(non-spatial) and continuous (spatial) structures can be fitted to a proximity
matrix to increase the information extracted about the relations among the row
and column objects vis-à-vis a representation featuring only a single
structure. However, using multiple discrete and continuous structures often
leads to ambiguous results that make it difficult to determine the most
faithful representation of the proximity matrix in question. We propose to
resolve this dilemma by using a nonmetric analogue of spectral matrix
decomposition, namely, the decomposition of the proximity matrix into a sum
of equally-sized matrices, restricted only to display an order-constrained
patterning, the anti-Robinson (AR) form. Each AR matrix captures a unique
amount of the total variability of the original data. As our ultimate goal, we
seek to extract a small number of matrices in AR form such that their sum
allows for a parsimonious, but faithful reconstruction of the total
variability among the original proximity entries. Subsequently, the AR matrices
are treated as separate proximity matrices. Their specific patterning lends
them immediately to the representation by a single (discrete non-spatial)
ultrametric cluster dendrogram and a single (continuous spatial) unidimensional
scale. Because both models refer to the same data base and involve the same
number of parameters, estimated through least-squares, a direct comparison of
their differential fit is legitimate. Thus, one can readily
determine whether the amount of variability associated which each AR matrix is
most faithfully represented by a discrete or a continuous structure, and which
model provides in sum the most appropriate representation of the original
proximity matrix. We propose an extension of the order-constrained
anti-Robinson decomposition of square-symmetric proximity matrices to the
analysis of individual differences of three-way data, with the third way
representing individual data sources. An application to judgments of
schematic face stimuli illustrates the method.
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