I'm interested in testing for membership similarity between multiple categorical variables.
For instance, let's say that two categorical variables have the following membership:
Group1 = 1, 1, 2, 2, 3, 3
Group2 = 2, 2, 1, 1, 3, 3
I want to test how consistently samples group together across these variable. The name of (1, 2, 3) is arbitrary and strictly qualitative. In this instance, it would be a perfect match because the 2 matches bidirectionally to 1.
Is there a statistical test that I can apply to test this? I had read that $\chi^2$ might be appropriate, but I'm still a little fuzzy on how to interpret it in my application, because I don't think it would account for the bidirectional relationship between 1 and 2 in the different groups.
Any suggestions?