Psychometrika

, Volume 55, Issue 1, pp 123–149

Least squares metric, unidimensional scaling of multivariate linear models

  • Keith T. Poole
Article

Abstract

The squared error loss function for the unidimensional metric scaling problem has a special geometry. It is possible to efficiently find the global minimum for every coordinate conditioned on every other coordinate being held fixed. This approach is generalized to the case in which the coordinates are polynomial functions of exogenous variables. The algorithms shown in the paper are linear in the number of parameters. They always descend and, at convergence, every coefficient of every polynomial is at its global minimum conditioned on every other parameter being held fixed. Convergence is very rapid and Monte Carlo tests show the basic procedure almost always converges to the overall global minimum.

Key words

city block scaling metric unfolding constrained coordinates 

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References

  1. Asher, Herbert B., & Herbert F. Weisberg. (1978). Voting change in congress: Some dynamic perspectives on an evolutionary process.American Journal of Political Science, 22, 391–425.Google Scholar
  2. Carrol, J. Douglas, Pruzansky, Sandra, & Kruskal, Joseph B. (1980). CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters.Psychometrika, 45, 3–24.Google Scholar
  3. Clausen, Aage. (1973).How congressmen decide: A policy focus. New York: St. Martin's Press.Google Scholar
  4. Defays, D. (1978). A short note on a method of seriation.British Journal of Mathematical and Statistical Psychology, 31, 49–53.Google Scholar
  5. de Leeuw, Jan. (1984). Differentiability of Kruskal's stress at a local minimum.Psychometrika, 49, 111–114.Google Scholar
  6. DeSarbo, Wayne, & Carroll, J. Douglas. (1984). Three-way metric unfolding via alternating weighted least squares.Psychometrika, 50, 275–300.Google Scholar
  7. Eckart, Carl, & Young, Gale. (1936). The approximation of one matrix by another of lower rank.Psychometrika, 1, 211–218.Google Scholar
  8. Fenno, Richard F. (1978).Home style: House members in their districts Boston: Little, Brown and Co.Google Scholar
  9. Fiorina, Morris. (1974).Representatives, roll calls, and constituencies. Lexington, MA: Heath.Google Scholar
  10. Heiser, Willem J. (1981).Unfolding analysis of proximity data. Leiden: University of Leiden, Department of Psychology.Google Scholar
  11. Hinich, Melvin J., & Roll, Richard. (1981). Measuring nonstationarity in the parameters of the market model.Research in Finance, 3, 1–51.Google Scholar
  12. Hubert, Lawrence, & Arabie, Phipps. (1986). “Unidimensional Scaling and Combinatorial Optimization.” In de Leeuw et al. (Eds.),Multidimensional data analysis. Leiden: DSWO Press.Google Scholar
  13. Hubert, Lawrence, & Arabie, Phipps. (1988). Relying on necessary conditions for optimization: Unidimensional scaling and some extensions. InClassification and related methods of data analysis. Amsterdam: North-Holland.Google Scholar
  14. Kritzer, Herbert M. (1978). Ideology and American political elites.Public Opinion Quarterly, 42, 484–502.Google Scholar
  15. Kruskal, Joseph B. (1964a) Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 29, 1–28.Google Scholar
  16. Kruskal, Joseph B. (1964b). Nonmetric multidimensional scaling: A numerical method.Psychometrika, 29, 115–130.Google Scholar
  17. Peltzman, Sam. (1984). Constituent interest and congressional voting.Journal of Law and Economics, 27, 181–210.Google Scholar
  18. Poole, Keith T. (1981). Dimensions of interest group evaluation of the U.S. Senate, 1969–1978.American Journal of Political Science, 25, 49–67.Google Scholar
  19. Poole, Keith T. (1984). Least squares metric, unidimensional unfolding.Psychometrika, 49, 311–323.Google Scholar
  20. Poole, Keith T., & Daniels, R. Steven. (1985). Ideology, party, and voting in the U.S. Congress, 1959–80.American Political Science Review, 79, 373–399.Google Scholar
  21. Poole, Keith T., & Rosenthal, Howard. (1986). The dynamics of interest group evaluations of Congress (GSIA Working Paper 1-86-87). Pittsburgh, PA: Carnegie Mellon University.Google Scholar
  22. Ramsay, James O. (1977). Maximum likelihood estimation in multidimensional scaling.Psychometrika, 42, 241–266.Google Scholar

Copyright information

© The Psychometric Society 1990

Authors and Affiliations

  • Keith T. Poole
    • 1
  1. 1.Graduate School of Industrial AdministrationCarnegie-Mellon UniversityPittsburgh

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