Linear Regression using aggregate variable create multicollinearity or adjustment issue?

Question

my group project has a number of independent binary variables x1,x2,...,xm and a dependent binary variable y. My dataset contains some million of rows.

Since there are so many independent variables, our group decide to use aggregate variable to reduce the number of independent variables.

For a simpler example, we have (1=yes, 0=no): y = disease (1/0) x1 = obesity (1/0) x2 = asthma (1/0) x3 = high blood pressure (1/0) x123 = {1 if any of x1,x2, or x3 =1},or ={0 otherwise}

My group is deciding to fit a logistic regression model:

proc logistic;
class x1 x2 x3;
model y=x1 x2 x3 x123/ expb;
run;

If we were to use the above model, it seems that there is multicollinearity issue due to x123 depends on x1, x2, and x3. However my classmate argues that our x1, x2, and x3 are independent and uncorrelated to each other, so it would okay to include all four variables in the model.

Might someone be willing to explain if the model in the above SAS makes sense? Or should we drop x123?

Thank you

Answer 1

You could calculate the Variance Inflation Factors (VIF) to check for multicollinearity. You can do that by using linear regression instead of logistic regression.

Apply Multiple Linear Regression with the same independent variables x1, x2, x3 and x123. Then, calculate the VIFs. This will give you some insight into whether your model might suffer from multicollinearity.

If the VIF is high (greater than 5), your logistic regression will probably suffer from the same problem.