Title: Bayesian estimation of independence and compositional interaction in a two-way classification
Authors: Mi Ortego - Universitat Politecnica de Catalunya (Spain) [presenting]
Juan Jose Egozcue - Universitat Politecnica de Catalunya (Barcelona, Spain) (Spain)
Abstract: A two-way discrete classification is characterized by a table of probabilities. This table of probabilities can be assumed to be the parameters of a multinomial sampling. This two-way probability table can be interpreted as a composition in the simplex, and therefore the Aitchison geometry is a suitable structure for its analysis. In this geometry, the nearest independent table to a probability table is the one built from the geometric marginals. The compositional difference between the original and the independent table is called interaction table, which encloses the information about the dependence between the two classifications. The independent and the interaction table are an orthogonal decomposition of the table of probabilities and can be represented using the centered-logratio transformations of both tables. The square Aitchison norm of the interaction table is called simplicial deviance and it is a dependence measure. Starting from a contingency table under a multinomial sampling, a Bayesian procedure is proposed in order to obtain the orthogonal decomposition into its independent and interaction tables. Interaction tables can be simplified in order to improve their interpretation. A Bayesian assessment of independence is used to this end.