What distribution should I use to model bounded count data (that also feels a bit like a proportion)? I am trying to determine the correct model for my data. I want to model the effect of two categorical independent variables (and their interaction) on my outcome variable. I am using SAS (proc genmod, probably).
The outcome variable can range from 0-6 and is the count of items in a sample of 6 with a particular feature. I do not think a Poisson model is appropriate because of the bounds of the count.
Consider, for example, an individual selecting a sample of 6 pieces of fruit from a basket of apples and oranges, with a goal to select a representative sample. I want to model the number of apples selected. I'm not sure if the goal is relevant for the model determination, but it seems like it should be relevant: there is a normative response of 2.4 apples (obviously, the sample is discrete, but this is what I'd expect on average), based on the proportion of apples in the basket (2/5).
What distribution should I use to model this outcome? I have ruled out normal (failed Shapiro-Wilk test for normality) and I think I should also rule out Poisson (because of the bound between 0 and 6).

Sounds like a job for the binomial distribution, with 6 trials and a “success” probability that denotes the probability of selecting the feature of interest.
