# Multiple regression when one predictor is the sum of another predictor and a third non-included variable

Assuming three variables are exactly related as:

C = A + B

Is it reasonable to run a logistic regression including both C and A as independent variables? If it is reasonable, how would the estimate for each be interpreted?

For example, for some binary outcome, I'd like to estimate the effect of the number of cardiovascular related hospitalizations in the year prior, ("A" in the above equation) controlling for the total number of hospitalizations (any type) in the prior year ("C" in the above equation). "B" in the above equation would be "number of non-cardiovascular related hospitalizations, and would not be included in the model. This model "runs" without error, but I am really struggling with interpretation.

I have also considered categorizing the counts - (e.g. estimating the effect of a three category variable: 0, low, or high number of cardiovascular related hospitalizations, while controlling for 0, low, or high number of overall hospitalizations).

• Your assumed data generating process doesn't make a lot of sense to me. It isn't stochastic, ie, there seems to be no randomness at all. Beyond that, I'm not sure if I really understand what you are getting at. May 16, 2017 at 19:36
• My apologies for not being more clear! The equation above is not the data generating process, but rather the relationship between two independent variables I'd like to include in the model, and a third variable (not in included in the model). - I have edited for hopefully more clarity.
– aghl
May 16, 2017 at 21:58

It will work.

If your goal is prediction, your predictions will be just as accurate as they would have been using A and B instead of A and C.

However, if your goal is explanation, what matter are estimation of coefficients -to measure the effect of each predictor and to decide which predictors matter- and here collinearity might be a problem.

The issue is expanded in an answer to a related question in https://stats.stackexchange.com/a/285065/123561

Theoretically, it won't work. VIF will pop because in your case, C is perfectly linearly dependent on A. If you run glm, most likely one will be tossed out automatically.

• It does run, and gives very valid looking estimates for both predictors. I don't think C is perfectly linearly dependent on A, I think it's linearly dependent on A and B, but B is left out of the model.
– aghl
May 16, 2017 at 22:01