Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-07T18:47:57.909Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Symmetric, Invariant Measures of Multivariate Association

Published online by Cambridge University Press:  01 January 2025

Elliot M. Cramer  and
W. Alan Nicewander
Elliot M. Cramer*
Affiliation:
University of North Carolina at Chapel Hill
W. Alan Nicewander
Affiliation:
University of Oklahoma
*
Requests for reprints should be sent to Elliot M. Cramer, The L. L. Thurstone Psychometric Laboratory, Davie Hall, The University of North Carolina at Chapel Hill, Chapel Hill, NC, 27514.
Get access Rights & Permissions [Opens in a new window]

Abstract

A distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables. Several measures of multivariate association between two sets of variables are examined. It is shown that all of these measures are generalizations of the (univariate) squared-multiple correlation; all are functions of the canonical correlations, and all are invariant under linear transformations of the original sets of variables. It is further shown that the measures can be considered to be symmetric and are strictly ordered for any two sets of observed variables. It is suggested that measures of multivariate relationship may be used to generalize the concept of test reliability to the case of vector random variables.

Keywords

canonical correlationmultivariate analysisassociationredundancy
Type
Original Paper
Information
Psychometrika , Volume 44 , Issue 1 , March 1979 , pp. 43 - 54
Copyright
Copyright © 1979 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 authors wish to thank Dr. Scott Maxwell for his comments and suggestions on an earlier version of this manuscript. The work of the second author was completed while serving as Visiting Associate Professor, L. L. Thurstone Psychometric Laboratory.

References

Anderson, T. W. An introduction to multivariate statistical analysis, 1958, New York: Wiley.Google Scholar
Conger, A. J., & Lipshitz, R. Measures of reliability for profiles and test batteries. Psychometrika, 1973, 38, 411–427.CrossRefGoogle Scholar
Cooley, W. W., & Lohnes, P. R. Multivariate data analysis, 1971, New York: Wiley.Google Scholar
Coxhead, P. Measuring the relationship between two sets of variables. British Journal of Mathematical and Statistical Psychology, 1974, 27, 205–212.CrossRefGoogle Scholar
Cramer, E. M. A generalization of vector correlation and its relation to canonical correlation. Multivariate Behavioral Research, 1974, 9, 347–352.CrossRefGoogle ScholarPubMed
Cramer, E. M. The use of highly correlated predictors in regression analysis. Multivariate Behavioral Research, 1974, 9, 241–244.CrossRefGoogle ScholarPubMed
Darlington, R. B. Weinberg, S. L., & Walberg, J. J. Canonical variate analysis and related techniques. Review of Educational Research, 1973, 43, 433–454.CrossRefGoogle Scholar
Gleason, T. C. On redundancy in canonical analysis. Psychological Bulletin, 1976, 83, 1004–1006.CrossRefGoogle Scholar
Hotelling, H. Relations between two sets of variates. Biometrika, 1936, 28, 321–377.CrossRefGoogle Scholar
Miller, J. K., & Farr, S. D. Bimultivariate redundancy: A comprehensive measure of interbattery relationship. Multivariate Behavioral Research, 1971, 6, 313–324.CrossRefGoogle Scholar
Rozeboom, W. W. Linear correlation between sets of variables. Psychometrika, 1965, 30, 57–71.CrossRefGoogle ScholarPubMed
Selby, S. M. Standard Mathematical Tables, 17th ed., Cleveland, Ohio: Chemical Rubber Co., 1969.Google Scholar
Shaffer, J. P., & Gillo, M. W. A multivariate extension of the correlation ratio. Educational and Psychological Measurement, 1974, 34, 521–524.CrossRefGoogle Scholar
Stewart, D., & Love, W. A general canonical correlation index. Psychological Bulletin, 1968, 70, 160–163.CrossRefGoogle ScholarPubMed
Timm, N. H. Multivariate analysis with applications in education and psychology, 1975, Monterey, California: Brooks/Cole.Google Scholar