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Covariance structure analysis of ordinal ipsative data

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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.

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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.

    Article  Google 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.

    Google 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.

    Google Scholar 

  • Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation.Psychometrika, 50, 229–242.

    Article  Google 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.

    Article  Google Scholar 

  • Brady, H. E. (1990). Dimensional analysis of ranking data.American Journal of Political Science, 34, 1017–1048.

    Google Scholar 

  • Burros, R. H., & Gibson, W. A. (1954). A solution for Case III of the law of comparative judgment.Psychometrika, 19, 57–64.

    Article  Google Scholar 

  • Chan, W. (1995).Covariance structure analysis of ipsative data. Unpublished doctoral dissertation, UCLA.

  • Chan, W., & Bentler, P. M. (1993). The covariance structure analysis of ipsative data.Sociological Methods & Research, 22, 214–247.

    Google 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.

    Article  Google Scholar 

  • Chan, W., & Bentler, P. M. (1997).Covariance structure analysis of multiplicative ipsative data. Manuscript submitted for publication.

  • 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.

    Google 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.

    Google Scholar 

  • Clemans, W. V. (1966). An analytical and empirical examination of some properties of ipsative measures.Psychometric Monographs, 14.

  • Critchlow, D. E. (1985).Metric methods for analyzing partially ranked data. New York: Springer-Verlag.

    Google Scholar 

  • Critchlow, D. E., Fligner, M. A., & Verducci, J. S. (1991). Probability models on rankings.Journal of Mathematical Psychology, 35, 294–318.

    Article  Google Scholar 

  • De Soete, G., & Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model.Psychometrika, 48, 553–566.

    Google Scholar 

  • Diaconis, P. (1988).Group representation in probability and statistics. Hayward, CA: Institute of Mathematical Statistics.

    Google Scholar 

  • Fligner, M. A., & Verducci, J. S. (1986). Distance-based ranking models.Journal of the Royal Statistical Society B, 48, 859–869.

    Google Scholar 

  • Gibson, W. A. (1953). A least-squares solution for Case IV of the law of comparative judgment.Psychometrika, 18, 15–21.

    Google Scholar 

  • Gong, G., & Samaniego, F. J. (1981). Pseudo maximum likelihood estimation: Theory and applications.The Annals of Statistics, 9, 861–869.

    Google 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.

    Google 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.

    Article  Google 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.

    PubMed  Google Scholar 

  • MacKay, D. B., & Chaiy, S. (1982). Parameter estimation for the Thurstone Case III model.Psychometrika, 47, 353–358.

    Article  Google Scholar 

  • Mallows, C. L. (1957). Non-null ranking models, I.Biometrika, 44, 114–130.

    Google 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.

    Google Scholar 

  • Noether, G. E. (1960). Remarks about a paired comparison model.Psychometrika, 25, 357–367.

    Article  Google Scholar 

  • Parke, W. R. (1986). Pseudo maximum likelihood estimation: The asymptotic distribution.The Annals of Statistics, 14, 355–357.

    Google Scholar 

  • Poon, W. Y., & Lee, S. Y. (1987). Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients.Psychometrika, 52, 409–430.

    Article  Google 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.

    Article  Google Scholar 

  • Poon, W. Y., & Leung, Y. P. (1993). Analysis of structural equation models with interval and polytomous data.Statistics & Probability Letters, 17, 127–137.

    Article  Google 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.

    Google 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.

    Google Scholar 

  • Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone's model of comparative judgment.Japanese Psychological Research, 22, 188–196.

    Google 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.

    Google Scholar 

  • Wald, A. (1950). A note on the identification of economic relations. In T. C. Koopmans (Ed.),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 A. S. Goldberger & O. D. Duncan (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.

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Authors and Affiliations

  1. University of California, Los Angeles

    Peter M. Bentler

  2. Department of Psychology, The Chinese University of Hong Kong, Shatin, Hong Kong

    Wai Chan

Authors
  1. Wai Chan
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  2. Peter M. Bentler
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Additional information

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.

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Chan, W., Bentler, P.M. Covariance structure analysis of ordinal ipsative data. Psychometrika 63, 369–399 (1998). https://doi.org/10.1007/BF02294861

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