Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T08:05:22.697Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian Metric Multidimensional Scaling

Published online by Cambridge University Press:  04 January 2017

Ryan Bakker  and
Keith T. Poole
Ryan Bakker*
Affiliation:
School of Public and International Affairs, Department of Political Science, University of Georgia, Athens, GA
Keith T. Poole
Affiliation:
School of Public and International Affairs, Department of Political Science, University of Georgia, Athens, GA e-mail: ktpoole@uga.edu
*
e-mail: rbakker@uga.edu (corresponding author)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article, we show how to apply Bayesian methods to noisy ratio scale distances for both the classical similarities problem as well as the unfolding problem. Bayesian methods produce essentially the same point estimates as the classical methods, but are superior in that they provide more accurate measures of uncertainty in the data. Identification is nontrivial for this class of problems because a configuration of points that reproduces the distances is identified only up to a choice of origin, angles of rotation, and sign flips on the dimensions. We prove that fixing the origin and rotation is sufficient to identify a configuration in the sense that the corresponding maxima/minima are inflection points with full-rank Hessians. However, an unavoidable result is multiple posterior distributions that are mirror images of one another. This poses a problem for Markov chain Monte Carlo (MCMC) methods. The approach we take is to find the optimal solution using standard optimizers. The configuration of points from the optimizers is then used to isolate a single Bayesian posterior that can then be easily analyzed with standard MCMC methods.

Type
Regular Articles
Information
Political Analysis , Volume 21 , Issue 1 , Winter 2013 , pp. 125 - 140
Copyright
Copyright © The Author 2013. Published by Oxford University Press on behalf of the Society for Political Methodology 

Footnotes

Authors' note: Supplementary materials for this article are available on the Political Analysis Web site.

