Generalized linear models: General theory

Chapters 1-3 focused on the most common class of models used in applications: linear models. Despite their versatility, linear models do not apply in all situations. In particular, they are not designed to deal with binary or count responses. In Chapter 4, we introduce generalized linear models (GLMs), a generalization of linear models that encompasses a wide variety of incredibly useful models, including logistic regression and Poisson regression.

We’ll start Chapter 4 by introducing exponential dispersion models (Section 19  Exponential dispersion model (EDM) distributions), a generalization of the Gaussian distribution that serves as the backbone of GLMs. Then we formally define a GLM, demonstrating logistic regression and Poisson regression as special cases (Section 20  GLM definition). Next, we discuss maximum likelihood inference in GLMs (Section 21  Parameter estimation). Finally, we discuss how to carry out statistical inference in GLMs (Section 22  Inference in GLMs).