I have a dataset of 28 nominal and ordinal categorical variables, 161 observations. If I use chi sq tests of association between variables, for example I test var1 and var2, var1 and var3 and var1 and var4, do I violate the independence assumption of the test (I know I must use Bonferoni or similar) because var 1 is common to all three tests? If so, could someone explain why this is the case and propose an alternative? I don’t think I need to give a data sample here as it’s a theory question? Also, some of the tests have expected cell values less than 5 but are in bigger than 2$\times$2 contingency tables. My understanding is that Fishers exact is only for 2$\times$2. Can someone suggest a good alternative for, say, 4$\times$9 table categorical variables? Thanks.
Sounds to me just a multiple testing question. It does not seem specific to chi-square test. Even if you do t-test, you can test one group against multiple other groups respectively. It is fine, as long as you do the appropriate multiplicity adjustment.
Independence between the variables is the null hypothesis you are testing. It is not an assumption?