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Ordinal Data, Ordered Scale Points, and Order Statistics

Published online by Cambridge University Press:  01 January 2025

Pieter Vijn
Pieter Vijn*
Affiliation:
University of Amsterdam
*
Reprint requests should be addressed to Pieter Vijn, Psychological Laboratory, University of Amsterdam, Weesperplein 8, kr. 456, 1018 XA Amsterdam.
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Abstract

This paper concerns ordinal responses. An ordered Dirichlet distribution describes prior and posterior beliefs about the cumulative probabilities of response categories. Associating the response categories with intervals of a latent random variable then induces a distribution on the order statistics of that variable. The psychometrician can use the asymptotic theory of order statistics to learn how distributional assumptions about the latent variable effect inference. An example relates the skewness of a latent variable to the proportional odds and proportional hazards models of McCullagh [1980].

Keywords

ordered Dirichlet priorordered cumulative probabilitiesordered scalesorder statisticslatent populationcoverageslog oddsratiostochastic orderingcomplementary log-log scalelogistic scale
Type
Original Paper
Information
Psychometrika , Volume 48 , Issue 3 , September 1983 , pp. 437 - 449
Copyright
Copyright © 1983 The Psychometric Society

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Footnotes

The author thanks Gideon J. Mellenbergh and Hendrikus Kelderman for comments, Jan Hoeksma for computational assistance and Erna Bot for typing the manuscript.

References

Aitchison, J. & Dunsmore, I. R. Statistical prediction analysis, Cambridge: University Press, 1975.CrossRefGoogle Scholar
Andersen, E. B. Discrete statistical models with social science applications, Amsterdam: North-Holland, 1980.Google Scholar
Andersen, J. A. & Philips, P. R. Regression, discrimination and measurement models for ordered categorical variables. Applied Statistics, 1981, 30, 22–31.CrossRefGoogle Scholar
Andersen, E. B. & Madsen, M. Estimating the parameters of the latent population distribution. Psychometrika, 1977, 42, 357–374.CrossRefGoogle Scholar
Ananda-Ordaz, F. J. On two families of transformations to additivity for binary response data. Biometrika, 1981, 68, 357–363.CrossRefGoogle Scholar
Altham, P. M. E. Exact Bayesian analysis of a 2 × 2 contingency table, and Fisher's “Exact” significance test. Journal of the Royal Statistical Society B, 1969, 31, 261–269.CrossRefGoogle Scholar
Barlow, D., Bremmer, J. & Brunk, H. Statistical inference under order restrictions, New York: John Wiley, 1972.Google Scholar
Bishop, Y. M. M., Fienberg, S. E. & Holland, P. W. Discrete multivariate analysis. Theory and practice, Cambridge: MIT Press, 1975.Google Scholar
Bock, R. D. Multivariate Statistical Methods in Behavioral Research, New York: McGraw-Hill, 1975.Google Scholar
Box, G. E. P. & Tiao, G. C. Bayesian Inference in statistical problems, Massachussets: Adisson-Wesley, 1973.Google Scholar
Gibbons, J. D. Nonparametric statistical inference, New York: McGraw-Hill, 1971.Google Scholar
Goldstein, H. Dimensionality, bias, independence and measurement scale problems in latent trait test score models. British Journal of Mathematical and Statistical Psychology, 1980, 33, 234–246.CrossRefGoogle Scholar
Goodman, L. A. Simple models for the analysis of association in cross classifications having ordered categories. Journal of the American Statistical Association, 1979, 74, 537–552.CrossRefGoogle Scholar
Grey, D. R. & Morgan, B. J. T. Some aspects of ROC curve fitting: normal and logistic models. Journal of Mathematical Psychology, 1972, 9, 128–139.CrossRefGoogle Scholar
Haberman, S. J. Log-linear models for frequency tables with ordered classifications. Biometrics, 1974, 30, 589–600.CrossRefGoogle Scholar
Johnson, N. L. & Kotz, S. Distributions in statistics: Continuous univariate distributions—1–2, New York: Wiley, 1970.Google Scholar
Leonard, T. A Bayesian approach to some multinomial estimation and pretesting problems. Journal of the American Statistical Association, 1977, 72, 869–874.Google Scholar
Lord, F. M. An analysis of the verbal scholastic aptitude test using Birnbaum's three parameter logistic model. Educational and Psychological Measurement, 1968, 28, 989–1020.CrossRefGoogle Scholar
Mann, N. R., Schafer, R. E.& Singpurwalla, N. D. Methods for statistical analysis of reliability and life data, New York: Wiley, 1974.Google Scholar
McCullagh, P. Regression models for ordinal data. Journal of the Royal Statistical Society B, 1980, 42, 109–142.CrossRefGoogle Scholar
Nishisato, S. Analysis of categorical data: Dual Scaling and its applications, Toronto: University of Toronto Press, 1980.CrossRefGoogle Scholar
Novick, M. R. & Jackson, P. H. Statistical methods for education and psychological research, New York: McGraw-Hill, 1974.Google Scholar
Pettitt, A. N. Inference for the linear model using a likelihood based on ranks. Journal of the Royal Statistical Society B, 1982, 44(2), 234–243.CrossRefGoogle Scholar
Pregibon, D. Goodness of link tests for generalized linear models. Applied Statistics, 1980, 29, 115–123.CrossRefGoogle Scholar
Torgerson, W. S. Theory and methods of scaling, New York: John Wiley, 1958.Google Scholar
Vijn, P. & Molenaar, J. W. Robustness regions for dichotomous decisions. Journal of Educational Statistics, 1981, 6(3), 205–235.CrossRefGoogle Scholar
Weisberg, H. I. Bayesian comparison of two ordered multinomial populations. Biometrics, 1972, 28, 859–867.CrossRefGoogle Scholar
Wilks, S. S. Mathematical statistics, New York: Wiley, 1962.Google Scholar