Editing General linear model (section)

== Comparison to generalized linear model ==
The general linear model and the [[generalized linear model]] (GLM)<ref name=":0">{{Cite book |last1=McCullagh |first1=P. |author1-link=Peter McCullagh |last2=Nelder |first2=J. A. |author2-link=John Nelder |date=January 1, 1983 |chapter=An outline of generalized linear models |title=Generalized Linear Models |pages=21–47 |publisher=Springer US |isbn=9780412317606 |doi=10.1007/978-1-4899-3242-6_2 |doi-broken-date=13 December 2024}}</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 general 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 GLM loosens this assumption and allows for a variety of other [[Distribution (mathematics)|distributions]] from the [[exponential family]] for the residuals.<ref name=":0"/> The general linear model is a special case of the GLM 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 GLM family. Commonly used models in the GLM 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 |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 GLM 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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|Typical estimation method
|[[Least squares]], [[best linear unbiased prediction]]
|[[Maximum likelihood]] or [[Bayesian probability|Bayesian]]
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|Examples
|[[ANOVA]], [[ANCOVA]], [[linear regression]]
|[[linear regression]], [[logistic regression]], [[Poisson regression]], gamma regression,<ref name=":02">{{cite book |last1=McCullagh |first1=Peter |author1-link=Peter McCullagh |last2=Nelder |first2=John |author2-link=John Nelder |year=1989 |title=Generalized Linear Models |edition=2nd |publisher=Boca Raton: Chapman and Hall/CRC |isbn=978-0-412-31760-6 |ref=McCullagh1989}}</ref> general linear model
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|Extensions and related methods
|[[Multivariate analysis of variance|MANOVA]], [[Multivariate analysis of covariance|MANCOVA]], [[Mixed model|linear mixed model]]
|[[generalized linear mixed model]] (GLMM), [[Generalized estimating equation|generalized estimating equations]] (GEE)
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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,
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|[[MATLAB]] function
|mvregress()
|glmfit()
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|[[SAS (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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|[[Stata]] command
|regress
|glm
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|[[SPSS]] command
|[https://stats.idre.ucla.edu/spss/output/regression-analysis/ regression], [https://stats.idre.ucla.edu/spss/library/spss-librarymanova-and-glm-2/ glm]
|genlin, logistic
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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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|[[EViews]] command
|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>
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|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]
|}