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Sunday, February 4, 2018

Magnetism - Questions and Answers ​in MRI
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The general linear model or multivariate regression model is a statistical linear model. It may be written as

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

where Y is a matrix with series of multivariate measurements (each column being a set of measurements on one of the dependent variables), X is a matrix of observations on independent variables that might be a design matrix (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 errors (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.


Video General linear model



Multiple linear regression

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

Y i = ? 0 + ? 1 X i 1 + ? 2 X i 2 + ... + ? p X i p + ? i . {\displaystyle 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, Yi is the ith observation of the dependent variable, Xij is ith observation of the jth independent variable, j = 1, 2, ..., p. The values ?j represent parameters to be estimated, and ?i is the ith independent identically distributed normal error.


Maps General linear model



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.


Generalized linear model - YouTube
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See also

  • Bayesian multivariate linear regression
  • Comparison of general and generalized linear models

Clinical Features of Celiac Disease: A Prospective Birth Cohort ...
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Notes


MInitab - basic two-way ANOVA using a general linear model - YouTube
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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. 

Source of article : Wikipedia