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Generalized Linear Models

Statistical Theory and Methods

  1. Bent Jørgensen

Published Online: 15 JAN 2013

DOI: 10.1002/9780470057339.vag010.pub2

Encyclopedia of Environmetrics

Encyclopedia of Environmetrics

How to Cite

Jørgensen, B. 2013. Generalized Linear Models . Encyclopedia of Environmetrics. 3.

Author Information

  1. University of Southern Denmark, Odense, Denmark

Publication History

  1. Published Online: 15 JAN 2013

Abstract

Generalized linear models provide a general framework for handling regression modeling for normal and non-normal data, including multiple linear regression, ANOVA, logistic regression, Poisson regression and log-linear models for contingency tables. All the major statistical packages include facilities for fitting generalized linear models. A generalized linear model is defined by choosing a link function and a variance function, along with choosing a response variable and a set of explanatory variables. The link function transforms the mean of the response variable to a scale where the model is linear. The variance function describes how the variance behaves as a function of the mean. Each choice of variance function corresponds to a certain deviance function, and model fitting is accomplished by minimizing the deviance, generalizing least squares fitting. Inference on parameters and hypothesis testing is performed by means of analysis of deviance, generalizing the classical ANOVA method. Estimation and analysis of deviance are based on quasi-likelihood methods, requiring only second-moment assumptions, thereby providing a certain robustness against misspecification of the probability model. The choice of link and variance functions may be checked by means of residual analysis.

Keywords:

  • analysis of deviance;
  • hypothesis testing;
  • non-normal data analysis;
  • parameter estimation;
  • quasi-likelihood;
  • residual analysis;
  • regression modeling