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Covariance Structure Analysis of Ordinal Ipsative Data

Published online by Cambridge University Press:  01 January 2025

Wai Chan and
Peter M. Bentler
Wai Chan*
Affiliation:
The Chinese University of Hong Kong
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Correspondence concerning this article should be addressed to Wai Chan, Department of Psychology, The Chinese University of Hong Kong, Shatin, HONG KONG.
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Abstract

Data are ipsative if they are subject to a constant-sum constraint for each individual. In the present study, ordinal ipsative data (OID) are defined as the ordinal rankings across a vector of variables. It is assumed that OID are the manifestations of their underlying nonipsative vector y, which are difficult to observe directly. A two-stage estimation procedure is suggested for the analysis of structural equation models with OID. In the first stage, the partition maximum likelihood (PML) method and the generalized least squares (GLS) method are proposed for estimating the means and the covariance matrix of Acy, where Ac is a known contrast matrix. Based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, the generalized least squares method is used to estimate the structural parameters in the second stage. A goodness-of-fit statistic is given for testing the hypothesized covariance structure. Simulation results show that the proposed method works properly when a sufficiently large sample is available.

Keywords

covariance structure analysisordinal ipsative datapartition maximum likelihood methodpseudo maximum likelihood methodtwo-stage estimation

Information

Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 4 , December 1998, pp. 369 - 399
Copyright
Copyright © 1998 The Psychometric Society

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Footnotes

This research was supported by National Institute on Drug Abuse Grants DA01070 and DA10017. The authors are indebted to Dr. Lee Cooper, Dr. Eric Holman, Dr. Thomas Wickens for their valuable suggestions on this study, and Dr. Fanny Cheung for allowing us to use her CPAI data set in this article. The authors would also like to acknowledge the helpful comments from the editor and the two anonymous reviewers.

