GRM Introductory Overview - Model Building in GRM

Unlike the multiple regression model, which is used to analyze designs with continuous predictor variables, the general linear model can be used to analyze any ANOVA design with categorical predictor variables, any ANCOVA design with both categorical and continuous predictor variables, as well as any regression design with continuous predictor variables. Effects for categorical predictor variables can be coded in the design matrix X using either the overparameterized model or the sigma-restricted model.

Only the sigma-restricted parameterization can be used for model-building
True to its description as general, the general linear model can be used to analyze designs with effects for categorical predictor variables which are coded using either parameterization method. In many uses of the general linear model, it is arbitrary whether categorical predictors are coded using the sigma-restricted or the overparameterized coding. When one desires to build models, however, the use of the overparameterized model is unsatisfactory; lower-order effects for categorical predictor variables are redundant with higher-order containing interactions, and therefore cannot be fairly evaluated for inclusion in the model when higher-order containing interactions are already in the model.

This problem does not occur when categorical predictors are coded using the sigma-restricted parameterization, so only the sigma-restricted parameterization is available in GRM.

Designs that cannot be represented using the sigma-restricted parameterization
The sigma-restricted parameterization can be used to represent most, but not all types of designs. Specifically, the designs which cannot be represented using the sigma-restricted parameterization are designs with nested effects, such as nested ANOVA and separate slope designs, and mixed-model designs with random effects. Any other type of ANOVA, ANCOVA, or regression design can be represented using the sigma-restricted parameterization, and can therefore be analyzed in GRM.
Model building for designs with multiple dependent variables
Stepwise and best-subset model-building techniques are well-developed for regression designs with a single dependent variable (e.g., see Cooley and Lohnes, 1971; Darlington, 1990; Hocking Lindeman, Merenda, and Gold, 1980; Morrison, 1967; Neter, Wasserman, and Kutner, 1985; Pedhazur, 1973; Stevens, 1986; Younger, 1985). Using the sigma-restricted parameterization and general linear model methods, these model-building techniques can be readily applied to any ANOVA design with categorical predictor variables, any ANCOVA design with both categorical and continuous predictor variables, as well as any regression design with continuous predictor variables. Building models for designs with multiple dependent variables, however, involves considerations that are not typically addressed by the general linear model. Model-building techniques for designs with multiple dependent variables are available in the Structural Equation Modeling (SEPATH) module.

Other GRM Introductory Overview Topics

A detailed discussion of univariate and multivariate ANOVA techniques can also be found in the Introductory Overview of the ANOVA/MANOVA module; a discussion of Multiple Regression methods is provided in the Overviews. Discussion of the ways in which the linear regression model is extended by the general linear model can be found in the Introductory Overview of the GLM module.