General linear model

The general linear model or general multivariate regression model is a compact way of simultaneously writing several <a href="/facts/Multiple_linear_regression/5n998IhK">multiple linear regression</a> models. In that sense it is not a separate statistical <a href="/facts/Linear_model/x1we3e1d">linear model</a>. The various multiple linear regression models may be compactly written as

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

where Y is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> with series of multivariate measurements (each column being a set of measurements on one of the <a href="/facts/Dependent_variable/dINfzIF2">dependent variables</a>), X is a matrix of observations on <a href="/facts/Independent_variable/dINfzIF2">independent variables</a> that might be a <a href="/facts/Design_matrix/yRTdMhf4">design matrix</a> (each column being a set of observations on one of the independent variables), B is a matrix containing parameters that are usually to be estimated and U is a matrix containing <a href="/facts/Errors_and_residuals_in_statistics/Ef0LgAS4">errors</a> (noise). The errors are usually assumed to be uncorrelated across measurements, and follow a <a href="/facts/Multivariate_normal_distribution/2Xfegqz2">multivariate normal distribution</a>. If the errors do not follow a multivariate normal distribution, <a href="/facts/Generalized_linear_model/Sf1htm5W">generalized linear models</a> may be used to relax assumptions about Y and U.
The general linear model incorporates a number of different statistical models: <a href="/facts/Analysis_of_variance/HHR2jkDk">ANOVA</a>, <a href="/facts/Analysis_of_covariance/yJB0SUy0">ANCOVA</a>, <a href="/facts/Multivariate_analysis_of_variance/YacRVgl9">MANOVA</a>, <a href="/facts/Multivariate_analysis_of_covariance/vkr3Pu6v">MANCOVA</a>, ordinary <a href="/facts/Linear_regression/5n998IhK">linear regression</a>, <a href="/facts/T-test/dMH8uC28">t-test</a> and <a href="/facts/F-test/Kq8lzNqY">F-test</a>. The general linear model is a generalization of multiple linear regression to the case of more than one dependent variable. If Y, B, and U were <a href="/facts/Column_vector/pPuRr9Xk">column vectors</a>, the matrix equation above would represent multiple linear regression.
Hypothesis tests with the general linear model can be made in two ways: <a href="/facts/Multivariate_statistics/tDuNcWrL">multivariate</a> or as several independent <a href="/facts/Univariate/PYwC7u5H">univariate</a> 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.

General linear model open-in-new

General linear model