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A New Heterogeneous Multidimensional Unfolding Procedure

Abstract

A variety of joint space multidimensional scaling (MDS) methods have been utilized for the spatial analysis of two- or three-way dominance data involving subjects’ preferences, choices, considerations, intentions, etc. so as to provide a parsimonious spatial depiction of the underlying relevant dimensions, attributes, stimuli, and/or subjects’ utility structures in the same joint space representation. We demonstrate that care must be taken with respect to a key assumption in existent joint space MDS models such that all estimated dimensions are utilized by each and every subject in the sample, as this assumption can lead to serious distortions with respect to the derived joint spaces. We develop a new Bayesian dimension selection methodology for the multidimensional unfolding model which accommodates heterogeneity with respect to such dimensional utilization at the individual subject level for the analysis of two or three-way dominance data. A consumer psychology application regarding the preference for Over-the-Counter (OTC) analgesics is provided. We conclude by discussing the practical implications of the results, as well as directions for future research.

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Notes

  1. 1.

    Note that PREFSCAL assumes Euclidean distance while all other models, including the proposed model, assume squared Euclidean distance. As such, we inputted Euclidean distances and added the same amount of error for PREFSCAL analysis.

  2. 2.

    We thank an anonymous reviewer for this point.

  3. 3.

    These derived joint spaces are usually referred as degenerate solutions meaning that the derived joint space is extremely uninformative despite good fit to the data (Heiser 1989). In MDU models, this usually takes the form of a wide separation between row and column points. Note that we use 0.00001 for default convergence criteria for ALSCAL as the default convergence option would keep the final solution close to the initial solution.

  4. 4.

    One reviewer suggested a different specification of PREFSCAL to mimic the proposed model with a three-way dimension weighting option combined with almost missing values. Here, one can replicate the two-way data I=100 times where row i of the i-th replication has a weight one and very small numbers (i.e., 1e–06) elsewhere. We tried this approach but didn’t find any significant improvement over the more traditional PREFSCAL approach.

  5. 5.

    Theoretically, the weighted unfolding model should be able to account for such dimensional heterogeneity where dimensions not utilized for a particular subject would have an associated weight equal to zero. However, we observed from the small synthetic example presented earlier that this does not always occur in practice.

  6. 6.

    As noted by one reviewer, many nonmetric unfolding models can yield degenerate solutions when the transformation can become a constant combined with a solution that has all between set distances equal to the same constant (see, e.g., Borg & Groenen 2005). The proposed model avoids this type of degeneracy problem as there is no slope parameter on the utility U ir .

  7. 7.

    Also note that BIC (Bayesian information criterion) could be a rough approximation to the logarithm of the Bayes factor (see Kass & Raftery 1995, for complete review on Bayes factor).

  8. 8.

    Note that the proposed HDMDU is uniquely determined as discussed in Section 3.1 As such, no transformation was applied to the results of HDMDU model.

  9. 9.

    Bayes factor between HDMDU and BSMDU would not be significant (see Kass & Raftery 1995, for review of the Bayes factor).

  10. 10.

    The computing time for this data is approximately 3 hours run on a 3.3 GHZ computer with a Windows operating system.

References

  1. Alba, J.W., & Hutchinson, J.W. (1987). Dimensions of consumer expertise. Journal of Consumer Research, 13, 411–454.

  2. Baumgartner, H., & Steenkamp, J.E.M. (2001). Response styles in marketing research: A cross-national investigation. Journal of Marketing Research, 38, 143–156.

  3. Benzécri, J.P. (1973). L’analyse des données: Tome II. Analyse de correspondances. Paris: Dunod.

  4. Benzécri, J.P. (1992). Correspondence analysis handbook. New York: Dekker.

  5. Bettman, J.R., Luce, M.F., & Payne, J.W. (1998). Constructive consumer choice processes. Journal of Consumer Research, 25, 187–217.

  6. Borg, I., & Groenen, P.J.F. (2005). Modern multidimensional scaling: Theory & application (2nd ed.). New York: Springer.

  7. Bradlow, E.T., & Schmittlein, D.C. (2000). The little engines that could: Modeling the performance of world wide web search engines. Marketing Science, 19(1), 43–62.

  8. Brucks, M. (1985). The effects of product class knowledge on information search behavior. Journal of Consumer Research, 12, 1–16.

  9. Busing, F.M.T.A., Groenen, P.J.F., & Heiser, W.J. (2005). Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70, 71–98.

