Three-way association measure decompositions: The Delta index

In this paper we describe a general framework to decompose three-way association measures for contingency tables, in particular symmetric and non-symmetric measures will be discussed, like Pearson's index, Marcotorchino's index, whose special case is the Gray-Williams index, and a new non-symmetric one will be proposed, called the Delta index. After showing the orthogonal decomposition of these indices, practical examples illustrating the different decompositions will be given.