What is variance argument in quasi-generalized linear models? The glm function in R can estimate models from a number of families, including the quasi family. The quasi family takes a variance parameter.  
What is the variance function for?
 A: In a generalized linear model (GLM), the distributional family entails a specific variance function $V(\mu)$, and in some cases also the dispersion parameter, $\phi$ (e.g. for the Poisson and binomial, which have $\phi=1$). 
The link and variance functions are really what drives the fitting of a GLM.
In particular, the variance function $V(\mu)$ specifies how the conditional variance is related to the conditional mean, $\mu$. Let $\mu(x)=E(Y|x)$; then $\text{Var}(Y|x)=\phi V(\mu)$ for some function $V$. If you specify a distribution family that determines $V$ and sometimes $\phi$. If you specify the quasi family you choose $V$ yourself.
In that case, rather than specify a specific distributional family, you could specify a quasi model, which allows you to specify a link and a variance function of your preference (for example, you might perhaps do this if your preferred combination is not available in one of the families). It also has $\phi$ free rather than setting it to any specific value. You can pair link and variance functions that wouldn't normally go together.
You can even write your own variance function, should none of the built-in options suit you.
What you get is a model within the exponential family that has the specified variance function and link function. This offers a great deal of flexibility (though with such flexibility it's possible to pair things that may sometimes be difficult to achieve convergence with, or may make little sense).
