Finding predictive power of variables with many missing values I'm aggregating several large surveys with similar/same questions, but from different years. While the questions are often similar in consecutive years and within eras, few questions were asked in every year that the survey was taken. Unfortunately, imputation is not an option for most of the variables. 
I'm interested in understanding the predictive power of a good portion of these variables on one of the questions that was asked on every year of the survey. 
Does anyone have suggestions for a good methodology? Standard regressions seem to be out -- although maybe someone is familiar with a type of "bagged" regression algorithm, similar to trees. 
A tree type algorithm seems like the best fit, although I'm interested if anyone has strong opinions here on the type. 
Anything else I should be considering? 
 A: Bayesian regression models can deal with missing values in a principled manner. 
For theoretical background, there is a chapter on it in Bayesian Data Analysis by Gelman et al.. A bit easier to read as an introduction is Statistical Rethinking by McElreath, in which missing values are mentioned as one of the topics where Bayesian methods can shine potentially. 
Prefabricated solutions like rstanarm or brms in R don't deal with it out-of-the box, allthough there are some running discussions on Github devoted to the topic, but depending on how complicated your model is, you can code it yourself in Stan, or find someone to help you a bit with that. 
This is all about the missing values problem, and should set you up to create a solid regression model, also able to handle ordinal questions, and hierarchical structure (within subjects variation). You could then interpret correlation between the answers of the questions
If your question is actually more about analysing "predictive power", which is maybe not that clear defined, things get more complicated, and less well-studied. There are people working on Graphical Models, possibly even with a time-dependency. You can start with this link for some nice pictures, but this is a large field, if not as mature as standard regression analysis. 
