Double Skew-Dual Scaling: A Conjoint Scaling of Two Sets of Objects Related by a Dominance Matrix

Consider two sets of objects, {alpha(1), em leader, alpha(n)} and {beta(1), em leader, beta(m), such as n subjects solving m tasks, or n stimuli presented first and m stimuli presented second in a pairwise comparison experiment. Let any pair (alpha(i), beta(j)) be associated with a real number a(ij), interpreted as the degree of dominance of alpha(i) over beta(j) (e.g., the probability of alpha(i) relating in a certain way to beta(j)). Intuitively, the problem addressed in this paper is how to conjointly, in a "naturally" coordinated fashion, characterize the alpha-objects and beta-objects in terms of their overall tendency to dominate or be dominated. The gist of the solution is as follows. Let A denote the nxm matrix of a(ij) values, and let there be a class of monotonic transformations straight phi with nonnegative codomains. For a given straight phi, a complementary matrix B is defined so that straight phi(a(ij))+straight phi(b(ij))=const, and one computes vectors D(alpha) and D(beta) (the dominance values for alpha-objects and beta-objects) by solving the equations straight phi(A) straight phi(D(beta))/Sigma;straight phi(D(beta))=straight phi(D(alpha)) and straight phi(B(T)) straight phi(D(alpha))/Sigmastraight phi(D(alpha))=straight phi(D(beta)), where (T) is transposition, Sigma is the sum of elements, and straight phi applies elementwise. One also computes vectors S(alpha) and S(beta) (the subdominance values for alpha-objects and beta-objects) by solving the equations straight phi(B) straight phi(S(beta))/Sigmastraight phi(S(beta))=straight phi(S(alpha)) and straight phi(A(T)) straight phi(S(alpha))/Sigmastraight phi(S(alpha))=straight phi(S(beta)). The relationship between S-vectors and D-vectors is complex: intuitively, D(alpha) characterizes the tendency of an alpha-object to dominate beta-objects with large dominance values, whereas S(alpha) characterizes the tendency of an alpha-objects to fail to dominate beta-objects with large subdominance values. For classes containing more than one straight phi-transformation, one can choose an optimal straight phi as the one maximizing some measure of discrimination between individual elements of vectors straight phi(D(alpha)), straight phi(D(beta)), straight phi(S(alpha)), and straight phi(S(beta)), such as the product or minimum of these vectors' variances. The proposed analysis of dominance matrices has only superficial similarities with the classical dual scaling (Nishisato, 1980). Copyright 1999 Academic Press.