GLM Introductory Overview - Mixed Model ANOVA and ANCOVA
Designs containing random effects for one or more categorical predictor variables are called mixed-model designs. 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 that 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 squared residual as the error term. In mixed-model designs, between effects are tested using relevant error terms based on the covariation of random sources of variation in the design. Specifically, this is done using Satterthwaite's method of denominator synthesis (Satterthwaite, 1946), which finds the linear combinations of sources of random variation that serve as appropriate error terms for testing the significance of the respective effect of interest. A basic discussion of these types of designs, and methods for estimating variance components for the random effects, can also be found in the Introductory Overview of the Variance Components and Mixed Model ANOVA/ANCOVA module.
Mixed-model designs, like nested designs and separate slope designs, are designs in which the sigma-restricted coding of categorical predictors is overly restrictive. Mixed-model designs require estimation of the covariation between the levels of categorical predictor variables, and the sigma-restricted coding of categorical predictors suppresses this covariation. Thus, only the overparameterized model is used to represent mixed-model designs (some programs will use the sigma-restricted approach and a so-called "restricted model" for random effects; however, only the overparameterized model as implemented in GLM applies to both balanced and unbalanced designs, as well as designs with missing cells; see Searle, Casella, & McCulloch, 1992, p. 127).
It is important to recognize, however, that sigma-restricted coding can be used to represent any between design, with the exceptions of mixed-model, nested, and separate slope designs. Furthermore, some types of hypotheses can only be tested using the sigma-restricted coding (i.e., the effective hypothesis, Hocking, 1996), thus the greater generality of the overparameterized model for representing between designs does not justify it being used exclusively for representing categorical predictors in the general linear model.
Between-Subject Designs:
Within-Subject (Repeated Measures) Designs:
Multivariate Designs: