Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-02-07T06:24:26.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Confidence Regions for Multidimensional Scaling Analysis

Published online by Cambridge University Press:  01 January 2025

J. O. Ramsay
J. O. Ramsay*
Affiliation:
McGill University
*
Requests for reprints should be sent to J. O. Ramsay, Department of Psychology, McGill University, 1205 McGregor Avenue, Montreal, Quebec, Canada H3A 1B1.
Get access Rights & Permissions [Opens in a new window]

Abstract

Techniques are developed for surrounding each of the points in a multidimensional scaling solution with a region which will contain the population point with some level of confidence. Bayesian credibility regions are also discussed. A general theorem is proven which describes the asymptotic distribution of maximum likelihood estimates subject to identifiability constraints. This theorem is applied to a number of models to display asymptotic variance-covariance matrices for coordinate estimates under different rotational constraints. A technique is described for displaying Bayesian conditional credibility regions for any sample size.

Keywords

conditional credibility regionsBayesian analysisasymptotic distributionmaximum likelihood estimationvariance-covariance of configuration
Type
Original Paper
Information
Psychometrika , Volume 43 , Issue 2 , June 1978 , pp. 145 - 160
Copyright
Copyright © 1978 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

The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.

References

Reference Notes

Ramsay, J. O. Further developments in maximum likelihood multidimensional scaling. Paper submitted for publication, 1977.CrossRefGoogle Scholar
Ramsay, J. O. Asymptotic results for singular information matrices with internal and external identifiability constraints. Paper submitted for publication, 1977.Google Scholar

References

Box, G. E. P., & Tiao, G. C. Bayesian inference in statistical analysis, 1973, Reading, Massachusetts: Addison-Wesley.Google Scholar
Carroll, J. D., & Chang, J. J. Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition. Psychometrika, 1970, 35, 283–319.CrossRefGoogle Scholar
Christian, J. Psychological differentiation and definition of the self: Multidimensional scaling approach. Unpublished doctoral dissertation, McGill University, 1976.Google Scholar
De Groot, M. H. Optimal statistical decisions, 1970, New York: McGraw-Hill.Google Scholar
Novick, M. R., & Jackson, P. H. Statistical methods for educational and psychological research, 1974, New York: McGraw-Hill.Google Scholar
Ramsay, J. O. Economical method of analyzing perceived color differences. Journal of the Optical Society of America, 1968, 58, 19–22.CrossRefGoogle ScholarPubMed
Ramsay, J. O. Maximum likelihood estimation in multidimensional scaling. Psychometrika, 1977, 42, 241–266.CrossRefGoogle Scholar
Rao, C. R. Linear statistical inference and its applications, 2nd ed., New York: Wiley, 1973.CrossRefGoogle Scholar
Searle, S. R. Linear models, 1971, New York: Wiley.Google Scholar
Walker, A. M. On the asymptotic behavior of posterior distributions. Journal of the Royal Statistical Society, Series B, 1969, 31, 80–88.CrossRefGoogle Scholar
Wilks, S. S. Mathematical statistics, 1962, New York: Wiley.Google Scholar