Generalized Linear Models
Generalized linear models are a class of models that generalize the linear models used for regression and analysis of variance. They allow for more general mean structures and more general distributions than regression and analysis of variance. Generalized linear models were first suggested by Neider and Wedderburn (1972). An extensive treatment is given by McCullagh and Neider (1989). Generalized linear models include logistic regression as a special case. Another special case, Poisson regression, provides the same analysis for count data as log-linear models. The discussion here involves more distribution theory than has been required elsewhere in this book; in particular, it makes extensive use of the exponential family of distributions and the gamma distribution. Information on these distributions can be obtained from many sources, e.g., Cox and Hinkley (1974). Section 1 presents the family of distributions used in generalized linear model theory. Estimation of the linear parameters is dealt with in Section 2. Model fitting and estimation of dispersion are examined in Section 3; both of these topics involve a version of the likelihood ratio test statistic called the deviance. Section 4 contains a summary and discussion. We begin with a brief review of some ideas from regression and analysis of variance and three examples of generalized linear models.
KeywordsGeneralize Linear Model Maximum Likelihood Estimate Exponential Family Saturated Model Likelihood Ratio Test Statistic
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