I am running an experiment where I am testing the effects of three interventions (A,B,C) and measuring participants' performance via a multiple choice test with 5 questions. To perform hypothesis testing, I would like to use a 3-way ANOVA, however, I am faced with the challenge that I can't really assume my data is normal, since it can take on only 6 possible values (0-5 inclusive). I would like to use a Generalized linear model for my data instead, however, I'm not sure what kind of GLM I should use.
My first guess would be that this is a binomial distribution (how many successes out of n trials). However, my problem is that this would assume that all questions have equal difficulty/probability of success, which is not necessarily true. I have seen that Poisson and Inverse Binomial are often applied to count data, but my problem is that these distributions do not have an upper limit, while my data does. What distribution would be best for a GLM, or is there an alternative method that would be better?