  10. Carlin, B.P., & Chib, S. (1995). Bayesian model choice via Markov chain Monte Carlo. Journal of the Royal Statistical Society, Series B, 57(3), 473–484.

  11. Carroll, J.D. (1972). Individual differences and multidimensional scaling. In R.N. Shepard, A.K. Romney, & S. Nerlove (Eds.), Multidimensional scaling: Theory & applications in the behavior sciences: Theory (Vol. I, pp. 105–155). New York: Seminar Press.

  12. Carroll, J.D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance) data. In E.D. Lantermann & H. Feger (Eds.), Similarity and choice (pp. 234–289). Vienna: Hans Huber Publishers.

  13. Carroll, J.D., & Arabie, P. (1980). Multidimensional scaling. Annual Review of Psychology, 31, 607–649.

  14. Chib, S. (2002). Markov chain Monte Carlo methods. In S.J. Press (Ed.), Subjective and objective Bayesian statistics (2nd ed., pp. 119–171) New York: Wiley.

  15. Chintagunta, P.K. (1994). Heterogeneous logit model implications for brand positioning. Journal of Marketing Research, 31(2), 304–311.

  16. Churchill, G.A., & Peter, J.P. (1984). Research design effects on the reliability of rating scales: A meta analysis. Journal of Marketing Research, 21(4), 360–375.

  17. Coombs, C.H. (1960). A theory of data. Psychological Review, 67(3), 143–159.

  18. Cox, T.F., & Cox, M.A. (2001). Multidimensional scaling (2nd ed.). London: Chapman & Hall.

  19. DeSarbo, W.S., & Carroll, J.D. (1985). Three-way metric unfolding via weighted least-squares. Psychometrika, 50, 275–300.

  20. DeSarbo, W.S., & Hoffman, D. (1986). Simple and weighted unfolding MDS threshold models for the spatial analysis of binary data. Applied Psychological Measurement, 10, 247–264.

  21. DeSarbo, W.S., Kim, Y., & Fong, D.K.H. (1999). A Bayesian multidimensional scaling procedure for the spatial analysis of revealed choice data. Journal of Econometrics, 89(1–2), 79–108.

  22. DeSarbo, W.S., Kim, Y., Wedel, M., & Fong, D.K.H. (1998). A Bayesian approach to the spatial representation of market structure from consumer choice data. European Journal of Operational Research, 111(2), 285–305.

  23. 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, 2, 147–168.

  24. DeSarbo, W.S., & Rao, V.R. (1986). A constrained unfolding methodology for product positioning. Marketing Science, 5(1), 1–19.

  25. DeSarbo, W.S., Young, M.R., & Rangaswamy, A. (1997). A parametric multidimensional unfolding procedure for incomplete nonmetric preference/choice set data in marketing research. Journal of Marketing Research, 34, 499–516.

  26. Diebolt, J., & Robert, C.P. (1994). Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B, 56(2), 363–375.

  27. Elrod, T. (1988). Choice map: Inferring a product-market map from panel data. Marketing Science, 7(1), 21–40.

  28. Elrod, T., & Keane, M.P. (1995). A factor-analytic model for representing the market structure in panel data. Journal of Marketing Research, 32, 1–16.

  29. Erdem, T. (1996). A dynamic analysis of market structure based on panel data. Marketing Science, 15(4), 359–378.

  30. Feldman, J.M., & Lynch, J.G. (1988). Self-generated validity and other effects of measurement on belief, attitude, intention, and behavior. Journal of Applied Psychology, 73(3), 421–435.

  31. Fong, D.K.H., DeSarbo, W.S., Park, J., & Scott, C.J. (2010). A Bayesian vector multidimensional scaling procedure for the analysis of ordered preference data. Journal of the American Statistical Society, 105(490), 482–492.

  32. Friedman, H.H., Friedman, L.W., & Gluck, B. (1988). The effects of scale-checking styles on responses to a semantic differential scale. Journal of the Market Research Society, 30(4), 477–481.

  33. Gelfand, A.E., & Smith, A.F.M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.

  34. Gelman, A., Gilks, W.R., & Roberts, G.O. (Eds.) (1996). Efficient Metropolis jumping rules (Vol. 5). Oxford: Oxford University Press.

  35. George, E.I., & McCulloch, R.E. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88, 881–889.

  36. George, E.I., & McCulloch, R.E. (1997). Approaches for Bayesian variable selection. Statistica Sinica, 7(2), 339–373.

