Generalized linear model: Difference between revisions

Revision as of 06:18, 31 May 2023

Revision as of 01:30, 15 September 2023

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In a generalized linear model (GLM), each outcome '''Y''' of the [[dependent variable]]s is assumed to be generated from a particular [[probability distribution|distribution]] in an [[exponential family]], a large class of [[probability distributions]] that includes the [[normal distribution|normal]], [[binomial distribution|binomial]], [[poisson distribution|Poisson]] and [[gamma distribution|gamma]] distributions, among others. The mean, '''''μ''''', of the distribution depends on the independent variables, '''X''', through:

: <math>\operatorname{E}(\mathbf{Y}|\mid\mathbf{X}) = \boldsymbol{\mu} = g^{-1}(\mathbf{X}\boldsymbol{\beta}) </math>

where E('''Y'''&nbsp;|&nbsp;'''X''') is the [[expected value]] of '''Y''' [[conditional expectation|conditional]] on '''X'''; '''X''&beta;''''' is the ''linear predictor'', a linear combination of unknown parameters '''''&beta;'''''; ''g'' is the link function.

In this framework, the variance is typically a function, '''V''', of the mean:

:<math> \operatorname{Var}(\mathbf{Y}|\mid\mathbf{X}) = \operatorname{V}(g^{-1}(\mathbf{X}\boldsymbol{\beta})). </math>

It is convenient if '''V''' follows from an exponential family of distributions, but it may simply be that the variance is a function of the predicted value.

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| [[multinomial distribution|Multinomial]]

| ''K''-vector of integer: <math>[0,N]</math> || count of occurrences of different types (1, ..., ''K'') out of ''N'' total ''K''-way occurrences

|}