Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation

Psychometrika

Volumn 70, Issue 1, 2005, Pages 71-98

  Busing, Frank M T A ^a   Groenen, Patrick J K ^b   Heiser, Willem J ^a  

^a LEIDEN UNIVERSITY   (Netherlands)

^b ERASMUS UNIVERSITY ROTTERDAM   (Netherlands)

Author keywords

Degeneracy; Iterative majorization; Penalty; PREFSCAL; Stress; Unfolding

Indexed keywords

EID: 14944353240     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11336-001-0908-1     Document Type: Article

References (52)

1. Bennett, J.F., & Hayes, W.L. (1960). Multidimensional unfolding: Determining the dimensionality of ranked preference data. Psychometrika, 25, 36-48.
2. Borg, I., & Bergermaier, R. (1982). Degenerationsprobleme im Unfolding und Ihre Lösung. Zeitschrift für Sozialpsychologie, 13, 287-299.
3. Borg, I., & Groenen, P.J.F. (1997). Modern Multidimensional Scaling: Theory and Applications. New York: Springer.
4. Borg, I., & Lingoes, J. (1987). Multidimensional Similarity Structure Analysis. Berlin: Springer.
5. Busing, F.M.T.A., & Heiser, W.J. (2003). PREFSCAL Progress Report: two-way models (Tech. Rep.). Leiden, The Netherlands: Leiden University, (working paper)
6. Carroll, J.D. (1972). In R.N. Shepard, A.K. Romney, & S.B. Nerlove (Eds.), Multidimensional Scaling: Theory and Applications in the Behavioral Sciences (Vol. 1, pp. 105-155). New York: Seminar Press.
7. Coombs, C.H. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57, 148-158.
8. Coombs, C.H. (1964). A Theory of Data. New York: Wiley.
9. Coombs, C.H., & Kao, R.C. (1960). On a connection between factor analysis and multidimensional unfolding. Psychometrika, 25, 219-231.
10. Dagpunar, A. (1988). Principles of Random Variate Generation. Oxford: Clarendon Press.
11. De Leeuw, J. (1977). Applications of convex analysis to multidimensional scaling. In J.R. Barra, F. Brodeau, G. Romier, & B. van Cutsem (Eds.), Recent Developments in Statistics (pp. 133-145). Amsterdam, The Netherlands: North-Holland.
12. De Leeuw, J. (1983). On Degenerate Nonmetric Unfolding Solutions (Tech. Rep.). Department of Data Theory, FSW/RUL.
13. De Leeuw, J., & Heiser, W.J. (1980). Multidimensional scaling with restrictions on the configuration. In P. R. Krishnaiah (Ed.), Multivariate Analysis (Vol. 5, pp. 501-522). Amsterdam, The Netherlands: North-Holland.
14. Dennis, J.E., & Schnabel, R.B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, New Jersey: Prentice-Hall.
15. (Republished by SIAM, Philadelphia, in 1996 as Vol. 16 of Classics in Applied Mathematics)
16. DeSarbo, W.S., & Carroll, J.D. (1985). Three-way metric unfolding via alternating weighted least squares. Psychometrika, 50(3), 275-300.
17. DeSarbo, W.S., & Rao, V.R. (1984). GENFOLD2: A set of models and algorithms for the GENeral unFOLDing analysis of preference/dominance data. Journal of Classification, 1, 147-186.
18. DeSarbo, W.S., Young, M.R., & Rangaswamy, A. (1997). A parametric Multidimensional Unfolding Procedure for Incomplete Nonmetric Preference/Choice Set Data in Marketing Research (Tech. Rep.). The Pennsylvania State University. (Working paper)
19. Dinkelbach, W. (1967). On nonlinear fractional programming. Management Science, 13, 492-498.
20. Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multi variate analysis. Biometrika, 53, 325-338.
21. Green, P.E., & Rao, V. (1972). Applied multidimensional scaling. Hinsdale, IL: Dryden Press.
22. Groenen, P.J.F. (1993). The majorization Approach to Multidimensional Scaling: Some Problems and Extensions. Leiden, The Netherlands: DSWO Press.
23. Groenen, P.J.F., & Heiser, W.J. (1996). The tunneling method for global optimization in multidimensional scaling. Psychometrika, 61, 529-550.
24. Hardy, G.H., Littlewood, J.E., & Polya, G. (1952). Inequalities. Cambridge: Cambridge University Press.
25. Hayes, W.L., & Bennett, J.F. (1961). Multidimensional unfolding: Determining configuration from complete rank order preference data. Psychometrika, 26, 221-238.
26. Heiser, W.J. (1981). Unfolding analysis of proximity data. Unpublished doctoral dissertation, Leiden University.
27. Heiser, W.J. (1987). Joint ordination of species and sites: The unfolding technique. In P. Legendre & L. Legendre (Eds.), Developments in Numerical Ecology (pp. 189-221). Berlin, Heidelberg: Springer.
