Generalized Linear Models

Generalized linear models (GLMs) are a large class of statistical models for relating responses to linear combinations of predictor variables, including many commonly encountered types of dependent variables and error structures as special cases. In addition to regression models for continuous dependent variables, models for rates and proportions, binary, ordinal and multinomial variables and counts can be handled as GLMs. The GLM approach is attractive because it (1) provides a general theoretical framework for many commonly encountered statistical models; (2) simplifies the implementation of these different models in statistical software, since essentially the same algorithm can be used for estimation, inference and assessing model adequacy for all GLMs. The canonical treatment of GLMs is McCullagh and Nelder (1989), and this review closely follows their notation and approach. Begin by considering the familiar linear regression model, yi = xib + ei, where i = 1, . . . , n, yi is a dependent variable, xi is a vector of k independent variables or predictors, b is a k-by-1 vector of unknown parameters and the ei are zero-mean stochastic disturbances. Typically, the ei are assumed to be independent across observations with constant variance r2, and distributed normal. That is, the normal linear regression model is characterized by the following features: