IDEAS home Printed from https://ideas.repec.org/p/ems/eureri/8045.html
   My bibliography  Save this paper

Multidimensional Scaling with Regional Restrictions for Facet Theory: An Application to Levi's Political Protest Data

Author

Listed:
  • Groenen, P.J.F.
  • van der Lans, A.

Abstract

Multidimensional scaling (MDS) is often used for the analysis of correlation matrices of items generated by a facet theory design. The emphasis of the analysis is on regional hypotheses on the location of the items in the MDS solution. An important regional hypothesis is the axial constraint where the items from different levels of a facet are assumed to be located in different parallel slices. The simplest approach is to do an MDS and draw the parallel lines separating the slices as good as possible by hand. Alternatively, Borg and Shye (1995) propose to automate the second step. Borg and Groenen (1997, 2005) proposed a simultaneous approach for ordered facets when the number of MDS dimensions equals the number of facets. In this paper, we propose a new algorithm that estimates an MDS solution subject to axial constraints without the restriction that the number of facets equals the number of dimensions. The algorithm is based on constrained iterative majorization of De Leeuw and Heiser (1980) with special constraints. This algorithm is applied to Levi’s (1983) data on political protests.

Suggested Citation

  • Groenen, P.J.F. & van der Lans, A., 2006. "Multidimensional Scaling with Regional Restrictions for Facet Theory: An Application to Levi's Political Protest Data," ERIM Report Series Research in Management ERS-2006-057-MKT, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    • Handle: RePEc:ems:eureri:8045
    as

    Download full text from publisher

    File URL: https://repub.eur.nl/pub/8045/ERS-2006-057-MKT.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Shlomit Levy, 1983. "A cross-cultural analysis of the structure and levels of attitudes towards acts of political protest," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 12(3), pages 281-309, April.
    2. Mathar, Rudolf, 1990. "Multidimensional scaling with constraints on the configuration," Journal of Multivariate Analysis, Elsevier, vol. 33(2), pages 151-156, May.
    3. Ingwer Borg & James Lingoes, 1980. "A model and algorithm for multidimensional scaling with external constraints on the distances," Psychometrika, Springer;The Psychometric Society, vol. 45(1), pages 25-38, March.
    4. Patrick Groenen & Bart-Jan Os & Jacqueline Meulman, 2000. "Optimal scaling by alternating length-constrained nonnegative least squares, with application to distance-based analysis," Psychometrika, Springer;The Psychometric Society, vol. 65(4), pages 511-524, December.
    5. Norbert Gaffke & Rudolf Mathar, 1989. "A cyclic projection algorithm via duality," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 36(1), pages 29-54, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. F. Deutsch & W. Li & J. Swetits, 1999. "Fenchel Duality and the Strong Conical Hull Intersection Property," Journal of Optimization Theory and Applications, Springer, vol. 102(3), pages 681-695, September.
    2. Mark Groves, 1992. "Beyond spatial representation," Quality & Quantity: International Journal of Methodology, Springer, vol. 26(1), pages 49-59, February.
    3. de Leeuw, Jan & Mair, Patrick, 2009. "Multidimensional Scaling Using Majorization: SMACOF in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 31(i03).
    4. Chao Ding & Hou-Duo Qi, 2017. "Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation," Computational Optimization and Applications, Springer, vol. 66(1), pages 187-218, January.
    5. K. Van Deun & P. J. F. Groenen, 2005. "Majorization Algorithms for Inspecting Circles, Ellipses, Squares, Rectangles, and Rhombi," Operations Research, INFORMS, vol. 53(6), pages 957-967, December.
    6. Ingwer Borg & Rene Bergermaier, 1981. "Some comments on ‘the structure of subjective well-being in nine western societies’ by andrews and inglehart," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 9(3), pages 265-278, September.
    7. Wayne DeSarbo & Vithala Rao, 1984. "GENFOLD2: A set of models and algorithms for the general UnFOLDing analysis of preference/dominance data," Journal of Classification, Springer;The Classification Society, vol. 1(1), pages 147-186, December.
    8. J. Carroll, 1985. "Review," Psychometrika, Springer;The Psychometric Society, vol. 50(1), pages 133-140, March.
    9. Frank de Meijer & Renata Sotirov, 2021. "SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1262-1276, October.
    10. Chin How Jeffrey Pang, 2019. "Dykstra’s Splitting and an Approximate Proximal Point Algorithm for Minimizing the Sum of Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 1019-1049, September.
    11. Gnot, Stanislaw & Grzadziel, Mariusz, 2002. "Nonnegative Minimum Biased Quadratic Estimation in Mixed Linear Models," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 217-233, February.
    12. van Deun, K. & Groenen, P.J.F. & Delbeke, L., 2005. "VIPSCAL: A combined vector ideal point model for preference data," Econometric Institute Research Papers EI 2005-03, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    13. Herden, Gerhard & Pallack, Andreas, 2005. "Adequateness and interpretability of objective functions in ordinal data analysis," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 19-69, May.
    14. Gérard d'Aubigny, 1990. "Reviews," Psychometrika, Springer;The Psychometric Society, vol. 55(1), pages 184-188, March.
    15. James Lingoes & Ingwer Borg, 1986. "On evaluating the equivalency of alternative MDS representations," Quality & Quantity: International Journal of Methodology, Springer, vol. 20(2), pages 249-256, June.
    16. J M Brown, 1983. "The Structure of Motives for Moving: A Multidimensional Model of Residential Mobility," Environment and Planning A, , vol. 15(11), pages 1531-1544, November.
    17. Herden, Gerhard & Pallack, Andreas, 2002. "Consistency in ordinal data analysis I," Mathematical Social Sciences, Elsevier, vol. 43(1), pages 79-113, January.
    18. Jacqueline Meulman, 2003. "Prediction and classification in nonlinear data analysis: Something old, something new, something borrowed, something blue," Psychometrika, Springer;The Psychometric Society, vol. 68(4), pages 493-517, December.
    19. repec:jss:jstsof:31:i03 is not listed on IDEAS

    More about this item

    Keywords

    Axial Partitioning; Constrained Estimation; Facet Theory; Iterative Majorization; Multidimensional Scaling; Regional Restrictions;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • C49 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Other
    • M - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics
    • M31 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Marketing and Advertising - - - Marketing

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:ems:eureri:8045. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: RePub (email available below). General contact details of provider: https://edirc.repec.org/data/erimanl.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.