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A generalized definition of the polychoric correlation coefficient
Uppsala universitet, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Samhällsvetenskapliga fakulteten, Institutionen för informationsvetenskap, Statistik.
(Engelska)Manuskript (Övrigt vetenskapligt)
Abstract [en]

We generalize the polychoric correlation coefficient to a large class of parametric families of bivariate distributions. The generalized definition agrees with the conventional definition on the family of bivariate normal distributions, and with the generalized tetrachoric correlation coefficient for dichotomous variables. Furthermore, we provide some suggestions for goodness-of-fit tests. The theory is illustrated with examples, which show that the distributional assumption can have a substantial impact on the conclusions of the association analysis.

Nyckelord [en]
polychoric correlation, generalization, contingency table, ordinal variables, measure of association, robustness, goodness-of-fit
Nationell ämneskategori
Sannolikhetsteori och statistik
Forskningsämne
Statistik
Identifikatorer
URN: urn:nbn:se:uu:diva-100695OAI: oai:DiVA.org:uu-100695DiVA, id: diva2:210883
Tillgänglig från: 2009-04-06 Skapad: 2009-04-06 Senast uppdaterad: 2011-03-29Bibliografiskt granskad
Ingår i avhandling
1. Contributions to the Theory of Measures of Association for Ordinal Variables
Öppna denna publikation i ny flik eller fönster >>Contributions to the Theory of Measures of Association for Ordinal Variables
2009 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

In this thesis, we consider measures of association for ordinal variables from a theoretical perspective. In particular, we study the phi-coefficient, the tetrachoric correlation coefficient and the polychoric correlation coefficient. We also introduce a new measure of association for ordinal variables, the empirical polychoric correlation coefficient, which has better theoretical properties than the polychoric correlation coefficient, including greatly enhanced robustness.

In the first article, entitled ``On the relation between the phi-coefficient and the tetrachoric correlation coefficient'', we show that under given marginal probabilities there exists a continuous bijection between the two measures of association. Furthermore, we show that the bijection has a fixed point at zero for all marginal probabilities. Consequently, the choice of which of these measures of association to use is for all practical purposes a matter of preference only.

In the second article, entitled ``A generalized definition of the tetrachoric correlation coefficient'', we generalize the tetrachoric correlation coefficient so that a large class of parametric families of bivariate distributions can be assumed as underlying distributions. We also provide a necessary and sufficient condition for the generalized tetrachoric correlation coefficient to be well defined for a given parametric family of bivariate distributions. With examples, we illustrate the effects on the polychoric correlation coefficient of different distributional assumptions.

In the third article, entitled ``A generalized definition of the polychoric correlation coefficient'', we generalize the polychoric correlation coefficient to a large class of parametric families of bivariate distributions, and show that the generalized and the conventional polychoric correlation coefficients agree on the family of bivariate normal distributions. With examples, we illustrate the effects of different distributional assumptions on the polychoric correlation coefficient. In combination with goodness-of-fit p-values, the association analysis can be enriched with a consideration of possible tail dependence.

In the fourth article, we propose a new measure of association for ordinal variables, named the empirical polychoric correlation coefficient. The empirical polychoric correlation coefficient relaxes the fundamental assumption of the polychoric correlation coefficient so that an underlying joint distribution is only assumed to exist, not to be of a particular parametric family. We also provide an asymptotical result, by which the empirical polychoric correlation coefficient converges almost surely to the true polychoric correlation under very general conditions. Thus, the proposed empirical polychoric correlation coefficient has better theoretical properties than the polychoric correlation coefficient.

Ort, förlag, år, upplaga, sidor
Uppsala: Acta Universitatis Upsaliensis, 2009. s. 32
Serie
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Social Sciences, ISSN 1652-9030 ; 50
Serie
Nationell ämneskategori
Sannolikhetsteori och statistik
Forskningsämne
statistik
Identifikatorer
urn:nbn:se:uu:diva-100735 (URN)978-91-554-7498-0 (ISBN)
Disputation
2009-05-15, Sal IV, Universitetshuset, 753 12, Uppsala, 10:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2009-04-24 Skapad: 2009-04-06 Senast uppdaterad: 2009-04-24Bibliografiskt granskad

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