  37. George, E.I., McCulloch, R.E., & Tsay, R.S. (1995). Two approaches to Bayesian model selection with applications. In D. Berry, K. Chaloner, & J. Geweke (Eds.), Bayesian statistics and econometrics: Essays in honor of Arnold Zellner (pp. 339–348). New York: Wiley.

  38. Geweke, J. (1996). Monte Carlo simulation and numerical integration. In H.M. Amman, D.A. Kendrick, & J. Rust (Eds.), Handbook of computational economics (Vol. 1, pp. 731–800). New York: Elsevier.

  39. Gifi, A. (1990). Nonlinear multivariate analysis. New York: Wiley.

  40. Gilbride, T.J., Allenby, G.M., & Brazell, J.D. (2006). Models for heterogeneous variable selection. Journal of Marketing Research, 43(3), 420–430.

  41. Greenacre, M.J. (1984). Theory and applications of correspondence analysis. New York: Academic Press.

  42. Greenacre, M.J., & Browne, M.W. (1986). An efficient alternating least-squares algorithm to perform multidimensional unfolding. Psychometrika, 51, 241–250.

  43. Harshman, R.A., & Lundy, M.E. (1984). Data preprocessing and the extended PARAFAC model. In H.G. Law, C.W. Snyder, J.A. Hattie, & R.P. McDonald (Eds.), Research methods for multimode data analysis (pp. 216–284). New York: Praeger.

  44. Harshman, R.A., & Lundy, M.E. (1985). The preprocessing controversy: An exchange of papers between Kroonenberg, Harshman and Lundy (Technical Report). London, Ontario: University of Western Ontario, Department of Psychology.

  45. Heiser, W.J. (1981). Unfolding analysis of proximity data (Unpublished doctoral dissertation). University of Leiden.

  46. Heiser, W.J. (1989). The city-block model for three-way multidimensional scaling. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 395–404). Amsterdam: North-Holland.

  47. Houston, D.A., & Sherman, S.J. (1995). Cancellation and focus: The role of shared and unique features in the choice process. Journal of Experimental Social Psychology, 31(4), 357–378.

  48. Hutchinson, W.J., & Alba, J.W. (1991). Ignoring irrelevant information: Situational determinants of consumer learning. Journal of Consumer Research, 18, 325–345.

  49. Isen, A.M. (1993). Positive affect and decision making. In M. Lewis & J. Haviland (Eds.), Handbook of emotions (pp. 261–273). New York: Guilford.

  50. Isen, A.M., Daubman, K.A., & Nowicki, G.P. (1987). Positive affect facilitates creative problem solving. Journal of Personality and Social Psychology, 52(6), 1122–1131.

  51. Johnson, K.E., & Mervis, C.B. (1997). Effects of varying levels of expertise on the basic level of categorization. Journal of Experimental Psychology: General, 126, 248–277.

  52. Kass, R.E., & Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–795.

  53. Kunda, Z. (1990). The case for motivated reasoning. Psychological Bulletin, 108, 480–498.

  54. Kruskal, J.B. (1964a). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.

  55. Kruskal, J.B. (1964b). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.

  56. Kruskal, J.B., & Carroll, J.D. (1969). Geometric models and badness of fit functions. In P.R. Krishnaiah (Ed.), Multivariate analysis (Vol. II). New York: Academic Press.

  57. Kuo, L., & Mallick, B. (1988). Variable selection for regression models. Sankhya: The Indian Journal of Statistics, Series B, 60, 65–81.

  58. Lingoes, J.C. (1972). A general survey of the Guttman–Lingoes nonmetric program series. In R.N. Shepard, A.K. Romney, & S. Nerlove (Eds.), Theory and applications in the behavior sciences: Theory (Vol. I, pp. 49–68). New York: Seminar Press.

  59. Lingoes, J.C. (1973). The Guttman–Lingoes nonmetric program series. Ann Arbor: Mathesis Press.

  60. Miller, K.E., & Ginter, J.L. (1979). An investigation of situational variation in brand choice behavior and attitude. Journal of Marketing Research, 16, 11–123.

  61. Mitchell, T.J., & Beauchamp, J.J. (1988). Bayesian variable selection in linear regression. Journal of the American Statistical Association, 83, 1023–1032.

  62. Newton, M.A., & Raftery, A.E. (1994). Approximate Bayesian inference with the weighted likelihood bootstrap. Journal of the Royal Statistical Society, Series B, 56, 3–48.

  63. Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press.

