Abstract
Monte Carlo procedures were used to investigate the properties of a nonmetric multidimensional scaling algorithm when used to scale an incomplete matrix of dissimilarities. Various recommendations for users who wish to scale incomplete matrices are made: (a) recovery was found to be satisfactory provided that the “degrees of freedom” ratio exceeded 3.5, irrespective of error level; (b) cyclic designs were found to provide best recovery, although random patterns of deletion performed almost as well; and (c) strongly locally connected designs, specifically overlapping cliques, were generally inferior. These conclusions are based on 837 scaling solutions and are applicable to stimulus sets containing more than 30 objects.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Clatworthy, W. H. Partially balanced incomplete block designs with two associate classes and two treatments per block.Journal of Research, National Bureau of Standards, 1955,54, 177–190.
Clatworthy, W. H.Tables of two-associate-class partially balanced designs. National Bureau of Standards Applied Mathematics Series #63, 1973.
David, H. A. The structure of cyclic paired-comparison designs.The Journal of the Australian Mathematical Society, 1963 (a),3, 117–127.
David, H. A.The method of paired comparisons. London: Griffin, 1963 (b).
Degerman, R. L. The geometric representation of some simple structures. In Shepard, R. N., Romney, A. K., & Nerlove, S. B. (Eds.)Multidimensional scaling: Theory and applications in the behavioral sciences. Vol. I Theory. New York: Seminar Press, 1972.
Domoney, D. W. Nonmetric multidimensional scaling: A Monte Carlo study of planned missing data. Unpublished M. A. thesis, University of Western Ontario, 1973.
Harary, F.Graph Theory. Reading, Mass.: Addison-Wesley, 1969.
John, J. A., Wolock, F. W. & David, H. A.Cyclic designs. National Bureau of Standards Applied Mathematics Series #62, 1972.
Johnson, R. M. & Van Dyk, G. T. A resistance analogy for pairwise experimental designs. Unpublished manuscript, Market Facts Inc., Chicago, 1974.
Kendall, M. G.Rank correlation methods (3rd Ed.). London: Griffin, 1962.
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 1964,29, 1–27.
Kruskal, J. B. Special problems and possibilities. Mimeo, Bell-Penn Multidimensional Scaling Workshop, 1972.
Kruskal, J. B. & Carroll, J. D. Geometrical models and badness-of-fit functions. In Krishnaiah, P. R. (Ed.),Multivariate Analysis II. New York: Academic Press, 1969.
Lingoes, J. C. & Roskam, E. E. A mathematical and empirical analysis of two multidimensional scaling algorithms.Psychometric Monograph No. 19, 1973.
McKeon, J. J. Some cyclical incomplete paired comparison designs. L. L. Thurstone Psychometric Laboratory Report #24, University of North Carolina, 1960.
Ramsay, J. O. Some statistical considerations in multidimensional scaling.Psychometrika, 1969,34, 167–182.
Rao, V. R. & Katz, R. Alternative multidimensional scaling methods for large stimulus sets.Journal of Marketing Research, 1971,8, 488–494.
Sherman, C. R. Nonmetric multidimensional scaling: A Monte Carlo study of the basic parameters.Psychometrika, 1972,37, 323–355.
Spence, I. A Monte Carlo evaluation of three nonmetric multidimensional scaling algorithms.Psychometrika, 1972,37, 461–486.
Spence, I. & Graef, J. The determination of the underlying dimensionality of an empirically obtained matrix of proximities.Multivariate Behavioral Research, 1974,9, (in press).
Spence, I. & Janssen, H. Cyclic incomplete designs as an alternative to ISIS models in nonmetric scaling. Paper presented at the Spring Meetings of the Psychometric Society, Stanford University, 1974.
Torgerson, W. S.Theory and methods of scaling. New York: Wiley, 1958.
Tschudi, F. The latent, the manifest, and the reconstructed in multivariate data reduction models. Unpublished Ph.D. thesis, University of Oslo, 1972.
Turán, P. On the theory of graphs.Colloquium Mathematicum, 1954,3, 19–30.
Wagenaar, W. A. & Padmos, P. Quantitative interpretation of stress in Kruskal's multidimensional scaling technique.British Journal of Mathematical and Statistical Psychology, 1971,24, 101–110.
Young, F. W. A FORTRAN IV program for nonmetric multidimensional scaling. L. L. Thurstone Psychometric Laboratory Report #56, University of North Carolina, 1968.
Young, F. W. Nonmetric multidimensional scaling: Recovery of metric information.Psychometrika, 1970,35, 455–473.
Young, F. W. & Cliff, N. Interactive scaling with individual subjects.Psychometrika, 1972,37, 385–415.
Author information
Authors and Affiliations
Additional information
This research was supported by grant A8351 from the National Research Council of Canada to the first author, and is based in part on the M. A. thesis of the second author. The assistance, advice and criticisms of John C. Ogilvie and John Vranch are gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Spence, I., Domoney, D.W. Single subject incomplete designs for nonmetric multidimensional scaling. Psychometrika 39, 469–490 (1974). https://doi.org/10.1007/BF02291669
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02291669