Hi all and thanks for taking the time to read this question. I'll try to be as clear as possible!
The data: I have one sample of patients which was divided in two groups based on a binary variable for a condition - Condition A and Condition B. Gorup A contains approx. 500 patients, while B contains approx. 170. The patients records have a categorical variable with 5 different (mutually exclusive) outcomes, say $[o_1,o_2,o_3,o_4,o_5]$.
The problem: I want to compare the proportions of each of those outcomes between the groups, i.e., I want to know if the proportion of outcome o1 is different between groups A and B, AND I want to know if the proportion of outcome $o_2$ is different between A and B, and so on. There would be multiple $H_0$'s in this case, in the line of: "Outcome $o_1$ is the same between groups".
Additional details: Some of the outcomes have very small observed proportions (e.g. 1 in 170, 7 in 500).
My thoughts: I've compared groups like this before, but never had to determine which outcome was different, only if the overall outcome proportions were dependent on group (Fisher's test and Chi-Squared test, etc). The groups do form contingency tables, since a patient cannot have more than one outcome. Is there any specific test for this situation? Or should I use something like a Chi-Squared test for each outcome and then perform a multiplicity adjustment? I found this thread, which seems to be similar, however I couldn't really undestand the answer given: Test of significance of multiple proportions in two groups (it seems to suggest multiple chi-squared tests, is that it?)
Sorry for bothering you guys! Any help would be much appreciated!