General linear model: Difference between revisions

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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 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.

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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''_''ik'' is ''ki''^th observation of the ''k''^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:

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== 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=1312 DecemberJuly 20242025}}</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.

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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