GLM Hypothesis Testing - Tests in Mixed Model Designs
Designs that contain random effects for one or more categorical predictor variables are called mixed-model designs. These types of designs, and the analysis of those designs, is also described in detail in the Introductory Overview of the Variance Components and Mixed Model ANOVA/ANCOVA module. Random effects are classification effects where the levels of the effects are assumed to be randomly selected from an infinite population of possible levels. The solution for the normal equations in mixed-model designs is identical to the solution for fixed-effect designs (i.e., designs which do not contain random effects). Mixed-model designs differ from fixed-effect designs only in the way in which effects are tested for significance. In fixed-effect designs, between effects are always tested using the mean square residual as the error term. In mixed-model designs, between effects are tested using relevant error terms based on the covariation of sources of variation in the design. Also, only the overparameterized model is used to code effects for categorical predictors in mixed-models, because the sigma-restricted model is overly restrictive.
The covariation of sources of variation in the design is estimated by the elements of a matrix called the Expected Mean Squares (EMS) matrix. This non-square matrix contains elements for the covariation of each combination of pairs of sources of variation and for each source of variation with Error. Specifically, each element is the mean square for one effect (indicated by the column) that is expected to be accounted by another effect (indicated by the row), given the observed covariation in their levels. Note that expected mean squares can be computing using any type of sums of squares from Type I through Type V. Once the EMS matrix is computed, it is used to the solve for the linear combinations of sources of random variation that are appropriate to use as error terms for testing the significance of the respective effects. This is done using Satterthwaite's method of denominator synthesis (Satterthwaite, 1946). Detailed discussions of methods for testing effects in mixed-models, and related methods for estimating variance components for random effects, can be found in the Introductory Overview of the Variance Components and Mixed Model ANOVA/ANCOVA module.
Whole Model Tests
Error Terms for Tests
Testing Hypotheses for Repeated Measures and Dependent Variables