Hostname: page-component-6bf8c574d5-gr6zb Total loading time: 0 Render date: 2025-02-24T07:44:12.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Boundary Conditions for a Monotone Analysis of Symmetric Matrices

Published online by Cambridge University Press:  01 January 2025

James C. Lingoes
James C. Lingoes*
Affiliation:
The University of Michigan
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matrices S, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements of S. A comparable and more easily proven theorem for the vector model is also introduced. At most n-2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.

Keywords

Boundary ConditionLower BoundPublic PolicyEuclidean SpaceStatistical Theory
Type
Original Paper
Information
Psychometrika , Volume 36 , Issue 2 , June 1971 , pp. 195 - 203
Copyright
Copyright © 1971 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

This research in nonmetric methods is supported in part by a grant from the National Science Foundation (GS-929 & -2850).

The very helpful comments and encouragement of Louis Guttman and J. Douglas Carroll are greatly appreciated. Finally, to that unknown, but not unsung, reviewer who helped in the clarification of the argument, I express my thanks.

References

Guttman, L. A definition of dimensionality and distance for graphs. (Unpubl. mimeograph), University of Michigan, 1965.Google Scholar
Guttman, L. The development of nonmetric space analysis: A letter to John Ross. Multivariate Behavioral Research, 1967, 2, 71–82.CrossRefGoogle ScholarPubMed
Guttman, L. A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 1968, 33, 469–506.CrossRefGoogle Scholar
Lingoes, J. C., & Guttman, L. Nonmetric factor analysis: A rank reducing alternative to linear factor analysis. Multivariate Behavioral Research, 1967, 2, 485–505.CrossRefGoogle Scholar
Lingoes, J. C. & Roskam, E. An empirical study of two multidimensional scaling algorithms. Multivariate Behavioral Research, 1970, 5, (in press).Google Scholar
Torgerson, W. S. Theory and Methods of Scaling, 1958, New York: John Wiley & Sons, Inc..Google Scholar