COMPOSITIONAL ANALYSIS OF CONTINGENCY TABLES
by
J. J. Egozcue and V. Pawlowsky-Glahn
Contingency tables contain in each cell counts of the corresponding events. In a
multinomial sampling scenario, the join distribution of counts in the cells can be
parametrised by a table of probabilities, which are a joint probability function of two
categorical variables. These probabilities can be assumed to be a composition. The
goal of the analysis is to decompose the probability table into an independent table
and an interaction table. The optimal independent table, in the sense of the Aitchison
geometry of the simplex, has been shown to be the product of geometric marginals,
better than the traditional arithmetic marginals. The decomposition is unique and the
independent part is an orthogonal projection of the probability table onto the
subspace of independent tables. Interaction table is analysed using its clrrepresentation which is directly related to cell interactions. A summary measure of
dependence is the simplicial deviance (the square Aitchison norm of the interaction
table). An example is presented.