I'm doing some imputation with the MICE package. The outcome variables I am using are zero-inflated, and in the absence of imputation, I would analyze them with a zero-inflated negative binomial regression (ZINB). I would like to do the same with the imputed data; however, I have to manually combine parameters since MICE or its add-ons (as far as I know), do not generate estimates for ZINB.
My question is whether Rubin's Rule for combining regression parameters (which, to my knowledge, is simply the mean) would apply to regression parameters for ZINB. I believe that regular linear multiple regression parameters can be combined in such a way because they are approximately normally distributed. Do regression parameters for ZINB (and for that matter, Poisson and logistic regression) approximate a normal distribution in the same way? In other words, is the Rubin's Rule for combining ZINB and other generalized linear regression parameters the same as combining regression parameters from a linear multiple regression?