Kronecker Products in ANOVA.A First Step

In the complete balanced model for the analysis of variance, the equivalence of sums of squares and quadratic forms is seen to imply well-fitting patterns involving Kronecker products of identity matrices and scalar multiples of matrices with all elements equal to 1.The questions of symmetry, idempotency, and orthogonality so central to this topic are answered by simple multiplications; ranks are determined from simple traces.The associations between the forms of the two-factor model are presented here in a way that is accessible to first-year students and makes generalizations to higher order models transparent.The lack of patterns in incomplete or unbalanced models is noted.Additional steps in design and analysis are suggested in the references.