Have you considered using Expectation Maximization to train in the values from initial guesses (where the initial guesses can be the means)?

This R-Bloggers post does a good job of explaining the procedure using off-the-shelf software.

http://www.r-bloggers.com/imputing-missing-data-with-expectation-maximization/

N.B. The problem you've come across is generally known as imputation.

Mohammad -

Have you considered looking for response propensity groups (RPGs)? Weighting is usually used with those, but as you are using individual data from there for modeling, you might want to use RPGs as a way of stratifying before imputing, to reduce bias.

As Pat said, you might want to use multiple imputation here. That might be for one RPG at a time.

Actually my favorite method for missing data, when estimating a total for those data is actually my goal, is to impute using regression for 'prediction.' It sounds like you have other related data to use for regressors for that. Then you could find the variance of the prediction error. (For an individual datum, see for example, top of page 5 at the attached link, and for totals, middle of page 4, though these may not be good examples for your particular latent variable case, but i think something should exist that is.) However, because your goal is apparently to use these data as input to a more complex model, these results for estimated variance of the prediction error in generating these imputed values only indicate a weakness in one of your latent variables. Hopefully it will be small enough and/or your need for that variable weak enough that it will not badly impact final results that you need. It might help in a sensitivity analysis to see how accurate your final results might be.

Perhaps 50% imputed data in your case, as Pat said, may not be too bad. But in the simpler applications of regression for establishment surveys that I saw, it generally would be very bad, so it would just make me nervous to see something like that if you did not investigate the impact on accuracy. But I only worked peripherally around econometrics problems, which I assume is related to your work, so maybe 50% is not as bad as it sounds to me. :-)   Also, trying to obtain more data, and just getting lower quality data, could be worse. 

If you do use multiple imputation, you could try other sensitivity analyses with that. 

Best wishes - Jim

Article Using Prediction-Oriented Software for Survey Estimation

Dear Hames and Patric,

Thank you both for your responses to my question and time allocation.