A general framework for implementing distances for categorical variables

The degree to which subjects differ from each other with respect to certain properties measured by a set of variables, plays an important role in many statistical methods. For example, classification, clustering, and data visualization methods all require a quantification of differences in the observed values. We can refer to the quantification of such differences, as distance. An appropriate definition of a distance depends on the nature of the data and the problem at hand. For distances between numerical variables, there exist many definitions that depend on the size of the observed differences. For categorical data, the definition of a distance is more complex, as there is no straightforward quantification of the size of the observed differences. Consequently, many proposals exist that can be used to measure differences based on categorical variables. In this paper, we introduce a general framework that allows for an efficient and transparent implementation of distances between observations on categorical variables. We show that several existing distances can be incorporated into the framework. Moreover, our framework quite naturally leads to the introduction of new distance formulations and allows for the implementation of flexible, case and data specific distance definitions. Furthermore, in a supervised classification setting, the framework can be used to construct distances that incorporate the association between the response and predictor variables and hence improve the performance of distance-based classifiers.