# Overdispersion vs Tweedie

I am dealing with data that could possibly be overdispersed and I am looking at fitting a GLM with a quasi distribution.
As far as I understand, when we fit a glm with a quasi distribution, we do not assume any particular distribution, just a relationship between the mean and the variance.
It follows that we do not have a real likelihood function, we cannot calculate the AIC value, and since we do not have a pdf we can't generate random values.
The Tweedie distribution assumes a relation between the mean and the variance of the values in the same way as the quasi distribution context. Moreover it is a real probability distribution.
It also leads to the same estimate values as in the example here:

library(tweedie)
library(statmod)

y <- c(4.0,5.9,3.9,13.2,10.0,9.0)
x <- 1:6
fit_quasi <- glm(y~x, family=quasi(link = "log", variance = "mu"))
summary(fit_tweedie)
summary(fit_quasi)


What I don't understand is why using quasi likelihood and fitting using the quasi family vs the tweedie distribution family. What are the advantages/disadvantages or using one or the other framework?

Thank you

The two models you have fitted are identical -- there is no difference.

Quasi families are only more general than regular glms when

1. they specify a variance function for which there is no true glm or

2. the true glm for that variance function is defined only for particular dispersion values (usually dispersion = 1).

If there is a regular two-parameter glm family for the specified variance function, then the quasi family offers nothing new. To take a few examples, you can't make the normal, gamma or inverse-Gaussian families more general by turning them into quasi families. In your example, you have specified a variance function for which a regular two-parameter glm exists (the gamma family), hence the quasi family is exactly the same as that glm. It will give exactly the same results as the glm family in all respects.

This is another way of saying that, while glm families do make distributional assumptions, they remain consistent under merely mean and variance moment assumptions. Indeed, they give the same results as you would get even if you only assumed moment assumptions. In this sense they are robust to deviations from the assumed distributional family.

The concept of "over-dispersion" refers to item 2 above, and the main examples are binomial and Poisson glms. In those cases, the glm families do not allow the dispersion parameter to be estimated because the variance of the reponse is fully determined a function of the mean. In those cases, to you can use a quasi glm with the same variance function to include a quaisi-dispersion parameter, in which case the quasi-dispersion parameter measures "over-dispersion".

For continuous gamma glms, there is no such thing as "over-dispersion". The gamma glm model can accommodate any variance, no matter how large, so it is impossible for the data to be "over-dispersed" relative to the gamma model.