Quasi-likelihood models

For all families the variance of the response will depend on the mean and will have the scale parameter as a multiplier. The form of dependence of the variance on the mean is a characteristic of the response distribution; for example for the poisson distribution $\hbox{Var}[y] = \mu$.

For quasi-likelihood estimation and inference the precise response distribution is not specified, but rather only a link function and the form of the variance function as it depends on the mean. Since quasi-likelihood estimation uses formally identical techniques to those for the gaussian distribution, this family provides a way of fitting gaussian models with non-standard link functions or variance functions, incidently.

For example, consider fitting the non-linear regression  $$y = \frac{\theta_1z_1}{z_2 - \theta_2} + e$$

(1)

this may be written alternatively as $$y = \frac{1}{\beta_1x_1 + \beta_2x_2} + e$$where x₁ = z₂/z₁ , x₂=-1/x₁ , $\beta_1=1/\theta_1$ and $\beta_2=\theta_2/\theta_1$. Supposing a suitable data frame to be set up we could fit this non-linear regression as

nlfit <- glm(yx1+x2-1,family=
quasi(link=inverse,variance=constant), data=biochem)

The reader is referred to the manual and the help document for further information, as needed.