Is there any "recipe" to remove confounding variables for linear regression? Is there any "recipe" to remove confounding variables for linear regression? Or "decompose" high level variables? I will elaborate my question by examples.
Suppose we are interested in predicting a person's income and we discover "features" / independent variables on the fly. i,e., many variables will be collected and calculated from data over time, and not all of them are available in the beginning.


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*For the first round, we may use some "high level" variables, such as "education (bachelor master phd)". The reason I say it is high level is that there are many things can be associated with "education", such as age, GPA, skills etc. Suppose we build a model and education is "significant".

*For the next round, we want to "decompose" high level variables. To know the exact reason inside of "education". We list all the variables related to education and observe if these variables are becoming "significant" and the "significant" level for "Education" comes down. 
Suppose we found, GPA and two skills are significant, and "education" is no longer significant. Can I say the GPA and skills are important variables inside of education?
 A: I'll give my 2 cents.
Are they the important ones (say within education)? You can say that. Are they the only important ones? Not necessarily. 


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*First, you need to make sure your coefficients (assuming a form of regression here) are correct with checking for multicollinearity. Often variable's that come from a patent category can be highly correlated among themselves thus possibly skewing results. 

*Second, it is prudent to be careful when signifying a variable as the important one (or a group). As further breaking down of he sub-variable's can yield the same pattern as they did to education. Maybe it's not GPA but the intelligence or a out of dedication that underlies it that truly matters. 

*Third, even if education becomes insignificant when introducing GPA (for whatever reason), it is still quite possible that education has an interaction effect with GPA Which is worth checking. It might prove important in such a way.
A: Usually we would use structural equation modeling for such purposes, especially in the context of linear models (i.e. regression). Say we want to investigate the effect of education on income controlling for the age. Education "consists" of three correlated variables such as GPA, university prestige, and skill level. A graphical representation of this structural equation model would be as follows:
. This allows us to simultaneously account for how well our "lower level" variables are related to the "higher level", composite variable and also estimate how strongly education is related to income when accounting for age, as one would do with a regular regression.  
