Multiple imputation for variables used to calculate regression weights

Question

My basic question: is there anything that you can't impute using MI?

My more complicated question:

Consider the regression $Y=\rho T+X'\beta+\epsilon$. For whatever reason, you want to weight the regression. Say that you are weighting by an inverse propensity score, assuming that such would get you to conditional independence and causality.

Specifically, $pr(T==1) = logit(Z'\gamma + \eta)$, which gives you weights as $$ w = (T_i/pr(T)+(1-T_i)/(1-pr(T))^.5 $$

Say you're missing elements of Z. Could you set up imputation models for them (using, say, a MICE approach), and use them to construct $w$ which you use to fit the main regression many times? After which you'd combine using Rubin's rules?

Is this invalid for any reason? Is there something special about missing model covariates such that those are the only things that can be imputed?

I've got a dataset with good coverage in the $X$'s, but a lot of missing values in the $Z$'s.

Answer 1

Good question. I suggest you take a look at the recent thesis of Shahab Jolani. He discusses various ways to combine the response model and the substantive model. See http://igitur-archive.library.uu.nl/dissertations/2012-1120-200602/Jolani.pdf

In general, the results are good (i.e. unbiased, appropriate coverage), and do not critically depend on the form of the parametrization chosen (chapter 2). Also, it is possible to put incomplete Y into the nonresponse model, thus opening up a way to deal with MNAR data by a MICE-like algorithm (chapter 4).