Imputing/instrumenting for missing variables in a case-control study I'm combining two surveys in a case-control design.  Survey B is drawn from the "case" population, and includes all the variables I need for analysis, plus some extras.  Survey A samples a general "control" population, and includes most (not all) of the variables I need, plus some extras.  For notation, let's say we need variable sets X and Y for analysis, variables Z are extra variables shared between the two data sets, and V and W are other variables contained only in A and B, respectively.
So the data structure looks like this.  Variables to the left of | are required for analysis.  Variables marked by periods are missing.
A:.Y|ZV.
B:XY|Z.W

My question: many of the variables in Z, V,and W could serve as proxies for the missing variables in X.  Is there a way to impute X in sample A using these variables?  What assumptions do we have to make in order for this to work?
For example, the strategy I tried first was to regress X on Z within sample B, then generate predicted X for in both samples, and include those values for X in the regression.  I think this works as long as we assume that the model X~Z is the same in both samples and error terms in Y~X are independent of Z.  These seem like moderately strong assumptions though, and this model fails to make use of data in V and W.  Since some of the best predictors of X are in these sets, this is a big loss.
A follow-up: this is a study on political participation. Previous studies of this kind have often used a two-stage least squares instrumental variable (2SLS-IV) approach: first estimating X from instrumental variables (e.g. contained in Z), then using predicted values in the final regression (e.g. Y~X). However, no previous study uses the two-sample approach I'm following. If we choose proxy variables carefully, the example above is identical to 2SLS-IV.  Are there "imputation" strategies that use V and W, AND preserve the casual interpretation of the 2SLS-IV approach?
It would be super helpful to find an imputation/instrumentation strategy that would work here.  My only other alternative for this analysis is to do another (time consuming and expensive) round of data collection. Any suggestions?
 A: Just about any form of imputation is going to require extra assumptions (because you will always need conditional models that you wouldn't need without the missing data). The stronger the assumptions you are willing to tolerate, the less impact the chosen imputation will generally have on the confidence of your final results.
Thus, if you have some reason to believe in a model linking your missing data to regressors that are indeed present: do go ahead, just mention it when you publish and if possible: check what happens if you change your assumptions somewhat (not always easy): this can give you an idea for how dependent the result is on your assumptions.
Finally, and more importantly: never use single imputation: whatever assumptions and models you use, impute a lot of times (since this typically happens in a random fashion, this will result in several similar but not identical datasets), run your analysis on all of the imputed datasets, and look what happens to your result (typically, average the results). This is called multiple imputation. There are also proper ways of accounting for the added uncertainty due to multiple imputation.
BTW: some of all this also depends on whether you data (missing and/or present) is categorical or continuous, or (worst case scenario): both.
All of the above can be found in the book "Statistical Analysis with Missing Data" by Little and Rubin. I suggest you read it. As an extra incentive: something in the back of my mind is screaming: I seem to recall reading about a situation very similar to yours in that book. Good luck!