  64. Okada, K., & Shigemasu, K. (2009). BMDS: A collection of R functions for Bayesian multidimensional scaling. Applied Psychological Measurement, 33(7), 570–571.

  65. Oh, M., & Raftery, A.E. (2001). Bayesian multidimensional scaling and choice of dimension. Journal of the American Statistical Association, 96(455), 1031–1044.

  66. Park, J., DeSarbo, W.S., & Liechty, J. (2008). A hierarchical Bayesian multidimensional scaling methodology for accommodating both structural and preference heterogeneity. Psychometrika, 73, 451–472.

  67. Payne, J.W., Bettman, J.R., & Johnson, E. (1993). The adaptive decision maker. Cambridge: Cambridge University Press.

  68. Payne, J.W., Bettman, J.R., & Luce, M.F. (1996). When time is money: Decision behavior under opportunity-cost time pressure. Organizational Behavior and Human Decision Processes, 66, 131–152.

  69. Petty, R.E., & Cacioppo, J.T. (1986). Communication and persuasion: Central and peripheral routes to attitude change. New York: Springer.

  70. Petty, R.E., Cacioppo, J.T., & Goldman, R. (1981). Personal involvement as a determinant of argument based persuasion. Journal of Personality and Social Psychology, 41, 847–855.

  71. Petty, R.E., Cacioppo, J.T., & Schumann, D. (1983). Central and peripheral routes for advertising effectiveness: The moderating role of involvement. Journal of Consumer Research, 10, 135–144.

  72. Roskam, E.E. (1973). Fitting ordinal relational data to a hypothesized structure (Technical Report No. 73MA06). Nijmegen: Catholic University.

  73. Schönemann, P.H. (1970). On metric multidimensional unfolding. Psychometrika, 35, 349–366.

  74. Schwarz, N., Strack, F., Muller, G., & Chassein, B. (1988). The range of response alternatives may determine the meaning of the question: Further evidence on informative functions of response alternatives. Social Cognition, 6, 107–117.

  75. Slater, P. (1960). Inconsistencies in a schedule of paired comparisons. Biometrika, 48, 303–312.

  76. Sterngold, A., Warland, R.H., & Herrmann, R. (1994). Do surveys overstate public concerns? Public Opinion Quarterly, 58, 255–263.

  77. Sujan, M. (1985). Consumer knowledge: Effects on evaluation strategies mediating consumer judgments. Journal of Consumer Research, 12, 31–46.

  78. Takane, Y., Young, F.W., & de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least-squares method with optimal scaling features. Psychometrika, 42(1), 7–67.

  79. Tanner, M.A., & Wong, W.H. (1987). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82, 528–540.

  80. Tucker, L.R. (1960). Intra-individual and inter-individual multidimensionality. In H. Gulliksen & S. Messick (Eds.), Psychological scaling: Theory and applications (pp. 110–123). New York: Wiley.

  81. Vincente, K.J., & Wang, J.H. (1998). An ecological theory of expertise effects in memory recall. Psychological Review, 105, 35–57.

  82. Wedel, M., & DeSarbo, W.S. (1996). An exponential-family multidimensional scaling mixture methodology. Journal of Business and Economic Statistics, 14(4), 447–459.

  83. Wedel, M., & Kamakura, W. (2000). Market segmentation: Conceptual and methodological foundations. Boston: Kluwer Academic.

  84. Wright, P., & Weitz, B. (1977). Time horizon effects on product evaluation strategies. Journal of Marketing Research, 14, 429–443.

  85. Yang, S., Allenby, G., & Fennell, G. (2002). Modeling variation in brand preference: The roles of objective environment and motivating conditions. Marketing Science, 21, 14–31.

  86. Young, F.W., & Hamer, R.M. (1987). Multidimensional scaling: History, theory and applications. Hillsdale: Lawrence Erlbaum Associates.

  87. Young, F.W., & Torgerson, W.S. (1967). TORSCA, a FORTRAN IV program for Shepard-Kruskal multidimensional scaling analysis. Behavioral Science, 12(6), 498.

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Correspondence to Joonwook Park.

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Park, J., Rajagopal, P. & DeSarbo, W.S. A New Heterogeneous Multidimensional Unfolding Procedure. Psychometrika 77, 263–287 (2012). https://doi.org/10.1007/s11336-012-9256-6

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Key words

  • multidimensional unfolding
  • dimension selection
  • Bayesian multidimensional scaling
  • consumer psychology
  • heterogeneity