General linear model

Not to be confused with generalized linear model.

The general linear model (GLM) is a statistical linear model. It may be written as^[1]

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

where Y is a matrix with series of multivariate measurements, X is a matrix that might be a design matrix, B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors or noise. The errors are usually assumed to follow a multivariate normal distribution. If the errors do not follow a multivariate normal distribution, generalized linear models may be used to relax assumptions about Y and U.

The general linear model incorporates a number of different statistical models: ANOVA, ANCOVA, MANOVA, MANCOVA, ordinary linear regression, t-test and F-test. If there is only one column in Y (i.e., one dependent variable) then the model can also be referred to as the multiple regression model (multiple linear regression).

Hypothesis tests with the general linear model can be made in two ways: 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.

Applications

An application of the general linear model appears in the analysis of multiple brain scans in scientific experiments where Y contains data from brain scanners, 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.^[2]

See also

• Comparison of general and generalized linear models
• Bayesian multivariate linear regression

Notes

1. ^ K. V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471252-5. 
2. ^ K.J. Friston, A.P. Holmes, K.J. Worsley, J.-B. Poline, C.D. Frith and R.S.J. Frackowiak (1995). "Statistical Parametric Maps in functional imaging: A general linear approach". Human Brain Mapping 2: 189–210. doi:10.1002/hbm.460020402. 

References

• Christensen, Ronald (2002). Plane Answers to Complex Questions: The Theory of Linear Models (Third ed.). New York: Springer. ISBN 0-387-95361-2. 
• Wichura, Michael J. (2006). The coordinate-free approach to linear models. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press. pp. xiv+199. ISBN 978-0-521-86842-6, ISBN 0-521-86842-4. MR2283455. 

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