What if value of one variable dictates (to an extent) the value of another? I'm working with a regression data set in which the dependent variable of interest is being predicted by two percentages: (a) the percentage of Caucasian residents in one's county and (b) the percentage of African American residents in one's county. Obviously, these are correlated (r = -.78; see Figure below), and I'd like to use both in my analyses. The issue that I'm running into here is that at some point the value on one variable precludes certain value of the other. For example, if the percentage of Caucasian residents is 80%, then it must be true that the percentage of African American residents is less than 20% (there must be some residents that are in neither group). I have several questions, and advice or references on any of these issues would be much appreciated:


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*Are there any statistical considerations that need to be made in these instances? I'm not sure that I've worked with variables in which it was not possible for values for one to occur at each value/level of the other.

*Might there be a better way than considering both together? Seems like a ratio might work, but my concern here is that this might obscure an association that is specific to one variable (and not the other).


Thanks!

 A: What linear regression can't handle is when one independent variable is an exact linear combination of the others. For example, if it were the case that all of your counties had only whites and blacks, then the percentage of whites would be 1 minus the percentage of blacks, so the coefficients would be unidentifiable. This problem would be fixed by simply removing one of the race terms. However, this isn't the case for you. So mostly all you can do is the usual methods for dealing with correlated predictors, such as regularization.
One thing to think about is the scale of your independent variables. Is it realistic for your dependent variable to be affected additively by percentages? This seems unlikely, because it would mean that a change from 50% to 55% would have the same effect as a change from 90% to 95%. I would suggest a transformation, such as the logit function $f(x) = \log \frac{x}{1 - x}$.
A: One other thing which you might consider is using the sum and difference of the two predictors. In this case the sum would differentiate between areas where most people were either White or African American versus areas where the other ethnic groups are more prominent and the difference would indicate how numerically dominant the more frequent of White and African American was. They may be less correlated but more importantly they might be more interpretable in terms of your substantive scientific question.
