Representation of individual differences in rectangular proximity data through anti-Q matrix decomposition

Decomposition of a rectangular proximity matrix into a sum of equal-sized matrices, each constrained to display a certain order pattern, called an anti-Q form, can be interpreted as a less restrictive analogue of singular value decomposition. Both decomposition techniques share the ultimate goal of identifying a parsimonious representation of the original matrix in the form of an approximation through a small sum of components. The specific patterning of the extracted anti-Q matrices lends to subsequent analyses steps (by treating each anti-Q component as a separate proximity matrix), and representations as a discrete two-mode ultrametric or a continuous unidimensional unfolding. Because both models entail the same number of estimated weights, a direct comparison of their fit values can be carried out. Thus, for each extracted anti-Q matrix we can distinguish whether a categorical (discrete) or a dimensional structure provides the better representation. A generalization of anti-Q decomposition is proposed to a cube formed by rectangular proximity matrices observed from multiple data sources, and therefore, as a way of representing individual differences. In addressing this task within a 'deviation-from-the-mean paradigm', the individual proximity matrices are decomposed against a reference structure derived from the aggregate body of data. Assessment of overall agreement with the reference structure for each data source, as well as discrete and continuous representations fit to each extracted confirmatory anti-Q matrix, provide a detailed inter- and intra-individual analysis of the predominate characterization of the relationships between row and column objects. As an illustrative application, we analyze criterion-based rankings given by males and females for various contraceptive measures.