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An Inner Product Model for the Multidimensional Scaling of Symmetric Layouts

Author(s):
Bechtel, Gordon G.
Publication Year:
1970
Report Number:
RB-70-22
Source:
ETS Research Bulletin
Document Type:
Report
Page Count:
19
Subject/Key Words:
Data Analysis, Mathematical Models, Multidimensional Scaling

Abstract

A multidimensional scaling analysis is presented for replicated symmetric layouts of pairwise compositions. The replicates may represent individuals, situations, or blocks which contain all pairwise responses generated from a distinct set of objects or treatments. The class of replicates and the class of objects are scaled in a joint space by means of an inner product model which weights each of the dimensions of the space. Least squares estimates of the replicates' and objects' coordinates, and of unscalability parameters, are obtained through a manipulation of the error sum of squares for the model. The solution involves the decomposition of a three-way least squares problem into three subproblems; two of these are trivial, and the third is solvable by classical least squares matrix factorization.

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