Multidimensional Unfolding by Nonmetric Multidimensional Scaling of Spearman Distances in the Extended Permutation Polytope

A multidimensional unfolding technique that is not prone to degenerate solutions and is based on multidimensional scaling of a complete data matrix is proposed: distance information about the unfolding data and about the distances both among judges and among objects is included in the complete matrix. The latter information is derived from the permutation polytope supplemented with the objects, called the preference sphere. In this sphere, distances are measured that are closely related to Spearman's rank correlation and that are comparable among each other so that an unconditional approach is reasonable. In two simulation studies, it is shown that the proposed technique leads to acceptable recovery of given preference structures. A major practical advantage of this unfolding technique is its relatively easy implementation in existing software for multidimensional scaling.

[1]  G. De Soete,et al.  Unfolding and consensus ranking: A prestige ladder for technical occupations , 1989 .

[2]  W. Hays,et al.  Multidimensional unfolding: Determining the dimensionality of ranked preference data , 1960 .

[3]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. I. , 1962 .

[4]  C. Coombs.  A theory of data. , 1965, Psychology Review.

[5]  I. Spence,et al.  Single subject incomplete designs for nonmetric multidimensional scaling , 1974 .

[6]  P. Groenen,et al.  Modern multidimensional scaling , 1996 .

[7]  J. Ramsay.  Maximum likelihood estimation in multidimensional scaling , 1977 .

[8]  Louis Guttman,et al.  The Guttman-Lingoes nonmetric program series , 1973 .

[9]  John G. Kemeny,et al.  Mathematical models in the social sciences , 1964 .

[10]  Patrick J. F. Groenen,et al.  The majorization approach to multidimensional scaling : some problems and extensions , 1993 .

[11]  David Kaplan,et al.  The Sage handbook of quantitative methodology for the social sciences , 2004 .

[12]  W. Kruskal.  Ordinal Measures of Association , 1958 .

[13]  Roger N. Shepard,et al.  Multidimensional scaling : theory and applications in the behavioral sciences , 1974 .

[14]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[15]  Wayne S. DeSarbo,et al.  A Quasi-Metric Approach to Multidimensional Unfolding for Reducing the Occurrence of Degenerate Solutions. , 1999, Multivariate behavioral research.

[16]  Willem J. Heiser,et al.  Interpreting degenerate solutions in unfolding by use of the vector model and the compensatory distance model , 2005 .

[17]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[18]  Paul E. Green,et al.  Applied multidimensional scaling : a comparison of approaches and algorithms , 1972 .

[19]  J. Marden.  Analyzing and Modeling Rank Data , 1996 .

[20]  Willem J. Heiser,et al.  Geometric representation of association between categories , 2004 .

[21]  Paul D. Feigin,et al.  Intergroup Diversity and Concordance for Ranking Data: An Approach via Metrics for Permutations , 1986 .

[22]  P. Groenen,et al.  Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation , 2005 .

[23]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[24]  Willem J. Heiser,et al.  Multidimensional Scaling and Unfolding of Symmetric and Asymmetric Proximity Relations , 2004 .

[25]  D. Critchlow.  Metric Methods for Analyzing Partially Ranked Data , 1986 .