# Pooling the results of random hot-deck imputation

I am using random hot-deck imputation on a repeated measures dataset.

I am tempted to use Rubin's rules for pooling the results of multiple imputation, in particular for regression coefficients. Intuitively it seems the average of the coefficient estimates could be used, but I really don't have any insight about pooling the standard errors and I have not seen any literature about this (Little & Rubin's book seems to be silent on the matter, unless I have missed something)

How can I pool regression coefficient estimates and their standard errors when performing random hot-deck imputation ? Pragmatic advice, theoretical justification and/or references to the literature would be most welcome.

Edit: To clarify, by "random hot deck imputation", I mean:

Hot deck imputation involves replacing missing values of one or more variables for a non-respondent (called the recipient) with observed values from a respondent (the donor) that is similar to the non-respondent with respect to characteristics observed by both cases. In some versions, the donor is selected randomly from a set of potential donors, which we call the donor pool; we call these methods random hot deck methods

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3130338/

-
Please define the process of random hot-deck imputation in your question. –  ttnphns Jan 14 '13 at 13:47
@ttnphns I have edited the question accordingly. –  Robert Long Jan 14 '13 at 15:57

## 1 Answer

One could be tempted to average the standard errors over the imputations, but this does not produce correct statistical inferences. The average standard error ignores the between-imputation variance. Moreover, it is known that the pooled parameter estimate has a t distribution, which is different from the normal distribution implied by the average standard error.

The correct procedure is described in Little and Rubin (2002), pages 86-87, and is sometimes referred to as "Rubin's rule". Chapter 3 of Rubin (1987) contains the justification of the procedure. The method can be applied for estimates that are approximately normally distributed over repeated samples, which covers a lot of statistics used in practice.

The pooling procedure does not depend on the imputation method used. Of course, the results only make sense if the imputation method makes sense. You could check whether that is the case on your data given knowledge you might have about the causes for the missing data.

-
Thanks a lot - actually I have implemented Rubin's rules now. It was very hard to impute these data because the main variable needing imputation (the outcome) is heavily zero-inflated. The analysis model is a zero inflated negative binomial mixed model (fitted with glmmADMB). This is why random hot-deck was used. Actually, I have an open question about the imputation method here: stats.stackexchange.com/questions/47674/… –  Robert Long Jan 17 '13 at 12:21
Funny, I just posted my answer to that question. –  Stef van Buuren Jan 17 '13 at 12:27