References

Anderson, J. C., & Gerbing, D. W. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49, 155–173.CrossRefGoogle Scholar
Arbuckle, J., & Nugent, J. H. (1973). A general procedure for parameter estimation for the law of comparative judgment. British Journal of Mathematical and Statistical Psychology, 26, 240–260.CrossRefGoogle Scholar
Bentler, P. M. (1995). EQS structural equations program manual, Encino, CA: Multivariate Software.Google Scholar
Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975). Discrete multivariate analysis, Cambridge: The MIT Press.Google Scholar
Böckenholt, U. (1992). Thurstonian representation for partial ranking data. British Journal of Mathematical and Statistical Psychology, 45, 31–49.CrossRefGoogle Scholar
Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation. Psychometrika, 50, 229–242.CrossRefGoogle Scholar
Bradley, R. A., & Terry, M. E. (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika, 39, 324–345.Google Scholar
Brady, H. E. (1989). Factor and ideal point analysis for interpersonally incomparable data. Psychometrika, 54, 181–202.CrossRefGoogle Scholar
Brady, H. E. (1990). Dimensional analysis of ranking data. American Journal of Political Science, 34, 1017–1048.CrossRefGoogle Scholar
Burros, R. H., & Gibson, W. A. (1954). A solution for Case III of the law of comparative judgment. Psychometrika, 19, 57–64.CrossRefGoogle Scholar
Chan, W. (1995). Covariance structure analysis of ipsative data. Unpublished doctoral dissertation, UCLA.Google Scholar
Chan, W., & Bentler, P. M. (1993). The covariance structure analysis of ipsative data. Sociological Methods & Research, 22, 214–247.CrossRefGoogle Scholar
Chan, W., & Bentler, P. M. (1996). Covariance structure analysis of partially additive ipsative data using restricted maximum likelihood estimation. Multivariate Behavioral Research, 31, 289–312.CrossRefGoogle ScholarPubMed
Chan, W., & Bentler, P. M. (1997). Covariance structure analysis of multiplicative ipsative data. Manuscript submitted for publication.Google Scholar
Cheung, F. M., Leung, K., Fan, R. M., Song, W. Z., Zhang, J. X., & Zhang, J. P. (1996). Development of the Chinese Personality Assessment Inventory. Journal of Cross-Cultural Psychology, 27, 181–199.CrossRefGoogle Scholar
Cheung, F. M., Leung, K., Song, W. Z., Zhang, J. X., & Zhang, J. P. (1993). The Chinese Personality Assessment Inventory, Hong Kong: The Chinese University of Hong Kong.Google Scholar
Clark, C. E. (1961). The greatest of a finite set of random variables. Operations Research, 9, 145–162.CrossRefGoogle Scholar
Clemans, W. V. (1966). An analytical and empirical examination of some properties of ipsative measures. Psychometric Monographs, 14.Google Scholar
Critchlow, D. E. (1985). Metric methods for analyzing partially ranked data, New York: Springer-Verlag.CrossRefGoogle Scholar
Critchlow, D. E., Fligner, M. A., & Verducci, J. S. (1991). Probability models on rankings. Journal of Mathematical Psychology, 35, 294–318.CrossRefGoogle Scholar
De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553–566.CrossRefGoogle Scholar
Diaconis, P. (1988). Group representation in probability and statistics, Hayward, CA: Institute of Mathematical Statistics.CrossRefGoogle Scholar
Fligner, M. A., & Verducci, J. S. (1986). Distance-based ranking models. Journal of the Royal Statistical Society B, 48, 859–869.CrossRefGoogle Scholar
Gibson, W. A. (1953). A least-squares solution for Case IV of the law of comparative judgment. Psychometrika, 18, 15–21.CrossRefGoogle Scholar
Gong, G., & Samaniego, F. J. (1981). Pseudo maximum likelihood estimation: Theory and applications. The Annals of Statistics, 9, 861–869.CrossRefGoogle Scholar
Graybill, F. A. (1969). Introduction to matrices with applications in statistics, Belmont, CA: Wadsworth.Google Scholar
Heiser, W., & de Leeuw, J. (1981). Multidimensional mapping of preference data. Mathematiques et Science Humaines, 19, 39–96.Google Scholar
Johnson, N. L., & Kotz, S. (1972). Distribution in statistics: continuous multivariate distributions, New York: Wiley.Google Scholar
Küsters, U. (1990). A note on sequential ML estimators and their asymptotic covariances. Statistical Papers, 31, 131–145.CrossRefGoogle Scholar
Lee, S. Y., Poon, W. Y., & Bentler, P. M. (1990). A three-stage estimation procedure for structural equation models with polytomous variables. Psychometrika, 55, 45–52.CrossRefGoogle Scholar
Lee, S. Y., Poon, W. Y., & Bentler, P. M. (1995). A two-stage estimation of structural models with continuous and polytomous variables. British Journal of Mathematical and Statistical Psychology, 48, 339–358.CrossRefGoogle ScholarPubMed
MacKay, D. B., & Chaiy, S. (1982). Parameter estimation for the Thurstone Case III model. Psychometrika, 47, 353–358.CrossRefGoogle Scholar
Mallows, C. L. (1957). Non-null ranking models, I. Biometrika, 44, 114–130.CrossRefGoogle Scholar
Marden, J. I. (1995). Analyzing and modeling rank data, London: Chapman & Hall.Google Scholar
McDonald, R. P., & Swaminathan, H. (1973). A simple matrix calculus with applications to multivariate analysis. General Systems, 18, 37–54.Google Scholar
Mosteller, F. (1951). Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations. Psychometrika, 16, 3–9.CrossRefGoogle Scholar
Noether, G. E. (1960). Remarks about a paired comparison model. Psychometrika, 25, 357–367.CrossRefGoogle Scholar
Parke, W. R. (1986). Pseudo maximum likelihood estimation: The asymptotic distribution. The Annals of Statistics, 14, 355–357.CrossRefGoogle Scholar
Poon, W. Y., & Lee, S. Y. (1987). Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients. Psychometrika, 52, 409–430.CrossRefGoogle Scholar
Poon, W. Y., Lee, S. Y., Afifi, A. A., & Bentler, P. M. (1990). Analysis of multivariate polytomous variates in several groups via the partition maximum likelihood approach. Computational Statistics & Data Analysis, 10, 17–27.CrossRefGoogle Scholar
Poon, W. Y., & Leung, Y. P. (1993). Analysis of structural equation models with interval and polytomous data. Statistics & Probability Letters, 17, 127–137.CrossRefGoogle Scholar
Radcliffe, J. A. (1963). Some properties of ipsative score matrices and their relevance for some current interest tests. Australian Journal of Psychology, 15, 1–11.CrossRefGoogle Scholar
SAS Institute (1989). SAS/IML software: Usage and reference, Version 6, first edition, Cary, NC: SAS Institute.Google Scholar
Siegel, S., & Castellan, N. J. (1988). Nonparametric statistics for the behavioral sciences, Singapore: McGraw-Hill.Google Scholar
Sjöberg, L. (1962). The law of comparative judgment: A case not assuming equal variances and covariances. Scandinavian Journal of Psychology, 3, 219–225.CrossRefGoogle Scholar
Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone's model of comparative judgment. Japanese Psychological Research, 22, 188–196.CrossRefGoogle Scholar
Takane, Y. (1987). Analysis of covariance structures and probabilistic binary choice data. Communications & Cognition, 20, 45–62.Google Scholar
Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
Wald, A. (1950). A note on the identification of economic relations. In Koopmans, T. C. (Eds.), Statistical inference in dynamic economic models (pp. 238–44). New York: Wiley.Google Scholar
Wiley, D. E. (1973). The identification problem for structural equations with unmeasured variables. In Goldberger, A. S., & Duncan, O. D. (Eds.), Structural equation models in the social sciences (pp. 69–83). New York: Academic Press.Google Scholar
Yao, K. P. G. (1995). Bayesian estimation of Thurstonian ranking models based on the Gibbs sampler. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign.Google Scholar