How Ordinal Is Ordinal in Ordinal Data Analysis?
Vahid Nassiri1, Jacques Tacq2
1Mathematics, Vrije Universiteit Brussel (VUB), Brussels, Belgium; 2Hogeschool-Universiteit Brussel (HUB), Brussels, Belgium

“To measure is to know”, Lord Kelvin said. In general, characteristics are measured on a quantitative scale (interval or ratio), on an ordinal scale or on a nominal scale. Among all of them ordinal scale plays an important role in humanities, specially in social sciences. There are many different methods developed in order to face with modeling ordinal data, like Kruskal-Wallis analysis of variance for ranks (Kruskal and Wallis (1952)), isotonic regression analysis (Barlow et al. (1972)), LISREL for ordinal data (Jöreskog (2005)), log-linear models for ordinal data (Goodman (1979)), polytomous Mokken scale analysis (Mokken (1971)), multidimensional unfolding techniques (MUDFOLD), Coombs (1964), the Gifi system for nonlinear multivariate analysis (Gifi (1990)), Spearman's-ρ, Kendall's-τ (Kendall (1962)), Somers'-D, ordinal path analysis (Smith (1974)) and many other measures and methods.

There are three main approaches in dealing with ordinal data. The first approach is lowering the scale and treat ordinal data as nominal, which is usually done by dummy coding, the second approach is using some appropriate re-scaling methods, and then treat ordinal data as if they are quantitative. The third approach is treating ordinal data as truly ordinal. While the third approach seems to be the most logical one, just few authors have used this approach, among them Kendall and Somers.

In this article we try to review many of the methods dealing with ordinal data analysis and study their weaknesses and strengths and show that few if any methods and measures like Kendall's-τ and Somers'-D which are respectful to the ordinality of ordinal data. In Section 1 we may discuss the importance of ordinal data, Section 2 will be dedicated to the review of several ordinal data analysis methods and measures. In this section we will show how most of the ordinal data analysis methods are in agreement with the first or second approach. The pros and cons of methods which follow the third approach is given in Section 3. Finally, the paper is concluded in Section 4.

References:

[1] Barlow, Bartholomew, Bremner and Brunk, Statistical Inference Under Oreder Restrections: The Theory and Application of Isotonic Regression, Wiley, 1972

[2] Coombs, A Theory of Data, Wiley, 1964

[3] Gifi, Nonlinear Multivariate Analysis, Wiley, 1990

[4] Goodman, 'Simple models for the analysis of association in cross-classifications having ordered categories', JASA, 76, 1979, pp. 537- 552

[5] Jöreskog, 'Structural Equation Modeling with ordinal variables using LISREL', 2005,http://www.ssicentral.com/lisrel/techdocs.ordinal.pdf

[6] Kendall, Rank Correlation Methods, Hafner, 1962

[7] Kruskal and Wallis, 'Use of ranks in non-criterion variance analysis', JASA, 47, 1952, pp. 583- 621

[8] Mokken, A Theory and Procedure of Scale Analysis, De Gruyter, 1971

[9] Smith, 'Continuities in ordinal path analysis, Social Forces, 53, 1974, pp. 200 - 253.

Keywords: Ordinal data analysis; Ordinal scale; Kendall's tau; rescaling

Biography: Jacques Tacq studied mathematics and sociology and defended his PhD-Thesis at Catholic University of Leuven in Belgium on a project about causality in philosophy and in sociological research. He is currently Professor of Sociology, Catholic University of Brussels. He was Visiting Professor at the Faculty of Social Sciences, Rotterdam, where he was co-ordinator of the training-programme for PhD-students and where he lectured a course on “Interdisciplinarity”. He is lecturer in the Advanced-Master in 'Quantitative Analysis in the Social Sciences' (QASS) at Catholic University of Brussels. He is also Visiting Professor in the Summer School in Social Science Data Analysis and collection at University of Essex.

He has publications in methodology of social science, in the philosophical sense (causality, theory formation) as well as in the statistical-technical sense (cluster analysis, multilevel analysis, loglinear modelling) and with both a quantitative and qualitative approach.