Generalized Linear Model (GLM) Introductory Overview - Summary of Computations
To conclude this discussion of the ways in which the general linear model extends and generalizes regression methods, the general linear model can be expressed as
YM = Xb + e
Here Y, X, b, and e are as described for the multivariate regression model and M is an m x s matrix of coefficients defining s linear transformation of the dependent variable. The normal equations are
X'Xb =X' YM
and a solution for the normal equations is given by
b = (X'X)`X' YM
Here the inverse of X'X is a generalized inverse if X'X contains redundant columns.
Add a provision for analyzing linear combinations of multiple dependent variable, add a method for dealing with redundant predictor variables and recoded categorical predictor variables, and the major limitations of multiple regression are overcome by the general linear model.
Matrix ill conditioning. The Generalized Linear Model (GLM) module provides a comprehensive set of techniques for analyzing designs with either full-rank or singularity design matrices. It should be noted, however, that in certain designs it is difficult to consistently identify matrix singularity and redundancy of columns in the design matrix. Specifically, numerical round-off in designs with very different variances of values in different columns of the design matrix (which can occur, for example, in factorial regression and polynomial regression designs) can and sometimes does lead to inconsistent results. Generalized Linear Model (GLM) will issue a warning when this condition is detected in an analysis. Re-scaling of continuous predictor variables so that they have similar variances (for example, by standardizing values on the predictor variables) will often prevent matrix ill conditioning.
Other Generalized Linear Model (GLM) Introductory Overview Topics
A detailed discussion of univariate and multivariate ANOVA techniques can also be found in Introductory Overview section of the ANOVA/MANOVA module; a discussion of multiple regression methods is also provided in the Introductory Overview of the Multiple Regression module.