General linear model: Difference between revisions

Revision as of 21:27, 4 August 2020

Revision as of 15:43, 16 August 2020

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{{distinguish|text=[[Multiple linear regression]], [[Generalized linear model]] or [[General linear methods]]}}

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The '''general linear model''' or '''general multivariate regression model''' is simply a compact way of simultaneously writing several [[Linear regression|multiple linear regression]] models. In that sense it is not a separate statistical [[linear model]]. The various multiple linear regression models may be compactly written as <ref name="MardiaK1979Multivariate">{{Cite book | author = [[K. V. Mardia]], J. T. Kent and J. M. Bibby | title = Multivariate Analysis | publisher = [[Academic Press]] | year = 1979 | isbn = 0-12-471252-5}}</ref>

: <math>\mathbf{Y} = \mathbf{X}\mathbf{B} + \mathbf{U},</math>

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The general linear model (GLM)<ref>Neter, J., Kutner, M. H., Nachtsheim, C. J., & Wasserman, W. (1996). ''Applied linear statistical models'' (Vol. 4, p. 318). Chicago: Irwin.</ref><ref name=":1">Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences.</ref> and the [[Generalized linear model|generalized linear model (GLiM)]]<ref name=":0">{{Citation|last=McCullagh|first=P.|title=An outline of generalized linear models|date=1989|work=Generalized Linear Models|pages=21–47|publisher=Springer US|isbn=9780412317606|last2=Nelder|first2=J. A.|doi=10.1007/978-1-4899-3242-6_2}}</ref><ref>Fox, J. (2015). ''Applied regression analysis and generalized linear models''. Sage Publications.</ref> are two commonly used families of [[Statistics|statistical methods]] to relate some number of continuous and/or categorical [[Dependent and independent variables|predictors]] to a single [[Dependent and independent variables|outcome variable]].

The main difference between the two approaches is that the GLM strictly assumes that the [[Errors and residuals|residuals]] will follow a [[Conditional probability distribution|conditionally]] [[normal distribution]],<ref name=":1" /> while the GLiM loosens this assumption and allows for a variety of other [[Distribution (mathematics)|distributions]] from the [[Exponentialexponential family|exponential]] family for the residuals<ref name=":0" />. Of note, the GLM is a special case of the GLiM in which the distribution of the residuals follow a conditionally normal distribution.

The distribution of the residuals largely depends on the type and distribution of the outcome variable; different types of outcome variables lead to the variety of models within the GLiM family. Commonly used models in the GLiM family include [[Logistic regression|binary logistic regression]]<ref>Hosmer Jr, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). ''Applied logistic regression'' (Vol. 398). John Wiley & Sons.</ref> for binary or dichotomous outcomes, [[Poisson regression]]<ref>Gardner, W., Mulvey, E. P., & Shaw, E. C. (1995). Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative binomial models. ''Psychological bulletin'', ''118''(3), 392.</ref> for count outcomes, and [[linear regression]] for continuous, normally distributed outcomes. This means that GLiM may be spoken of as a general family of statistical models or as specific models for specific outcome types.

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