General linear model

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In statistics , the general linear model ( ALM or English general linear model , short: GLM ), also multivariate linear model ( English multivariate linear model ) is a linear model in which the dependent variable is not a scalar but a vector . In this case, conditioned linearity is also assumed as in the classical model of linear multiple regression , but with a matrix that replaces the vector of the classical model of linear multiple regression. Multivariate counterparts to the common least squares method and the generalized least squares method have been developed. ${\displaystyle Y}$ ${\displaystyle \operatorname {E} (\mathbf {y} \mid \mathbf {X} )=\mathbf {X} \mathbf {B} }$ ${\displaystyle \mathbf {B} }$ ${\displaystyle {\boldsymbol {\beta }}}$

The general linear model should not be confused with the multiple linear regression model , as this is also (although rarely) referred to as the general linear model. Likewise, general linear models are not to be confused with generalized linear models , whose natural English abbreviation is also GLM , but, unlike general linear models, do not assume a normally distributed response variable.

Model description

The basic prerequisite for the application of such models in statistical practice is the assumption that there is a linear relationship between the observed data and the known influencing variables. In order for such models to be statistically observed at all, it is additionally assumed that the data cannot be observed directly, but rather is subject to errors. In contrast to multiple linear regression, the general linear model has many values for each observation , so that instead of a vector, a matrix is present. General linear models can then be formally represented by matrix equations of the form ${\displaystyle i}$ ${\displaystyle (i=1,\dots ,n)}$ ${\displaystyle r}$ ${\displaystyle y}$ ${\displaystyle n\times r}$ ${\displaystyle \mathbf {Y} }$

${\displaystyle \mathbf {Y} =\mathbf {X} \mathbf {B} +\mathbf {U} }$

represent.

literature

• J. Andres: The general linear model . In Edgar Erdfelder , Rainer Mausfeld , Thorsten Meiser, Georg Rudinger (eds.): Handbook of quantitative methods. Beltz, Weinheim 1996, ISBN 3-621-27280-1 , pp. 185-200. https://doi.org/10.25521/HQM15 (full text freely available online)
• H. Moosbrugger: Linear models: regression and variance analyzes. 4th edition. Verlag Hans Huber, Bern / Göttingen / Toronto / Seattle 2011, ISBN 978-3-456-84965-2 .
• J. Werner: Linear Statistics. Beltz, Weinheim 1997, ISBN 3-621-27371-9 .