The general linear model 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 be uncorrelated across measurements, and 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, ttest and Ftest. The general linear model is a generalization of multiple linear regression model to the case of more than one dependent variable. If Y, B, and U were column vectors, the matrix equation above would represent 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.
Contents

Multiple linear regression 1

Applications 2

See also 3

Notes 4

References 5
Multiple linear regression
Multiple linear regression is a generalization of linear regression by considering more than one independent variable, and a specific case of general linear models formed by restricting the number of dependent variables to one. The basic model for linear regression is

Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \ldots + \beta_p X_{ip} + \epsilon_i.
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_{ij} is i^{th} observation of the j^{th} independent variable, j = 1, 2, ..., p. The values β_{j} represent parameters to be estimated, and ε_{i} is the i^{th} independent identically distributed normal error.
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 massunivariate in this setting) and is often referred to as statistical parametric mapping.^{[2]}
See also
Notes

^

^ 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 (4): 189–210.
References

Christensen, Ronald (2002). Plane Answers to Complex Questions: The Theory of Linear Models (Third ed.). New York: Springer.

Wichura, Michael J. (2006). The coordinatefree approach to linear models. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press. pp. xiv+199.

Rawlings, John O.; Pantula, Sastry G.; Dickey, David A., eds. (1998). "Applied Regression Analysis". Springer Texts in Statistics.
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