# Alternative to Chi-square test for observations belonging to multiple groups?

Have a certain (high) number of rows corresponding to people. Have a bunch of independent variables which are group memberships and a dependent variable which is the outcome of a treatment. All are binary. Now, the group memberships are not exclusive -- a person can be a member of multiple groups, and the group memberships have some degree of collinearity. Feel like this is violating some assumptions of the $$\chi^2$$ test, namely that the rows and columns must sum to the number of observations.

What's a better test to check if the treatment of groups is different? Regression analysis (while looking at VIF?)?

Also the number of groups is too high (around 10) to enumerate all the possible combinations as new variables with distinct observation counts.