Editing General linear model (section)

{{Short description|Statistical linear model}}
{{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 a compact way of simultaneously writing several [[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 |last1=Mardia |first1=K. V. |author1-link=Kanti Mardia |last2=Kent |first2=J. T. |last3=Bibby |first3=J. M. |year=1979 |title=Multivariate Analysis |publisher=[[Academic Press]] |isbn=0-12-471252-5}}</ref>
: <math>\mathbf{Y} = \mathbf{X}\mathbf{B} + \mathbf{U},</math>

where '''Y''' is a [[Matrix (mathematics)|matrix]] with series of multivariate measurements (each column being a set of measurements on one of the [[dependent variable]]s), '''X''' is a matrix of observations on [[independent variable]]s that might be a [[design matrix]] (each column being a set of observations on one of the independent variables), '''B''' is a matrix containing parameters that are usually to be estimated and '''U''' is a matrix containing [[Errors and residuals in statistics|errors]] (noise). The errors are usually assumed to be uncorrelated across measurements, and follow a [[multivariate normal distribution]]. If the errors do not follow a multivariate normal distribution, [[generalized linear model]]s may be used to relax assumptions about '''Y''' and '''U'''.


The general linear model (GLM) encompasses several statistical models, including [[Analysis of variance|ANOVA]], [[Analysis of covariance|ANCOVA]], [[Multivariate analysis of variance|MANOVA]], [[Multivariate analysis of covariance|MANCOVA]], ordinary [[linear regression]]. Within this framework, both [[t-test|''t''-test]] and [[F-test|''F''-test]] can be applied. The general linear model is a generalization of multiple linear regression to the case of more than one dependent variable. If '''Y''', '''B''', and '''U''' were [[column vector]]s, the matrix equation above would represent multiple linear regression.

Hypothesis tests with the general linear model can be made in two ways: [[multivariate statistics|multivariate]] or as several independent [[univariate]] tests. In multivariate tests the columns of '''Y''' are tested together, whereas in univariate tests the columns of '''Y''' are tested independently, i.e., as multiple univariate tests with the same design matrix.