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Contributions to the Theory of Measures of Association for Ordinal Variables
Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Information Science, Statistics.
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
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.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis , 2009. , p. 32
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Social Sciences, ISSN 1652-9030 ; 50
Series
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
URN: urn:nbn:se:uu:diva-100735ISBN: 978-91-554-7498-0 (print)OAI: oai:DiVA.org:uu-100735DiVA, id: diva2:210896
Public defence
2009-05-15, Sal IV, Universitetshuset, 753 12, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2009-04-24 Created: 2009-04-06 Last updated: 2009-04-24Bibliographically approved
List of papers
1. On the relation between the phi-coefficient and the tetrachoric correlation coefficient
Open this publication in new window or tab >>On the relation between the phi-coefficient and the tetrachoric correlation coefficient
(English)Manuscript (Other academic)
Abstract [en]

We show existence of a continuous bijection between the tetrachoric correlation coefficient and the phi-coefficient under given marginal probabilities. Implications are that the tetrachoric correlation coefficient can be calculated using the assumptions of the phi-coefficient construction, and the phi-coefficient can be calculated using the assumptions of the tetrachoric correlation construction. As a consequence, whether to use the phi-coefficient or the tetrachoric correlation coefficient is a matter of preference only. The result can also be used to construct a numerical table of tetrachoric correlation coefficients, converted from the marginal probabilities and the phi-coefficient, which is easy to calculate by hand. Moreover, a mathematically rigorous definition of the tetrachoric correlation coefficient is provided, along with a proof that the coefficient is well defined.

Keywords
phi-coefficient, tetrachoric correlation coefficient, 2 x 2 contingency table, measure of association, dichotomous variables
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-100692 (URN)
Available from: 2009-04-06 Created: 2009-04-06 Last updated: 2011-03-29Bibliographically approved
2. A generalized definition of the tetrachoric correlation coefficient
Open this publication in new window or tab >>A generalized definition of the tetrachoric correlation coefficient
(English)Manuscript (Other academic)
Abstract [en]

We generalize the tetrachoric correlation coefficient to a large class of parametric families of bivariate distributions. We also show that the generalized definition agrees with the conventional definition on the family of bivariate normal distributions. Furthermore, we provide a necessary and sufficient condition for the generalized tetrachoric correlation coefficient to be well defined for a given family of distributions, and some sufficient criteria which can be useful for practical purposes. Moreover, we illustrate with examples how the distributional assumption can have a profound impact on the conclusions of the association analysis. Using S&P 100 stock data, we exemplify the fact that a correct distributional assumption is vitally important for the analysis. Consequently, it is concluded that the tetrachoric correlation coefficient is not robust to changes of the distributional assumption.

Keywords
tetrachoric correlation, generalization, 2 x 2 contingency table, dichotomous variables, measure of association, robustness
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-100694 (URN)
Available from: 2009-04-06 Created: 2009-04-06 Last updated: 2011-03-29Bibliographically approved
3. A generalized definition of the polychoric correlation coefficient
Open this publication in new window or tab >>A generalized definition of the polychoric correlation coefficient
(English)Manuscript (Other academic)
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.

Keywords
polychoric correlation, generalization, contingency table, ordinal variables, measure of association, robustness, goodness-of-fit
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-100695 (URN)
Available from: 2009-04-06 Created: 2009-04-06 Last updated: 2011-03-29Bibliographically approved
4. An empirical polychoric correlation coefficient
Open this publication in new window or tab >>An empirical polychoric correlation coefficient
(English)Manuscript (Other academic)
Abstract [en]

We propose a new measure of association for ordinal variables. The new measure of association, named the empirical polychoric correlation coefficient, builds upon the polychoric correlation coefficient, but relaxes its fundamental assumption so that an underlying distribution is only assumed to exist, not to be of a particular parametric family. The empirical polychoric correlation coefficient has properties that are superior to those of the polychoric correlation coefficient; it rests on weaker assumptions, it is well defined for every contingency table, and it converges almost surely to the correct theoretical polychoric correlation. Consequentially, the empirical polychoric correlation coefficient is theoretically robust to changes of the distributional assumption, unlike the polychoric correlation coefficient. A simulation study confirms that the empirical coefficient is considerably more robust than the polychoric correlation coefficient, and it also indicates that it has lower standard deviation.

Keywords
empirical polychoric correlation coefficient, polychoric correlation, contingency table, ordinal variables, measure of association, robust statistics
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-100696 (URN)
Available from: 2009-04-06 Created: 2009-04-06 Last updated: 2011-03-29Bibliographically approved

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