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Generalized Linear Models

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Definition

A generalization of general linear Models to include a variety of non-normal response distributions belonging to the exponential family of probability distributions. The Exponential family includes the normal, binomial, multinomial, Poisson, inverse Gaussian, gamma, geometric and negative Binomial Distributions.

In short Generalized linear Models (GLMs) were formed as a result of relaxing the assumption of normality and even that of linearity, common to general linear Models.

A generalized linear model is specified by the following components:
1. The random component which specifies the response variable and its probability distribution
2. The systematic component, which identifies the predictors or explanatory variables in the model and
3. The link function, which describes the functional relationship between the expected value of the random component and the systematic component.

A strength of GLMs is that by using the link component, we can model non-linear relationships as linear Models.

Examples

Ordinary least squares regression and Anova based on the normality assumption, are special cases of GLMs. The Logistic regression model is also a GLM, where the random component (response) is binomial, the link function is the logit, written as logit(π) = log[π/(1-π)]. The model is written as

logit(π) = a + b*X.

Thus, using the logit function allows us to write the functional relationship between Y = π(x) and X as a linear model, even though it is really non-linear (it is actually s-shaped).