Generalized linear model

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Generalized Linear Models ( GLM, also Generalized Linear Models ) are an introduced by John Nelder and Robert Wedderburn 1972 generalization of the classical linear regression model in the regression analysis. While it is assumed in linear models that the target variable is normally distributed, they can be used in GLMs have a distribution from the class of exponential families. This distribution class includes not only the normal distribution, the binomial, Poisson, gamma and inverse Gaussian distribution.

Model components

The GLMs consist of three components:

• Random component: As with the classical linear models, one is at a response and an independent Kovariablenvektoren, where interested. Here, the independent and have a distribution from the exponential family.

• Systematic component: Given Kovariablenvektoren which affect the distribution of only a linear function. This linear function is called the linear predictor, and is given in the following form:

• Parametric component link: The expectation vector is a differentiable, monotonic and thus invertible function of the linear predictor. In this case, the expected value of a response function with the linear predictor is linked:

Exponential Family

The distribution of a target variable belongs to the exponential family if the density function or probability function can be written in the following form:

For all distributions of the exponential family is:

Examples of distributions are exponential family: