What is an Analysis of Variance?

Abstract

The analysis of variance is usually regarded as being concerned with sums of squares of numbers and independent quadratic forms of random variables. In this paper, an alternative interpretation is discussed. For certain classes of dispersion models for finite or infinite arrays of random variables, a form of generalized spectral analysis is described and its intuitive meaning explained. The analysis gives a spectral decomposition of each dispersion in the class, incorporating an analysis of the common variance, and an associated orthogonal decomposition of each of the random variables. One by-product of this approach is a clear understanding of the similarity between the spectral decomposition for second-order stationary processes and the familiar linear models with random effects.