The most popular ways to test for independence of two ordinal random variables are by means of Kendall's tau and Spearman's rho. However, such tests are not consistent, only having power for alternatives with "monotonic" association. In this paper, we introduce a natural extension of Kendall's tau, called τ * , which is non-negative and zero if and only if independence holds, thus leading to a consistent independence test. Furthermore, normalization gives a rank correlation which can be used as a measure of dependence, taking values between zero and one. A comparison with alternative measures of dependence for ordinal random variables is given, and it is shown that, in a well-defined sense, τ * is the simplest, similarly to Kendall's tau being the simplest of ordinal measures of monotone association. Simulation studies show our test compares well with the alternatives in terms of average p-values.

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Polytomous latent scales for the investigation of the ordering of itemsLigtvoet, R.; van der Ark, L.A.; Bergsma, W.P.; Sijtsma, K. Published in: Psychometrika Document version:Publisher's PDF, also known as Version of record DOI:10.1007/s11336-010-9199-8 General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. We propose three latent scales within the framework of nonparametric item response theory for polytomously scored items. Latent scales are models that imply an invariant item ordering, meaning that the order of the items is the same for each measurement value on the latent scale. This ordering property may be important in, for example, intelligence testing and person-fit analysis. We derive observable properties of the three latent scales that can each be used to investigate in real data whether the particular model adequately describes the data. We also propose a methodology for analyzing test data in an effort to find support for a latent scale, and we use two real-data examples to illustrate the practical use of this methodology.

Models defined by a set of conditional independence restrictions play an important role in statistical theory and applications, especially, but not only, in graphical modeling. In this paper we identify a subclass of these consisting of hierarchical marginal log-linear models, as defined by Bergsma and Rudas (2002a). Such models are smooth, which implies the applicability of standard asymptotic theory and simplifies interpretation. Furthermore, we give a marginal loglinear parameterization and a minimal specification of the models in the subclass, which implies the applicability of standard methods to compute maximum likelihood estimates and simplifies the calculation of the degrees of freedom of chi-squared statistics to test goodness-offit. The utility of the results is illustrated by applying them to certain block-recursive Markov models associated with chain graphs.