Fit Analyses

Generalized Linear Models

Generalized linear models assume that the response y_i has a distribution from the exponential family (normal, inverse Gaussian, gamma, Poisson, binomial) and a function can be used to link the expected response mean and a linear function of the X effects. In SAS/INSIGHT software, a generalized linear model is written as

$y = {\mu} + {{\epsilon}}$

${\eta} = g({\mu}) = {\eta}_{0} + X {\beta}$

where y is the n×1 vector of responses, $\mu$ is the n×1 expected response means, and ${\epsilon}$ is the n×1 vector of unknown errors.

The monotone function g links the response mean $\mu$ with a linear predictor ${\eta}$ from the effects, and it is called the link function. The n×1 vector ${\eta}_{0}$ is the offset, X is the n×p design matrix, and $\beta$ is the p×1 vector of unknown parameters. The design matrix is generated the same way as for linear models.

You specify the response distribution, the link function, and the offset variable in the fit method options dialog.

Generalized Linear Models

The Exponential Family of Distributions

Link Function

The Likelihood Function and Maximum-Likelihood Estimation

Scale Parameter

Goodness of Fit

Quasi-Likelihood Functions