General linear model: Difference between revisions

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{{distinguishShort description|Statistical linear model}}
{{Distinguish|text=[[Multiple linear regression]], [[Generalized linear model]] or [[General linear methods]]}}

{{Regression bar}}

The '''general linear model''' or '''general multivariate regression model''' is simply a compact way of simultaneously writing multipleseveral [[Linear regression|multivariablemultiple linear regression]] models. In that sense it is not a separate statistical [[linear model]]. The multiplevarious multivariablemultiple linear regression models may be compactly written as <ref name="MardiaK1979Multivariate">{{Cite book |last1=Mardia author |first1= [[K. V. |author1-link=Kanti Mardia]], |last2=Kent |first2=J. T. Kent and|last3=Bibby |first3=J. M. Bibby |year=1979 |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 [[matrixMatrix (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 [[errorsErrors 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 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 incorporates(GLM) aencompasses number of differentseveral 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.

== Comparison to multiple linear regression ==

{{furtherFurther|Multiple linear regression}}

Multiple linear regression is a generalization of [[simple linear regression]] to the case of more than one independent variable, and a [[special case]] of general linear models, restricted to one dependent variable. The basic model for multiple linear regression is

:<math> Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \ldots + \beta_p X_{ip} + \epsilon_i</math> or more compactly <math>Y_i = \beta_0 + \sum \limits_{k=1}^{p} {\beta_k X_{ik}} + \epsilon_i</math>

for each observation ''i'' = 1, ... , ''n''.

In the formula above we consider ''n'' observations of one dependent variable and ''p'' independent variables. Thus, ''Y''_''i'' is the ''i''^th observation of the dependent variable, ''X''_''ijik'' is ''i''^th observation of the ''jk''^th independent variable, ''jk'' = 1, 2, ..., ''p''. The values ''ββk''_''j'' represent parameters to be estimated, and ''ε''_''i'' is the ''i''^th independent identically distributed normal error.

In the more general multivariate linear regression, there is one equation of the above form for each of ''m'' > 1 dependent variables that share the same set of explanatory variables and hence are estimated simultaneously with each other:

:<math> Y_{ij} = \beta_{0j} + \beta_{1j} X_{i1} + \beta_{2j}X_{i2} + \ldots + \beta_{pj} X_{ip} + \epsilon_{ij}</math> or more compactly <math>Y_{ij} = \beta_{0j} + \sum \limits_{k=1}^{p} { \beta_{kj} X_{ik}} + \epsilon_{ij}</math>

for all observations indexed as ''i'' = 1, ... , ''n'' and for all dependent variables indexed as ''j = 1'', ''...'' , ''m''.

Note that, since each dependent variable has its own set of regression parameters to be fitted, from a computational point of view the general multivariate regression is simply a sequence of standard multiple linear regressions using the same explanatory variables.

== Comparison to generalized linear model ==

The general linear model (GLM)<ref>Neter,and J.,the Kutner, M. H., Nachtsheim, C. J., & Wasserman, W. (1996). ''Applied[[generalized linear statistical models''model]] (Vol. 4, p. 318GLM). Chicago: Irwin.</ref><ref name=":10">Cohen,{{Cite J.,book Cohen,|last1=McCullagh |first1=P., West,|author1-link=Peter S.McCullagh G.,|last2=Nelder & Aiken, L|first2=J. SA. (2003).|author2-link=John AppliedNelder multiple regression/correlation analysis for the behavioral sciences.</ref> and the [[Generalized linear model|generalizeddate=January linear1, model1983 (GLiM)]]<ref name=":0">{{Citation|last=McCullagh|first=P.|titlechapter=An outline of generalized linear models |date=1989|worktitle=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 |doi-broken-date=12 July 2025}}</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 GLMgeneral linear model strictly assumes that the [[Errors and residuals|residuals]] will follow a [[Conditional probability distribution|conditionally]] [[normal distribution]],<ref name=":1">{{cite report |last1=Cohen |first1=J. |last2=Cohen |first2=P. |last3=West |first3=S. G. |last4=Aiken |first4=L. S. |author4-link=Leona S. Aiken |date=2003 |title=Applied multiple regression/correlation analysis for the behavioral sciences}}</ref> while the GLiMGLM 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" />. OfThe note,general thelinear GLMmodel is a special case of the GLiMGLM in which the distribution of the residuals follow a conditionally normal distribution.

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 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 GLiMGLM family. Commonly used models in the GLiMGLM 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>{{cite journal |last1=Gardner, |first1=W., |last2=Mulvey, |first2=E. P., & |last3=Shaw, |first3=E. C. (|date=1995). |title=Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative binomial models. ''|journal=Psychological bulletin'',Bulletin ''|volume=118''( |issue=3), |pages=392–404 |doi=10.1037/0033-2909.118.3.392. |pmid=7501743}}</ref> for count outcomes, and [[linear regression]] for continuous, normally distributed outcomes. This means that GLiMGLM may be spoken of as a general family of statistical models or as specific models for specific outcome types.

