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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.

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