Beta distribution, beta-binomial, and parameters

[Paul Wileyto]

I'm having to dive in to beta-binomial models, and I am seeing two different parameterizations of beta... If I look at beta in the stats functions, including random numbers, the parameters include a and b, two shape parameters. If I use betabin regression (Thank you James Hardin), I get back a different set of parameters for the beta, the binomial mean and some sort of over-dispersion parameter. I do not know how the two sets of parameters relate to one another, and how to translate one to the other. Right now, I can simulate based on the a/b parameters, and I cannot tell whether I am getting the same thing back.

Would appreciate it if someone pointed me in the right direction.

Thanks

Paul

[Maarten Buis]

mu = a/(a+b)
phi = a + b

[John Mullahy]

I don't know if you are considering conditional-on-x regression models but if so it might be noteworthy that there are some parameterizations of beta-binomial (negative hypergeometric) models where both a and b are specified as functions of covariates, a(x) and b(x). When a(x) and b(x) are specified as exp(x*alpha) and exp(x*beta) this is sometimes called a beta-logistic model, as its mean is the logistiic model's mean

exp(x*alpha)/(exp(x*alpha)+exp(x*beta)) = exp(x*gamma)/(1+exp(x*gamma)) (***)

where gamma=alpha-beta. (See e.g. https://www.jstor.org/stable/1828328)

However there are other parameterizations where a but not b is parameterized as a function of x. I seem to recall this is the parameterization used in Stata's –betabin–.