28. Heiser, W.J. (1989). Order invariant unfolding analysis under smoothness restrictions. In G. De Soete, H. Feger, & K.C. Klauer (Eds.), New Developments in Psychological Choice Modeling (pp. 3-31). Amsterdam: North-Holland.
29. Heiser, W.J. (1995). Convergent computation by iterative majorization: Theory and applications in multidimensional data analysis. In W.J. Krzanowski (Ed.), Recent Advances in Descriptive Multivariate Analysis (pp. 157-189). Oxford: Oxford University Press.
30. Heiser, W.J., & De Leeuw, J. (1979). How to Use SMACOF-III: A Program for Metric Multidimensional Unfolding (Tech. Rep.). Leiden University, Department of Data Theory.
31. Katsnelson, J., & Kotz, S. (1957). On the upper limits of some measures of variability. Archiv für Meterologie, Geophysik and Bioklimatologie (B), 8, 103.
32. Kim, C., Rangaswamy, A., & DeSarbo, W.S. (1999). A quasi-metric approach to multidimensional unfolding for reducing the occurence of degenerate solutions. Multivariate Behavioral Research, 34(2), 143-180.
33. Kruskal, J.B. (1964a). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 29, 1-27.
34. Kruskal, J.B. (1964b). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115-129.
35. Kruskal, J.B. (1977). Multidimensional scaling and other methods for discovering structure. In K. Enslein, A. Ralston, & H.S. Wilf (Eds.), Mathematical Methods for Digital Computers (Vol. 2, pp. 296-339). New York: Wiley.
36. Kruskal, J.B., & Carroll, J.D. (1969). Geometrical models and badness-of-fit functions. In P.R. Krishnaiah (Ed.), Multivariate Analysis (vol 2, pp. 639-671). New York: Academic Press.
37. Kruskal, J.B., Young, F.W., & Seery, J.B. (1978). How to Use KYST, A Very Flexible Program to do Multidimensional Scaling and Unfolding (Tech. Rep.). Murray Hill, NJ: Bell Laboratories.
38. L'Ecuyer, P. (1999). Tables of maximally equidistributed combined LFSR generators. Mathematics of Computing, 68, 261-269.
39. Marden, J.I. (1995). Analyzing and Modeling Rank Data. London: Chapman and Hall.
40. McClelland, G.H., & Coombs, C.H. (1975). ORDMET: A general algorithm for constructing all numerical solutions to ordered metric solutions. Psychometrika, 40, 269-290.
41. Pearson, K. (1896). Regression, heridity, and panmixia. Philosophical Transactions of the Royal Society of London, Series A, 187, 253-318.
42. Roskam, E.E.Ch.I. (1968). Metric Analysis of Ordinal Data. Voorschoten: VAM.
43. Shepard, R.N. (1974). Representation of structure in similarity data: Problems and prospects. Psychometrika, 39(4), 373-421.
44. Takane, Y., Young, F.W., & de Leeuw, J. (1977). Nonmetric individual differences MDS: An alternating least squares method with optimal scaling features. Psychometrika, 42, 7-67.
45. Torgerson, W.S. (1958). Theory and Methods of Scaling. New York: Wiley.
46. Trosset, M.W. (1998). A new formulation of the nonmetric STRAIN problem in multidimensional scaling. Journal of Classification, 15, 15-35.
47. Van Blokland-Vogelesang, A.W. (1989). Unfolding and consensus ranking: A prestige ladder for technical occupations. In G. De Soete, H. Feger, & K. C. Klauer (Eds.), New Developments in Psychological Choice Modeling (pp. 237-258). Amsterdam, The Netherlands: North-Holland.
48. Van Blokland-Vogelesang, A.W. (1993). A nonparametric distance model for unidimensional unfolding. In M. A. Fligner & J. S. Verducci (Eds.), Probability Models and Statistical Analyses for Ranking Data (pp. 241-276). New York: Springer-Verlag.
49. Wagenaar, W.A., & Padmos, P. (1971). Quantitative interpretation of Stress in Kruskal's method multidimensional scaling technique. British Journal of Mathematical and Statistical Psychology, 24, 101-110.
50. Winsberg, S., & Carroll, J. D. (1989). A quasi-nonmetric method for multidimensional scaling via an extended Euclidean model. Psychometrika, 54, 217-229.
51. Young, F. W. (1972). A model for polynomial conjoint analysis algorithms. In R.N. Shepard, A.K. Romney, & S.B. Nerlove (Eds.), Multidimensional Scaling, Theory (Vol. I, pp. 69-104). New York.
52. Young, F.W., & Torgerson, W.S. (1967). TORSCA, a Fortran IV program for Shepard-Kruskal multidimensional scaling analysis. Behavioral Science, 12, 498.