Definition:Generalized Linear Model

Definition

A generalized linear model is a mathematical model in which a function $\map g \mu$ of the expectation $\mu$ of a random variable $Y$ is a linear function of one or more cause variables.

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Link Function

Consider a generalized linear model which is a function $\map g \mu$ of the expectation $\mu$ of a random variable $Y$.

The function $\map g \mu$ is known as the link function.

Examples

One Variable

The generalized linear model for one cause variable $x$ is of the form:

$\map g \mu = \beta_0 + \beta_1 x$

where $\map g \mu$ is the link function.

Bernoulli Variable

Let $Y$ be a Bernoulli variable with expectation $\mu = p$.

Then the link function $\map g \mu$ of the generalized linear model for $Y$ is given by:

$\map g \mu = \map \ln {\dfrac p {1 - p} }$

Poission Distribution

Let $Y$ be a random variable with a Poisson distribution.

Let $Y$ have an expectation $\lambda$ which varies with $x$.

Then the link function $\map g \lambda$ of the generalized linear model for $Y$ is given by:

$\map g \lambda = \ln \lambda$

Probit Analysis

Probit analysis is a special case of a generalized linear model.

Note that, like least squares regression, it has been around since before the formulation of the generalized linear model.

Also see

• Results about generalized linear models can be found here.

Historical Note

The generalized linear model was conceived and developed by Robert William Maclagan Wedderburn and John Ashworth Nelder from $1972$.

Sources

• 1972: Robert Wedderburn and John Nelder: Generalized Linear Models (J.R. Stat. Soc. Ser. A Vol. 135, no. 3: pp. 370 – 384)  www.jstor.org/stable/2344614
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generalized linear models
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generalized linear models