Framework of multiple imputation I read this paper about "Multiple Imputation For Missing Data: What Is It And How Can I Use It?"

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*Does any one have information about step by step framework of
multiple imputation in general ?


*Do different multiple imputation packages in R  such as
(MICE,MI,Amelia ,aregimput) have the same framework or no ?
 A: I think your question is too broad, and better descriptions are available in the manuals for the packages you have cited. that being said..
The goto reference for the math is Rubin (1987)

Rubin, D. B. (1987). Multiple imputation for nonresponse in surveys, John Wiley & Sons. New York.

The approach is generally the same for the packages you mention (although I dont know aregimput). 1) Missing variables from different data types (ratio, integer, ordinal, nominal) are calculated based on a corresponding regression technique (e.g., linear regression, cumulative link regression, logistic regression). Basically the values you have are used to model the values you are missing. 2) These calculations start with random weights and are adjusted based on their error predicting the known values. 3) Chains of these calculations (weight adjustments, missing value predictions) are run until some convergence criterion is reached.
This paper was very helpful for me when learning the ropes in mi:

Su, Y. S., Yajima, M., Gelman, A. E., & Hill, J. (2011). Multiple imputation with diagnostics (mi) in R: Opening windows into the black box. Journal of Statistical Software, 45(2), 1-31.

Edit: one last suggestion that was quite helpful for me:

Schafer, J. L. (1997). Analysis of incomplete multivariate data. CRC press.

A: Multiple imputation creates $m > 1$ complete datasets. The $m$ results are pooled into a final point estimate plus standard error by simple pooling rules (“Rubin’s rules”), that means multiple imputation creates several complete versions of the data by replacing the missing values by plausible data values  and these plausible values are drawn from a distribution specifically modelled for each missing entry.
The analysis starts with observed, incomplete data.
The second step is to estimate the parameters of interest from each imputed dataset,  the results will differ because their input data differ, these differences are results of  the uncertainty about what value to impute.
The last step is to pool the $m$ parameter estimates into one estimate, and to estimate its variance. The variance combines the conventional sampling variance (within-imputation variance) and the extra variance caused by the missing data extra variance caused by the missing data (between-imputation variance).
This Figure portrays $m = 3$ imputed datasets. In practice, $m$ is often taken larger

For more details , See Flexible Imputation of Missing Data by Van Buuren. The micepackage is written also by Van Buuren. So the book also serves as tutorial on the practical application of mice.
