Psychometrika

, Volume 40, Issue 4, pp 525–548 | Cite as

Nonlinear programming approach to optimal scaling of partially ordered categories

  • Shizuhiko Nishisato
  • P. S. Arri
Article

Abstract

A modified technique of separable programming was used to maximize the squared correlation ratio of weighted responses to partially ordered categories. The technique employs a polygonal approximation to each single-variable function by choosing mesh points around the initial approximation supplied by Nishisato's method. The major characteristics of this approach are: (i) it does not require any grid refinement; (ii) the entire process of computation quickly converges to the acceptable level of accuracy, and (iii) the method employs specific sets of mesh points for specific variables, whereby it reduces the number of variables for the separable programming technique. Numerical examples were provided to illustrate the procedure.

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References

  1. Abrham, J. and Arri, P. S. Approximation of separable functions in convex programming.Canadian Journal of Operational Research and Information Processing, 1973,11, 245–252.Google Scholar
  2. Bock, R. D. The selection of judges for preference testing.Psychometrika, 1956,21, 349–366.Google Scholar
  3. Bock, R. D. Methods and applications of optimal scaling.The University of North Carolina Psychometric Laboratory Research Memorandum, No. 25, 1960.Google Scholar
  4. Bradley, R. A. Katti, S. K. & Coons, I. J. Optimal scaling for ordered categories.Psychometrika, 1962,27, 355–374.Google Scholar
  5. Dantzig, G. B.Linear programming and extensions. Princeton University Press, 1963.Google Scholar
  6. Edgerton, H. A. & Kolbe, L. E. The method of minimum variation for the combination of criteria.Psychometrika, 1936,1, 183–187.Google Scholar
  7. Fisher, R. A.Statistical methods for research workers. Edinburgh: Oliver and Boyd, 1938.Google Scholar
  8. Fox, L.An introduction to numerical linear algebra. Oxford: Oxford University Press, 1964.Google Scholar
  9. Guttman, L. The quantification of a class of attributes: a theory and method of scale construction. In P. Horst,et al., The prediction of personal adjustment. New York: Social Science Research Council, 1941, 319–348.Google Scholar
  10. Guttman, L. An approach for quantifying paired comparisons and rank order.Annals of Mathematical Statistics, 1946,17, 144–163.Google Scholar
  11. Guttman, L. The principal components of scale analysis. In S. A. Stouffer,et al., Measurement and prediction. Princeton: Princeton University Press, 1950, 312–361.Google Scholar
  12. Hadley, G.Linear programming. Addison-Wesley, 1962.Google Scholar
  13. Hadley, G.Nonlinear and dynamic programming. Addison-Wesley, 1964.Google Scholar
  14. Hayashi, C. On the quantification of qualitative data from the mathematico-statistical point of view.Annals of Institute of Statistical Mathematics,3, No. 1, 1950.Google Scholar
  15. Hayashi, C. On the prediction of phenomena from qualitative data and quantification of qualitative data from the mathematico-statistical point of view.Annals of Institute of Statistical Mathematics,3, No. 2, 1952.Google Scholar
  16. Horst, P. Obtaining a composite measure from a number of different measures of the same attribute.Psychometrika, 1936,1, 53–60.Google Scholar
  17. Johnson, P. O. The quantification of qualitative data in discriminant analysis.Journal of American Statistical Association, 1950,45, 65–70.Google Scholar
  18. Lord, F. M. Some relations between Guttman's principal components of scale analysis and other psychometric theory.Psychometrika, 1958,23, 291–296.Google Scholar
  19. McDonald, R. P. A unified treatment of the weighting problem.Psychometrika, 1968,33, 351–381.Google Scholar
  20. Miller, C. E. The simplex method for local separable programming. In Graves, R. and Wolfe P. (eds.)Recent advances in mathematical programming. McGraw-Hill, 1963.Google Scholar
  21. Mosteller, F. A theory of scalogram analysis, using noncumulative types of items: a new approach to Thurstone's method of scaling attitude.Harvard University Laboratory of Social Relations, Report No.9, 1949.Google Scholar
  22. Nishisato, S. Analysis of variance through optimal scaling.Proceeding of the First Canadian Conference on Applied Statistics. Montreal: Sir George Williams University Press, 1971, 306–316.Google Scholar
  23. Nishisato, S. Analysis of variance of categorical data through selective scaling.Proceedings of the 20th International Congress of Psychology, Tokyo, 1972, 279.Google Scholar
  24. Nishisato, S. Optimal scaling of partially ordered categories.Paper presented at the Spring Meeting of the Psychometric Society, 1973.Google Scholar
  25. Nishisato, S. and Inukai, Y. Partially optimal scaling of items with ordered categories.Japanese Psychological Research, 1972,14, 109–119.Google Scholar
  26. Shanno, D. F. and Weil, R. L. Management science: a view from nonlinear programming.Communications of the ACM, 1972,15, 542–549.Google Scholar
  27. Shiba, S. A method for scoring multicategory items.Japanese Psychological Research, 1965,7, 75–79.Google Scholar
  28. Slater, P. Canonical analysis of discriminance. In Eysenk, H. J. (Ed.)Experiments in Psychology, volume 2. London: Routledge and Kegan Paul, 1960, 256–270.Google Scholar
  29. Wilks, S. S. Weighting systems for linear function of correlated variables when there is no dependent variable.Psychometrika, 1938,3, 23–40.Google Scholar

Copyright information

© Psychometric Society 1975

Authors and Affiliations

  • Shizuhiko Nishisato
    • 1
  • P. S. Arri
  1. 1.The Ontario Institute for Studies in EducationCanada

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