General linear model

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, t-test and F-test. 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.

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 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 (4): 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. MR 2283455.
• Rawlings, John O.; Pantula, Sastry G.; Dickey, David A., eds. (1998). "Applied Regression Analysis". Springer Texts in Statistics. doi:10.1007/b98890. ISBN 0-387-98454-2.