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{{Distinguish2|[[generalized linear model]] or [[general linear methods]]}}
{{Regression bar}}
The ''general linear model'' (GLM) is a statistical [[linear model]].
It may be 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>


where '''Y''' is a matrix with series of multivariate measurements, '''X''' is a matrix that might be a [[design matrix]], '''B''' is a matrix containing parameters that are usually to be estimated and '''U''' is a matrix containing [[errors and residuals in statistics|errors or noise]].
The errors are usually assumed to follow a [[multivariate normal distribution]].  If the errors do not follow a multivariate normal distribution, [[generalized linear models]] may be used to relax assumptions about '''Y''' and '''U'''.


The general linear model incorporates a number of different statistical models: [[ANOVA]], [[ANCOVA]], [[MANOVA]], [[MANCOVA]], ordinary [[linear regression]], [[t-test]] and [[F-test]]. The general linear model is a generalization of multiple linear regression model to the case of more than one dependent variable. If '''Y''', '''B''', and '''U''' were column vectors, the matrix equation above would represent multiple linear regression.
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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.
 
== Multiple linear regression ==
Multiple linear regression is a generalization of [[linear regression]] by considering more than one independent variable, and a specific case of general linear models formed by restricting the number of dependent variables to one. The basic model for linear regression is
 
:<math> Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \ldots + \beta_p X_{ip} + \epsilon_i.</math>
 
In the formula above we consider ''n'' observations of one dependent variable and ''p'' independent variables. Thus, ''Y''<sub>''i''</sup> is the ''i''<sup>th</sup> observation of the dependent variable, ''X''<sub>''ij''</sub> is ''i''<sup>th</sup> observation of the ''j''<sup>th</sup> independent variable, ''j'' = 1, 2, ..., ''p''. The values ''&beta;''<sub>''j''</sub> represent parameters to be estimated, and ''&epsilon;''<sub>''i''</sub> is the ''i''<sup>th</sup> independent identically distributed normal error.
 
== Applications ==
An application of the general linear model appears in the analysis of multiple [[brain scan]]s in scientific experiments where '''Y''' contains data from brain scanners, '''X''' contains experimental design variables and confounds. It is usually tested in a univariate way (usually referred to a ''mass-univariate'' in this setting) and is often referred to as [[statistical parametric mapping]].<ref>{{Cite journal| doi = 10.1002/hbm.460020402| author = K.J. Friston, A.P. Holmes, K.J. Worsley, J.-B. Poline, C.D. Frith and R.S.J. Frackowiak| year = 1995| title = Statistical Parametric Maps in functional imaging: A general linear approach| journal = Human Brain Mapping| volume = 2| pages = 189–210| issue = 4}}</ref>
 
== See also ==
* [[Comparison of general and generalized linear models]]
* [[Bayesian multivariate linear regression]]
 
== Notes ==
{{Reflist}}
 
==References==
* {{cite book
|title=Plane Answers to Complex Questions: The Theory of Linear Models|last=Christensen|first=Ronald|location=New York|publisher=Springer|year=2002| edition=Third|isbn=0-387-95361-2}}
* {{cite book|last=Wichura|first=Michael J.|title=The coordinate-free approach to linear models|series=Cambridge Series in Statistical and Probabilistic Mathematics|publisher=Cambridge University Press|location=Cambridge|year=2006|pages=xiv+199|isbn=978-0-521-86842-6|mr=2283455|ref=harv}}
* {{cite doi | 10.1007/b98890 | noedit}}
 
{{statistics}}
 
[[Category:Multivariate statistics]]
[[Category:Regression analysis]]

Latest revision as of 23:46, 14 April 2014


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