Approximate sums of squares in analysis of variance

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Consider the two-way crossed classification model, in which there are a levels of the factor A, b levels of the factor B and nij observations y(i,j,k), k=1,⋯,n(i,j), for the (i,j)th cell, i=1,⋯,a, j=1,⋯,b. The sum of squares for testing interactions in this model can be written as Q=∑(i,j)n(i,j)(y(i,j,⋅)/n(i,j)−y(i,⋅,⋅)/n(i,⋅)−y(⋅,j,⋅)/n(j,⋅)+y(⋅,⋅,⋅)/n(⋅,⋅))^2, where y(i,j,⋅)=∑(k)y(i,j,k), y(i,⋅,⋅)=∑(j)y(i,j,⋅), y(⋅,j,⋅)=∑(i)y(i,j,⋅), y(⋅,⋅,⋅)=∑(i)y(i,⋅,⋅), n(i,⋅)=∑(j)n(i,j), n(⋅,j)=∑(i)n(i,j) and n(⋅,⋅)=∑(i)n(i,⋅). It is well known that if the numbers of observations are proportional, i.e., if (1) n(i,j)=n(i,⋅)n(⋅,j)/n(⋅,⋅) for all i=1,⋯,a and j=1,⋯,b, then the quadratic form Q(0)=∑(i,j)y(i,j,⋅)^2/n(i,j)−∑(i)y(i,⋅,⋅)^2/n(i,⋅)−∑(j)y^2(⋅,j,⋅)/n(⋅,j)+y^2(⋅,⋅,⋅)/n(⋅,⋅) is nonnegative definite, being then identical with Q. The author proves the converse of this implication; he shows that the nonnegative definiteness of Q0 implies the proportionality condition (1). He considers a similar problem also for the case of the three-way crossed classification model.