test interactions for multiple regression with many predictor variables I have a data set with around 25 predictor variables. If I am planning to build multi-regression model against this data set. What are the general approaches to test the interactions of these predictor variables. There are 25 variables, looks like it is not a good approach to test one with all others, then the next,...
 A: So, rather than answering the question (which is too dense now), I'd just outline and explain how I see this then perhaps you can make an informed decision.
Consider a regression model
$y = \beta_0 + \beta_1 Prog + \beta_2Male + \beta_3Age+ \beta_4Income$,
where the program participation (vs. control group with no program) is my  independent variable of interest and the rest are covariates I wish to adjust for.
If I were to set up some tests for interactions I'd consider two points:


*

*Does the interaction term involve the main independent variable of interest (aka Prog)?

*Does the interaction term make sense? In other words, if I am to test the interaction between Prog and Male, does it make any sense that the main effect of the program would vary depending on the participant's sex? Or, has there been literature suggesting such evidence?


If the answers to both questions are yes, then I'll test the interaction. The reason for point 2 is obvious and the reason for point 1 is that at the end of the day, I want to know if my program is effective. I don't want to walking around saying my program does not work, but not realizing that it works very well for younger age but not for the older because I didn't check their interaction. It's fine if Income and Age interact, I probably wouldn't care because they would unlikely affect the estimate for Prog.
This is how I usually plan around an explanatory model: with a framework, defined major independent variable, and planned analysis on speculated interactions.

And that's why I found it difficult to address the question; you seem to be running an exploration rather than operating with a conceptual framework. (Aka, to say it a bit informally, throwing 25 variables onto the wall and see what sticks.) Both are legitimate use of data and we could be having some paradigm differences here. So, I'm just listing what I consider are some dilemmas here, they may or may not be your actual concerns:


*

*I don't oppose exploring data freely (as long as we don't look at what stuck and then retrospectively make up a hypothesis/research question for them, or present them as if the results were backed by research questions). However, 400 is not enough. If there are no specific research questions the next "default" is probably testing all possible two-variable interactions. That would be a lot. And not to mention that it will already be a lot if they are all continuous; if some are categorical as you said, their interaction terms will take up a lot more degrees of freedom. (e.g. two 5-level categorical variables need $(5-1)\times(5-1)= 16$ interaction terms.)

*What about select the variables first then model interaction? Then the results may be prone to missing important independent variables. As one thing about interaction terms is that two variables can be non-predictive by themselves but when their interaction term is included all three of them can be statistically significant.
So, my (very scattered) recommendations are:


*

*Carefully evaluate your 25 variables and examine if you can build a conceptual framework around them. Think if interaction between any pair of them are likely, and only attempt to test a reasonably selected subset. One possible approach is to see if there are some major independent variables whose results you really want to tease out, then treat them like the Prog variable in the example above.

*Employ data reduction techniques to combine information from similar variables. This may help you decrease the original 25 somewhat.

*Get a lot more samples if you want to do the free-range exploratory expedition.
