Bivariate Dependence Orderings for Unordered Categorical Variables

Interest in assessing the degree of association between random variables, meaning the strengh of their mutual dependence or the dependence of one on the others, has a long history in the statistical literature. Usually the interest lies in how to measure the association: several measures are possible, which reflect different goals we may have in mind for specific applications. Various measures of dependence for ordered or unordered categorical random variables are also possible. Bickel and Lehmann (1975) have shifted the emphasis to the investigation of concepts of dependence and ways of comparing it. In this chapter some results dealing with dependence concepts for bivariate categorical variables are recalled and extended within the framework of stochastic orderings of multivariate discrete r.v.'s. An axiomatic approach is adopted: starting with desirable properties of one given type of association, these properties are translated into a mathematical definition of the stochastic order of interest, with implications on the most common indicators in use in the statistical literature. One very special type of association is the amount of agreement among different observers that classify the same group of statistical units into unordered classes: in the medical field this has led to widespread use of a well-known indicator called Cohen's kappa. We shall attempt to give an axiomatic definition of agreement and show its formal properties. Some criticism of Cohen's kappa will ensue.