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J.J. Egozcue, V. Pawlowsky-Glahn, M. Templ, K. Hron:
"Independence in Contingency Tables Using Simplical Geometry";
Communications in Statistics - Theory and Methods, 44 (2015), 18; 3978 - 3996.

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.

http://dx.doi.org/10.1080/03610926.2013.824980

http://www.tandfonline.com/doi/full/10.1080/03610926.2013.824980