Formula for multiple imputation (created 2009-07-24)

This page is moving to a new website.

I'm working on a project that involves multiple imputation, and I may have to program some of the work myself. I can use the R package MICE to generate the imputed data sets, but then I have to use a mixed linear model rather than a linear model. How do I combine the estimates from the multiple imputed data sets? The estimate is just the average of the individual estimates, but what about the standard error?

I found a simple answer in one of my books. If I run the model and get m estimates (rk) and m standard errors (sk) then the formula for the standard error of the combined estimate is:

This is found on page 30 of Allison, P (2002). Missing Data. Sage Publications, Thousand Oaks, CA.

You can find a similar formula at

Look for the question "How do I combine the results across the multiply imputed sets of data?"

When I have a chance, I'll illustrate how this works with some real data.