General linear model: Difference between revisions

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== Comparison to generalized linear model ==

The general linear model and the [[Generalized linear model|generalized linear model (GLM)]]<ref name=":0">{{Citation|lastlast1=McCullagh|firstfirst1=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 general linear model strictly assumes that the [[Errors and residuals|residuals]] will follow a [[Conditional probability distribution|conditionally]] [[normal distribution]],<ref name=":1">Cohen, J., Cohen, P., West, S. G., & [[Leona S. Aiken|Aiken, L. S.]] (2003). 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" /> Of note, 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. |title=Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative binomial models. |journal=Psychological Bulletin |date=1995 |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.

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== 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 = 10.1002/hbm.460020402|author1=K.J. Friston |author2=A.P. Holmes |author3=K.J. Worsley |author4=J.-B. Poline |author5=C.D. Frith |author6=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|s2cid=9898609 }}</ref>

== See also ==

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|title=Plane Answers to Complex Questions: The Theory of Linear Models|last=Christensen|first=Ronald|___location=New York|publisher=Springer|year=2020| edition=Fifth|isbn=978-3-030-32096-6}}

* {{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}}

* {{Cite journalbook | 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 | editor2-last = Pantula | editor3-last = Dickey }}

{{statistics}}