) claim: with MI we do not need a large number of repetitions for precise estimates.
Rubin (1987) showed that the efficiency of an estimate based on $m$ imputations, relative to one based on an infinite number, is $(1 + \lambda/m)^-1$, where $\lambda$ is the rate of missing information. In many practical applications, we have found that $m = 20$ imputations can effectively do this. Once a procedure for managing multiple versions of the data has been established, the additional time and effort required to handle $m = 20$ versions rather than $m = 10$ is often of little consequence.

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References:
Janssen, K. J., Donders, A. R. T., Harrell Jr, F. E., Vergouwe, Y., Chen, Q., Grobbee, D. E., & Moons, K. G. (2010). Missing covariate data in medical research: to impute is better than to ignore. Journal of clinical epidemiology, 63(7), 721-727.
Hawthorne, G., Hawthorne, G., & Elliott, P. (2005). Imputing cross-sectional missing data: Comparison of common techniques. Australian and New Zealand Journal of Psychiatry, 39(7), 583-590.
Royston, P. (2004). Multiple imputation of missing values. The Stata Journal, 4(3), 227-241.
Bennett, D. A. (2001). How can I deal with missing data in my study?. Australian and New Zealand journal of public health, 25(5), 464-469.
Schafer, J. L., & Graham, J. W. (2002). Missing data: our view of the state of the art. Psychological methods, 7(2), 147.‏‏‏
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I believe our current best practice is to use the two-step procedure described in von Hippel (2020) and his Statistical Horizons article, which is to estimate the fraction of missing information (FMI), which is distinct from the proportion of observations that are missing, and input that into the formula to compute the required number of imputations given the FMI and a user-supplied measure of the variability of the standard error estimate. That way, you can choose how to manage the tradeoff between time spent imputing and the precision of the resulting estimate of the standard error of your quantity of interest.
This methodology is also implemented in the R package howManyImputations.
