Chapter 13GENERALIZED LINEAR MODELS
13.1 AN OVERVIEW OF GENERAL LINEAR MODELS
In Chapter 11, the linear regression models we examined each had a continuous response variable. However, what happens if we want to build a regression model for a binary response instead? Or for a numeric discrete response? Luckily, there is a family of linear models that includes all three cases – continuous, numeric discrete, and binary – of regression response variables: General Linear Models (GLMs).
To explain how regression for three different kinds of responses can be related, we will briefly take another look at the parametric regression equations for each case. Once we establish how they are related, we will then use their descriptive versions, just as we did in Chapter 11.
Recall the parametric model for multiple regression, given here.
The sum β0 + β1x1 + β2x2 + ⋯ + βpxp is called the linear predictor. For brevity, we will write the linear predictor as Xβ. The formula that connects the linear predictor to the mean μ of the y variable at a set of given values of predictor variables is called the link function, g(μ).
Different link functions entail different regression models, with each link function associated with a particular response type. For each different response type we discuss, we will specify a particular g(μ) so that Xβ = g(μ), and solve for μ to obtain the final form of the model. ...