Rectangular Proximity Research Articles

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.

Internal multidimensional unfolding about a single-ideal—A probabilistic solution

A solution is presented for an internal multidimensional unfolding problem in which all the judgments of a rectangular proximity matrix are a function of a single-ideal object. The solution is obtained by showing that when real and ideal objects are represented by normal distributions in a multidimensional Euclidean space, a vector of distances among a single-ideal and multiple real objects follows a multivariate quadratic form in normal variables distribution. An approximation to the vector's probability density function (PDF) is developed which allows maximum likelihood (ML) solutions to be estimated. Under dependent sampling, the likelihood function contains information about the parametric distances among real object pairs, permitting the estimation of single-ideal solutions and leading to more robust multiple-ideal solutions. Tests for single- vs. multiple-ideal solutions and dependent vs. independent sampling are given. Properties of the proposed model and parameter recovery are explored. Empirical illustrations are also provided.

The estimation of ultrametric and path length trees from rectangular proximity data

A least-squares algorithm for fitting ultrametric and path length or additive trees to two-way, two-mode proximity data is presented. The algorithm utilizes a penalty function to enforce the ultrametric inequality generalized for asymmetric, and generally rectangular (rather than square) proximity matrices in estimating an ultrametric tree. This stage is used in an alternating least-squares fashion with closed-form formulas for estimating path length constants for deriving path length trees. The algorithm is evaluated via two Monte Carlo studies. Examples of fitting ultrametric and path length trees are presented.