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16.1: Generalized Linear Models

  • Page ID
    7276
  • Generalized Linear Models (GLMs) provide a modeling structure that can relate a linear model to response variables that do not have normal distributions. The distribution of YY is assumed to belong to one of an exponential family of distributions, including the Gaussian, Binomial, and Poisson distributions. GLMs are fit to the data by the method of maximum likelihood.

    Like OLS, GLMs contain a stochastic component and a systematic component. The systematic component is expressed as:

    \[η=α+β1Xi1+β2Xi2+…+βkXik(16.1)(16.1)η=α+β1Xi1+β2Xi2+…+βkXik\]

    However, GLMs also contain a link function" that relates the response variable, YiYi, to the systematic linear component, ηη. Table 16.1 shows the major exponential “families”" of GLM models, and indicates the kinds of link functions involved in each. Note that OLS models would fall within the Gaussian family. In the next section we focus on the binomial family, and on logit estimation in particular.

    families.jpg
    Figure \(\PageIndex{1}\): Exponential ‘Families’ of GLM Models