Visually compare binomial and beta-binomial appropriateness in R

I'm trying to choose the best distribution family for generalized linear regression. My outcome is cross-sectional, over-dispersed proportion data (# of behaviors/20-22 possible behaviors). I used the functions fitdistr and plotdist from the package fitdistrplus to visually rule out the binomial and poisson distributions, so now I'm choosing between the beta-binomial, quasibinomial, and negative binomial distributions.

Note: I don't really understand the beta-binomial distribution, but a lot of searching on this site suggested that it's often most appropriate for overdispersed proportion data.

Key info: I'm inclined to select the negative binomial distribution because

(1) I understand glm.nb better.

(2) glm.nb has the lowest AIC, provides similar results to both the beta-binomial and quasibinomial analyses, and my outcome closely aligns visually with the theoretical negbin distribution (see graphs below).

Questions:

1. The fitdistrplus package does not support the betabinomial distribution, so I don't know how to visually compare my data with its theoretical distribution. Is it possible to visually compare the binomial and beta-binomial distributions? How would I do that in R?
2. What are the pros and cons of selecting the negative binomial in this case? I read in Wagner (2015) that the negative binomial is supposed to approximate the beta-binomial distribution. What does that mean for my analysis?

References

Wagner, B., Riggs, B., & Mikulich-Gilbertson, S. 2015. The importance of distribution-choice in modeling substance use data: a comparison of negative binomial, beta binomial, and zero-inflated distributions. American Journal of Drug and Alcohol Abuse, 41(6), 489–497.

• (1) You don't seem to be considering the supports of the various distributions you're looking at. That of a normal is the real numbers; that of Poisson & negative binomial distributions, the non-negative integers; that of appropriate binomial & beta-binomial distributions, the integers from 0 to 22. It's a point in favour, at least, of the last. (2) You seem to be looking at the marginal distribution, but it's the distribution conditional on predictors that's relevant for regression. Commented Sep 2, 2017 at 20:25
• @Scortchi Do you mean binomial & beta-binomial for the last part of your comment (for integers 0 to 22)? Also, I believe I have been considering the supports, though I didn't know that's what they were called (I'll edit the question to remove the mention of the normal dist). If I go strictly by the logic of the supports/why these dists were designed, then, yes, the beta-binomial is best and the negative binomial is second best (not sure about quasibinomial). But is it ever reasonable to use negbin instead of betabin, if the distribution looks more like negbin? Commented Sep 2, 2017 at 20:45
• I did mean "binomial & beta-binomial" - thank you! (I've edited the comment). And it can be reasonable to use distributions with the wrong support when there's negligible probability mass in impossible regions - using the normal distribution to model heights is an example. Commented Sep 2, 2017 at 20:47
• @Scortchi Ah, I don't know how to look at the distribution conditional on predictors! How does one do that in R? Do you know of a link? I wouldn't know what to search. Commented Sep 2, 2017 at 20:49
• "diagnositics" + "generalized linear models" Commented Sep 2, 2017 at 20:54

First, the beta-binomial distribution is a compound distribution. While in a binomial experiment, the probability parameter $$p$$ is the same for every draw, in the beta-binomial, it is not. Before every draw, the $$p$$ is itself drawn from a beta distribution, independently each time. That gives rise to over-dispersion, as you have noted.