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 [[Exponential 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.

{| class="wikitable"

!

![[General linear model]]

![[Generalized linear model]]

|-

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|Examples

|[[ANOVA]], [[ANCOVA]], [[linear regression]]

|[[linear regression]], [[logistic regression]], [[Poisson regression]], gamma regression,<ref name=":02">{{cite book |titlelast1=GeneralizedMcCullagh Linear|first1=Peter Models,|author1-link=Peter SecondMcCullagh Edition|lastlast2=McCullaghNelder |firstfirst2=PeterJohn |author2-link=John Nelder, John|year=1989 |title=Generalized Linear Models |edition=2nd |publisher=Boca Raton: Chapman and Hall/CRC|year=1989 |isbn=978-0-412-31760-6 |ref=McCullagh1989|authorlink=Peter McCullagh|authorlink2=John Nelder}}</ref> [[general linear model]]

|-

|Extensions and related methods

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|[[R (programming language)|R]] package and function

|[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/lm.html lm()] in stats package (base R)

|[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/glm.html glm()] in stats package (base R) manova,

|-

|[[Matlab (programming language)|MatlabMATLAB]] function

|mvregress()

|glmfit()

|-

|[[SAS System(software)|SAS]] procedures

|[https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#glm_toc.htm PROC GLM], [https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#reg_toc.htm PROC REG]

|[https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#genmod_toc.htm PROC GENMOD], [https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#logistic_toc.htm PROC LOGISTIC] (for binary & ordered or unordered categorical outcomes)

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|[[Wolfram Language]] & [[Mathematica]] function

|LinearModelFit[]<ref>[http://reference.wolfram.com/language/ref/LinearModelFit.html LinearModelFit], Wolfram Language Documentation Center.</ref>

|GeneralizedLinearModelFit[]<ref>[http://reference.wolfram.com/language/ref/GeneralizedLinearModelFit.html GeneralizedLinearModelFit], Wolfram Language Documentation Center.</ref>

|-

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|ls<ref>[http://www.eviews.com/help/helpintro.html#page/content%2Fcommandcmd-ls.html ls], EViews Help.</ref>

|glm<ref>[http://www.eviews.com/help/helpintro.html#page/content%2Fcommandcmd-glm.html glm], EViews Help.</ref>

|-

|statsmodels Python Package

|[https://www.statsmodels.org/dev/user-guide.html#regression-and-linear-models regression-and-linear-models]

|[https://www.statsmodels.org/dev/glm.html GLM]

|}

== Applications ==

An application of the general linear model appears in the analysis of multiple [[brain scan]]s in scientific experiments where {{var|Y}} contains data from brain scanners, {{var|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 last1=Friston 10.1002/hbm.460020402|author1first1=K.J. Friston|last2=Holmes |author2first2=A.P. Holmes|last3=Worsley |author3first3=K.J. Worsley|last4=Poline |author4first4=J.-B. Poline|last5=Frith |author5first5=C.D. Frith|last6=Frackowiak |author6first6=R.S.J. Frackowiak | year =1995 1995| title = Statistical Parametric Maps in functional imaging: A general linear approach | journal = Human Brain Mapping | volume = 2| |pages = 189–210| |issue=4 |doi=10.1002/hbm.460020402 4|s2cid=9898609}}</ref>

== See also ==

* [[Bayesian multivariate linear regression]]

* [[F-test]]

* [[t-test]]

== Notes ==

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== References ==

* {{cite book |last1=Christensen |first1=Ronald |year=2020 |title=Plane Answers to Complex Questions: The Theory of Linear Models|last=Christensen|first=Ronald |___location=New York |publisher=Springer|year=2002| |edition=Third5th |isbn=0978-3-387030-9536132096-26}}

* {{cite book

* {{cite book |lastlast1=Wichura |firstfirst1=Michael J. |year=2006 |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}}

|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 journalbook | editor1-last = Rawlings | editor1-first = John O. |editor2-last=Pantula |editor2-first = Sastry G. |editor3-last=Dickey |editor3-first = David A. | doi year= 10.1007/b988901998 | title = Applied Regression Analysis | series = Springer Texts in Statistics | year = 1998 | isbn = 0-387-98454-2 | pmid doi= | pmc = | editor2-last = Pantula | editor3-last = Dickey 10.1007/b98890}}

* {{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 journal | editor1-last = Rawlings | editor1-first = John O. | editor2-first = Sastry G. | editor3-first = David A. | doi = 10.1007/b98890 | title = Applied Regression Analysis | series = Springer Texts in Statistics | year = 1998 | isbn = 0-387-98454-2 | pmid = | pmc = | editor2-last = Pantula | editor3-last = Dickey }}

{{statisticsStatistics}}

[[Category:Regression models]]