Programming Univariate and Multivariate Analysis of Variance

A formulation of analysis of variance based on a model for the subclass means is presented. The deficiency of rank in the model matrix is handled, not by restricting the parameters, but by factoring the matrix as a product of two matrices, one providing a column basis for the model and the other representing linear functions of the parameters. In terms of the column basis and a diagonal matrix of subclass or incidence numbers, a compact matrix solution is derived which provides for testing a hierarchy of hypotheses in the non-orthogonal case. Two theorems are given showing that a column basis for crossed and/or nested designs can be constructed from Kronecker products of equi-angular vectors, contrast matrices, and identity matrices. This construction can be controlled in machine computation by a symbolic representation of each degree of freedom for hypothesis in the analysis. Provision for a multivariate analysis of variance procedure for multiple response data is described. Analysis of covariance, both ...