# GLM for proportional data and underdispersion

I'd like asking your help to understand a statistical issue from my data set. I ran a GLM with proportional data, using a binomial distribution. However, I've found underdispersion in my model and I don't know how to deal with that. I'm aware that a solution for overdispersion is fit a model using a quasibinomial distribution, but I couldn't find a solution to my problem in the literature.

I'm comparing the differences between continuous forest sites and fragments regarding the proportion of richness and abundance of specialist species. So, the models are:

M1 <- glm(prop_rich_speci ~ LandscapeBin,
family = binomial,
weights=rich_total_sp,
data = envir.all)
M2 <- glm(prop_abu_speci ~ LandscapeBin,
family = binomial,
weights=abu_total_sp,
data = envir.all)


I think using a quasibinomial distribution I can solve my problem (underdispersion), as Ben suggested.

• What do you mean by proportional data? Do you mean the outcome is a proportion, as in continuous but bounded between 0 and 1? Jan 24, 2015 at 20:22
• If it's a continuous proportion then you might find a beta model rather than a binomial more suitable. That said, the variance function of a quasibinomial should work well in that case (in that the beta variance is a scaled version of p(1-p)), and will work for underdispersion as Ben suggests below. If your data aren't typically close to 0 or 1 it may do very well. Jan 24, 2015 at 23:26
• @Glen_b, sorry for digging-up this old thread but I face a rather similar situation except that my proportions (hatching rate) are underdispersed but also frequently reach 1 (complete success; i.e. all layed eggs have hatched) and sometimes even 0. In such a case, would the quasibinomial "family" still work well? You seem to suggest that it wouldn't, could you please explain why and direct me toward more suitable alternatives? Dec 16, 2022 at 14:45
• If the distribution of proportions between the ends is consistent with the proportions that the ends, it would work well, but in general this would not be the case. You might need a 0-inflated or even 0-1 inflated version of the model (just as you would if you treated the proportion as continuous within 0-1). Dec 16, 2022 at 23:58

If you just want to assume that $\textrm{variance} = \phi \cdot N p(1-p)$ with $\phi < 1$, quasi-likelihood estimation will work fine. Depending on the source of your underdispersion, how much you're concerned about modeling the details, other aspects of your data, you might want to look into ordinal or COM-Poisson models (both of these approaches have R packages devoted to them).