 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]}
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, ttest and Ftest. 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.
Contents
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. V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0124712525.
 ^ 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 0387953612.
 Wichura, Michael J. (2006). The coordinatefree approach to linear models. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press. pp. xiv+199. ISBN 9780521868426, ISBN 0521868424. MR2283455.
Summary tablesPearson productmoment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Partial correlation · Scatter plotBar chart · Biplot · Box plot · Control chart · Correlogram · Forest plot · Histogram · QQ plot · Run chart · Scatter plot · Stemplot · Radar chartData collection Designing studiesDesign of experiments · Factorial experiment · Randomized experiment · Random assignment · Replication · Blocking · Optimal designUncontrolled studiesStatistical inference Frequentist inferenceSpecific testsZtest (normal) · Student's ttest · Ftest · Pearson's chisquared test · Wald test · Mann–Whitney U · Shapiro–Wilk · Signedrank · Kolmogorov–Smirnov testCorrelation and regression analysis Errors and residuals · Regression model validation · Mixed effects models · Simultaneous equations modelsNonstandard predictorsPartition of varianceCategorical, multivariate, timeseries, or survival analysis Decomposition (Trend · Stationary process) · ARMA model · ARIMA model · Vector autoregression · Spectral density estimationApplications Categories: Statistics stubs
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