Definitions

# General linear model

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

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

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 residual is usually assumed to follow a multivariate normal distribution. If the residual is not 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 and mass-univariate.

## 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 mass-univariate way and is often referred to as statistical parametric mapping.

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