When is a Sum of Squares an Analysis of Variance?

Abstract

A sum of squares can be partitioned into sums of quadratic forms whose kernels are projections. If these projections are mutually orthogonal and add to the identity, then, under the classical fixed effects linear model, the terms of the decomposition are mutually independent and are distributed as multiples of chi-square. In this paper we exhibit necessary and sufficient conditions for a specified sum of squares decomposition to have this property in the case of the mixed model.