References

Abrajano, Marisa, and Poole, Keith T. 2011. A method of linking surveys using affective signature with an application to ethnic groups in the U.S. In Who Gets Represented, eds. Enns, Peter and Wlezien, Christopher. New York: Russell Sage.Google Scholar
Adams, James, Merrill, Samuel, and Grofman, Bernard. 2005. A unified theory of party competition: A cross-national analysis integrating spatial and behavioral factors. New York: Cambridge University Press.Google Scholar
Bakker, Ryan, and Poole, Keith T. 2012. Replication data for: Bayesian Metric Multidimensional Scaling. http://hdl.handle.net/1902.1/18965 IQSS Dataverse Network [Distributor] V1 [Version]. Also available at http://voteview.com/Bakker_Poole_Bayesian_MDS.htm.Google Scholar
Bradlow, Eric T., and Schmittlein, David C. 2000. The little engines that could: Modeling the performance of World Wide Web search engines. Marketing Science 19(1): 43–62.Google Scholar
Brady, Henry. 1990. Traits versus issues: Factor versus ideal-point analysis of candidate thermometer ratings. Political Analysis 2: 97–129.Google Scholar
Cahoon, Lawrence S., Hinich, Melvin J., and Ordeshook, Peter C. 1978. A statistical multidimensional scaling method based on the spatial theory of voting. In Graphical Representation of Multivariate Data, ed. Wang, P.C. New York: Academic Press.Google Scholar
Clinton, Joshua D., Jackman, Simon D., and Rivers, Douglas. 2004. The statistical analysis of roll call data: A unified approach. American Political Science Review 98: 355–70.Google Scholar
De Leeuw, Jan. 1977. Applications of convex analysis to multidimensional scaling. In Recent Developments in Statistics, eds. Barra, J.R., Brodeau, F., Romier, G., and van Cutsem, B., 133–45. Amsterdam, The Netherlands: North-Holland.Google Scholar
De Leeuw, Jan. 1988. Convergence of the majorization method for multidimensional scaling. Journal of Classification 5: 163–80.Google Scholar
De Leeuw, Jan, and Willem, Heiser. 1977. Convergence of correction-matrix algorithms for multidimensional scaling. In Geometric Representations of Relational Data, eds. Lingoes, J.C., Roskam, E.E., and Borg, I., 735–52. Ann Arbor, MI: Mathesis Press.Google Scholar
de Leeuw, Jan, and Mair, P. 2009. Multidimensional scaling using majorization: The R package smacof. Journal of Statistical Software 31(3): 1–30, http://www.jstatsoft.org/v31/i03/.Google Scholar
DeSarbo, Wayne S., Park, Joonwook, and Rao, Vithala R. 2011. Deriving joint space positioning maps from consumer preference ratings. Marketing Letters 22(1): 1–14.Google Scholar
Eckart, Carl H., and Young, Gale. 1936. The approximation of one matrix by another of lower rank. Psychometrika 1: 211–8.Google Scholar
Enelow, James M., and Hinich, Melvin J. 1984. The Spatial Theory of Voting. New York: Cambridge University Press.Google Scholar
Fong, Duncan K.H., DeSarbo, Wayne S., Park, Joonwook, and Scott, Crystal J. 2010. A Bayesian vector multidimensional scaling procedure for the analysis of ordered preference data. Journal of the American Statistical Association 105(490): 482–92.Google Scholar
Gormley, Isobel C., and Murphy, Thomas B. 2006. A latent space model for rank data. Statistical Network Analysis: Models, Issues, and New Directions: Lecture Notes in Computer Science. New York: Springer.Google Scholar
Hinich, Melvin J. 2005. A new method for statistical multidimensional unfolding. Communications in Statistics—Theory and Methods 34: 2299–310.Google Scholar
Hoff, Peter D., Raftery, Adrian E., and Handcock, Mark S. 2002. Latent space approaches to social network analysis. Journal of the American Statistical Association 97(460): 1090–98.Google Scholar
Jacoby, William G. 1982. Unfolding the party identification scale: Improving the measurement an important concept. Political Methodology 8: 33–60.Google Scholar
Kruskal, Joseph B. 1964a. Multidimensional scaling by optimizing a goodness of fit to a nonmetric hypothesis. Psychometrika 29: 1–27.Google Scholar
Kruskal, Joseph B. 1964b. Nonmetric multidimensional scaling: A numerical method. Psychometrika 29: 115–29.Google Scholar
Liu, Dong C., and Nocedal, Jorge. 1989. On the limited memory BFGS method. Mathematical Programming 45 (1–3): 503–28.Google Scholar
Londregan, John B. 2000. Estimating legislators' preferred points. Political Analysis 8(1): 35–56.Google Scholar
Martin, Andrew D., and Quinn, Kevin M. 2002. Dynamic ideal point estimation via Markov chain Monte Carlo for the U.S. Supreme Court, 1953–1999. Political Analysis 10: 134–53.Google Scholar
Merrill, Samuel III, and Grofman, Bernard. 1999. A unified theory of voting: Directional and proximity spatial models. New York: Cambridge University Press.Google Scholar
Navarro, Daniel J., and Lee, Michael D. 2003. Combining dimensions and features in similarity-based representations. In Advances in Neural Information Processing Systems, eds. Becker, S., Thrun, S., and Obermayer, K., Vol. 15, 59–66. Cambridge, MA: MIT Press.Google Scholar
Neal, Radford M. 2003. Slice sampling. Annals of Statistics 31(3): 705–67.Google Scholar
Nelder, John A., and Mead, Roger. 1965. A simplex method for function minimization. Computer Journal 7: 308–13.Google Scholar
Oh, Man-Suk, and Raftery, Adrian E. 2001. Bayesian multidimensional scaling and choice of dimension. Journal of the American Statistical Association 96(455): 1031–44.Google Scholar
Okada, Kensuke, and Shigemasu, Kazuo. 2010. Bayesian multidimensional scaling for the estimation of a Minkowski exponent. Behavior Research Methods 42(4): 899–905.Google Scholar
Park, Joonwook, DeSarbo, Wayne S., and Liechty, John. 2008. A hierarchical Bayesian multidimensional scaling methodology for accommodating both structural and preference heterogeneity. Psychometrika 73(3): 451–72.Google Scholar
Poole, Keith T., and Rosenthal, Howard. 1984. U.S. presidential elections 1968–1980: A spatial analysis. American Journal of Political Science 28: 282–312.Google Scholar
Poole, Keith T., and Rosenthal, Howard. 1997. Congress: A political-economic history of roll call voting. New York: Oxford University Press.Google Scholar
Pope, Jeremy C., and Treier, Shawn A. 2011. Reconsidering the great compromise at the federal convention of 1787: Deliberation and agenda effects on the senate and slavery. American Journal of Political Science 55: 289–306.Google Scholar
Powell, Michael J. D. 1973. On search directions for minimization algorithms. Mathematical Programming 4: 193–201.CrossRefGoogle Scholar
Rabinowitz, George. 1976. A procedure for ordering object pairs consistent with the multidimensional unfolding model. Psychometrika 45: 349–73.Google Scholar
Schönemann, Peter H. 1970. On metric multidimensional unfolding. Psychometrika 35: 349–66.CrossRefGoogle Scholar
Shepard, Roger N. 1962a. The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika 27: 125–39.Google Scholar
Shepard, Roger N. 1962b. The analysis of proximities: Multidimensional scaling with an unknown distance function. II. Psychometrika 27: 219–46.Google Scholar
Torgerson, Warren S. 1952. Multidimensional scaling: I. Theory and method. Psychometrika 17: 401–19.Google Scholar
Torgerson, Warren S. 1958. Theory and methods of scaling. New York: Wiley.Google Scholar
Wang, Ming-Mei, Schonemann, Peter H., and Rusk, Jerrold G. 1975. A conjugate gradient algorithm for the multidimensional analysis of preference data. Multivariate Behavioral Research 10: 45–80.Google Scholar
Weisberg, Herbert F., and Rusk, Jerrold G. 1970. Dimensions of candidate evaluation. American Political Science Review 64: 1167–85.Google Scholar
Young, Gale, and Householder, Alston S. 1938. Discussion of a set of points in terms of their mutual distances. Psychometrika 3: 19–22.Google Scholar
Supplementary material: PDF

Bakker and Poole supplementary material

Appendix

Download Bakker and Poole supplementary material(PDF)
PDF 437.3 KB