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 in the univariate and multivariate case, is shown to be most convenient computationally as part of the multivariate procedure.