There are many things you could do to select variables from multiply imputed data, but not all yield appropriate estimates. See Wood et al (2008) Stat Med for a comparison of various possibilities.
I have found the following two-step procedure useful in practice.
- Apply your preferred variable selection method independently to each of the $m$ imputed data sets. You will end up with $m$ different models. For each variable, count the number of times it appears in the model. Select those variables that appear in at least half of the $m$ models.
- Use the p-value of the Wald statistic or of the likelihood ratio test as calculated from the $m$ multiply-imputed data sets as the criterion for further stepwise model selection.
The pre-selection step 1 is included to reduce the amount of computation. See http://www.stefvanbuuren.nl/mi/FIMDmaterials/src/fimd6.r.txt (section 6.4.2) for a code example of the two-step method in R using mice(). In Stata, you can perform Step 2 (on all variables) with mim:stepwise.
