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.

1. The difference between the binomial distribution and quasi-binomial can be seen in their probability density functions (pdf), which characterize these distributions.

Binomial pdf:

$$P(X=k)={n \choose k}p^{k}(1-p)^{n-k}$$

Quasi-binomial pdf:

$$P(X=k)={n \choose k}p(p+k\phi)^{k-1}(1-p-k\phi)^{n-k}$$

The quasi-binomial distribution, while similar to the binomial distribution, has an extra parameter $\phi$ (limited to $|\phi| \le \min\{p/n, (1-p)/n\}$) that attempts to describe additional variance in the data that cannot be explained by a Binomial distribution alone.

(Note that the mean of the quasi-binomial distribution is $p \sum_{i=0}^n \frac{n!\phi^i}{(n-k)!}$ rather than $p$ itself.)

2. I am not sure on this one, perhaps the glm function in R adds weights in the quasibinomial mode in order to account for this?

3. The purpose of the extra parameter $\phi$ is to estimate extra variance in the data. Every generalized linear model (GLM) makes a distributional assumption for the outcome/response and maximizes the likelihood of the data based on this distribution. It is a choice the analyst makes, and if you feel you need to account for more variance in your data, then you can choose the quasi-binomial distirbution to model the response for your glm. A great way to test if we need to fit a quasi-binomial model instead of a binomial is to fit a quasi-binomial model, and test to see if the $\phi$ parameter is 0.

• Excellent Alejandro, now how can I test if the ϕ parameter is 0? – Juanchi Jul 12 '17 at 0:59
• Note that in R with glm.fit, binomial and quasibinomial are exactly the same, except that quasibinomial (1) removes the integer check, and (2) returns an AIC of NA. See this answer for more details. – miguelmorin Mar 21 '19 at 16:06
• -1 This kind of "quasi-binomial" distribution seems to be completely unrelated to quasi-binomial likelihoods in the context of glms, so it is hard to see why it got so many upvotes. – Jarle Tufto Sep 26 '19 at 12:35
• Totally agreing with @JarleTufto, this has nothing to do with the quasi binomial of a glm. – Elvis Nov 10 '20 at 18:22

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? – tim.farkas Mar 2 '15 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? – amoeba Sep 5 '16 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) – Glen_b Sep 5 '16 at 19:13