# What is quasi-binomial distribution (in the context of GLM)?

I'm hoping someone can provide an intuitive overview of what quasibinomial distribution is and what it does. I'm particularly interested in these points:

1. How quasibinomial differs to the binomial distribution.

2. When the response variable is a proportion (example values include 0.23, 0.11, 0.78, 0.98), a quasibinomial model will run in R but a binomial model will not.

3. Why quasibinomial models should be used when a TRUE/FALSE response variable is overdispersed.

The quasi-binomial isn't necessarily a particular distribution; it describes a model for the relationship between variance and mean in generalized linear models which is $\phi$ times the variance for a binomial in terms of the mean for a binomial.

There is a distribution that fits such a specification (the obvious one - a scaled binomial), but that's not necessarily the aim when a quasi-binomial model is fitted; if you're fitting to data that's still 0-1 it can't be scaled binomial.

So the quasi-binomial variance model, via the $\phi$ parameter, can better deal with data for which the variance is larger (or, perhaps, smaller) than you'd get with binomial data, while not necessarily being an actual distribution at all.

When the response variable is a proportion (example values include 0.23, 0.11, 078, 0.98), a quasibinomial model will run in R but a binomial model will not

To my recollection a binomial model can be run in R with proportions*, but you have to have it set up right.

* there are three separate ways to give binomial data to R that I'm aware of. I am pretty sure that's one.

• How is this related to quasilikelihood estimation? Commented Mar 2, 2015 at 22:25
• +1 (but I would love to see a more comprehensive answer!). The three ways to set up binomial GLM with proportions are probably these: stats.stackexchange.com/a/26779/28666 ? A link might be helpful. Also, how does what you said about "quasibinomial" not really being a distribution relates to the second answer in this thread? Commented Sep 5, 2016 at 18:54
• @amoeba you can write a distribution for it, as was stated in my answer (a scaled binomial) but that cannot be a distribution for count data (quasibinomial is not on all the integers unless the dispersion parameter is 1) nor for continuous data (it's discrete!). People generally use it for count data because of its variance-structure (but in which case there is no such distribution in the exponential family) Commented Sep 5, 2016 at 19:13

When the response variable is a proportion (example values include 0.23, 0.11, 0.78, 0.98), a quasibinomial model will run in R but a binomial model will not.

A binomial model can be estimated with proportions as outcome. See below:

prop.m1 <- glm(cbind(Successes, Total - Successes) ~ X1,
data = df,
family = binomial)

prop.m2 <- glm(Proportion ~ X1,
data = df,
family = binomial,
weights = Total) # provide prior weights