  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.
