# Literature request: (in)appropriateness of negative binomial for count data with an upper bound

I conducted an analysis where I used binomial logistic regression to analyze x successes in n trials (where n varies between observations) in aggregate (using the R syntax cbind(successes,failures) ~ predictor). To account for the overdispersion I observed in the data, I fitted a quasibinomial logistic regression.

Now I have come up against a reviewer asking why I did not use a negative binomial regression. In this answer, Ben Bolker suggests that one could use Poisson or negative binomial with an offset if the number of successes is low, i.e. the count does not approach the upper bound. This is not the case in my dataset, but I would like a reference to back this up in my response to the reviewer.

Does anyone have a reference that demonstrates the conceptual concerns in the answer above? I have already checked Agresti and McCullagh and Nelder.

• Describe the nature of failures. This may be a case for semiparametric models (e.g. proportional odds model) that don't care about the shape of the distribution of counts. – Frank Harrell Jun 29 '19 at 11:24
• @FrankHarrell in this case, we are comparing concordance between individual phylogenetic trees and one overall tree at individual nodes of the tree. So successes are where the nodes match, and failures are where they are in conflict. – NatWH Jun 29 '19 at 11:29
• Interesting. That would have a complex correlation structure in the data. I wonder how you handle that. – Frank Harrell Jun 29 '19 at 12:35
• @FrankHarrell it certainly could, although probably not in the classical way one handles phylogenetic structure. I think any structure would be more likely to be proportional to how close together the nodes were (e.g. how many branches separate them) but I am not sure of the best way to account for that. – NatWH Jun 29 '19 at 12:40
• With regard to your need for references, those claims about binomial and Poisson are "common knowledge". This paper might be what you're looking for: projecteuclid.org/euclid.aoap/1037125856 – Heteroskedastic Jim Jun 30 '19 at 14:43