Generalized linear models; families

Generalized linear modelling is a development of linear models to accommodate both non-normal response distributions and transformations to linearity in a clean and straightforward way. A generalized linear model may be described in terms of the following sequence of assumptions:

• There is a response, y , of interest and stimulus variables x₁ , x₂ , ...whose values influence the distribution of the response.
• The stimulus variables influence the distribution of y through a single linear function, only. This linear function is called the linear predictor, and is usually written $$\eta = \beta_1x_1 + \beta_2x_2 +\cdots+\beta_px_p$$hence x_i has no influence on the distribution of y if and only if $\beta_i=0$.
• The distribution of y is of the form $$f_Y(y;\mu,\varphi) = \exp\left[\frac{A}{\varphi}\left\{y\lam... ...gamma\left(\lambda(\mu)\right)\right\} + \tau(y,\varphi)\right]$$where $\varphi$ is a scale parameter, (possibly known), and is constant for all observations, A represents a prior weight, assumed known but possibly varying with the observations, and $\mu$ is the mean of y . So it is assumed that the distribution of y is determined by its mean and possibly a scale parameter as well.
• The mean, $\mu$, is a smooth invertible function of the linear predictor: $$\mu = m(\eta),\qquad \eta = m^{-1}(\mu) = \ell(\mu)$$and this inverse function, $\ell(.)$ is called the link function.

These assumptions are loose enough to encompass a wide class of models useful in statistical practice, but tight enough to allow the development of a unified methodology of estimation and inference, at least approximately. The reader is referred to any of the current reference works on the subject for full details, such as

Generalized linear models by Peter McCullagh and John A Nelder, 2nd edition, Chapman and Hall, 1989, or

An introduction to generalized linear models by Annette J Dobson, Chapman and Hall, 1990.