Bayesian estimation of the orthogonal decomposition of a contingency table

M.I. Ortego, J. J. Egozcue

Abstract


In a multinomial sampling, contingency tables can be parametrized by probabilities of each cell. These probabilities constitute the joint probability function of two or more discrete random variables. These probability tables have been previously studied from a compositional point of view. The compositional analysis of probability tables ensures coherence when analysing sub-tables. The main results are:
(1) given a probability table, the closest independent probability table is the product of their geometric marginals;
(2) the probability table can be orthogonally decomposed into an independent table and an interaction table;
(3) the departure of independence can be measured using simplicial deviance, which is the Aitchison square norm of the interaction table.

In previous works, the analysis has been performed from a frequentist point of view. This contribution is aimed at providing a Bayesian assessment of the decomposition. The resulting model is a log-linear one, which parameters are the centered log-ratio transformations of the geometric marginals and the interaction table.
Using a Dirichlet prior distribution of multinomial probabilities, the posterior distribution of multinomial probabilities is again a Dirichlet distribution. Simulation of this posterior allows to study the distribution of marginal and interaction parameters, checking the independence of the observed contingency table and cell interactions.

The results corresponding to a two-way contingency table example are presented.


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References


Aitchison J (1986). The Statistical Analysis of Compositional Data. Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). (Reprinted in 2003 with

additional material by The Blackburn Press). ISBN 0-412-28060-4. 416 p.

Bayarri MJ, Berger JO (2000). P-values for composite null models. Journal of the American Statistical Association, 95, 1127-1142.

Chambers RL, Welsh AH (1993). Log-linear Models for Survey Data with Non-ignorable Non-response. J. R. Statist. Soc. B, 55(1), 157-170.

Darroch JN, Lauritzen SL, Speed TP (1980). Markov fields and Log-linear interaction models for contingency tables. The Annals of Statistics, 8(3), 522-539.

Daunis-i Estadella J, Martín-Fernández JA (eds.) (2008). Proceedings of CODAWORK'08, The 3rd Compositional Data Analysis Workshop. University of Girona, Girona. ISBN 978-

-8458-272-4.

Egozcue JJ (2009). Reply to "On the Harker variation diagrams; ..." by J. A. Cortés." Mathematical Geosciences, 41(7), 829-834.

Egozcue JJ, Díaz-Barrero JL, Pawlowsky-Glahn V (2008). Compositional analysis of bivariate discrete probabilities. In Daunis-i Estadella and Martín-Fernández (2008), pp. 1-11.

ISBN 978-84-8458-272-4, URL http://hdl.handle.net/10256/717.

Egozcue JJ, Pawlowsky-Glahn V, Templ M, Hron K (2015). Independence in contingency tables using simplicial geometry. Communications in Statistics- Theory and methods, 44(18), 3978-3996. doi:10.1080/03610926.2013.824980.

Egozcue JJ, Tolosana-Delgado R, Ortego MI (eds.) (2011). Proceedings of the 4th International Workshop on Compositional Data Analysis. Departament de Matemàtica Aplicada III (Universitat Politècnica de Catalunya), CIMNE, Barcelona, Spain. ISBN 978-84-87867- 76-7, URL http://congress.cimne.com/codawork11.

Everitt BS (1977). The Analysis of Contingency Tables. John Wiley & Sons, Inc. New York, New York, USA. ISBN 0-470-71135-3.

Fácevicová K, Hron K (2013). Statistical analysis of compositional 2 X 2 tables. In Hron, Filzmoser, and Templ (2013). ISBN 978-3-200-03103-6.

Gallo M (2011). Tree-way compositional data analysis. In Egozcue, Tolosana-Delgado, and Ortego (2011), pp. 1-13. ISBN 978-84-87867-76-7, URL http://congress.cimne.com/

codawork11.

Gelman A, Meng XL, Stern H (1996). Posterior predictive assessment of model fitness via realized discrepancies (with discussion). Statistica Sinica, 6, 733-807.

Goodman LA (1996). A Single General Method for the Analysis of Cross-Classified Data: Reconciliation and Synthesis of Some Methods of Pearson, Yule, and Fisher, and Also Some Methods of Correspondence Analysis and Association Analysis. Journal of the American

Statistical Association, 91(433), 408-428.

Hron K, Filzmoser P, Templ M (eds.) (2013). Proceedings of the 5th International Workshop on Compositional Data Analysis. TU Wien, TU Wien. ISBN 978-3-200-03103-6.

Kenett RS (1983). On an Exploratory Analysis of Contingency Tables. The Statistician, 32(3), 395-403.

McCullagh P, Nelder JA (1983). Generalized Linear Models. Chapman and Hall, London, UK. 522 p.

Meng XL (1994). Posterior predictive p-values. Annals of Statistics, 22, 1142-1160.

Nelder JA (1974). Log linear models for contingency tables: a generalization of classical least squares. Appl. Statist., 23, 323-329.

Nelder JA, Wedderburn RWM (1972). Generalized linear models. Journal of the Royal Statistical Society, series A, 135, 370-384.

Ortego MI (2015). Estimación Bayesiana de cópulas extremales en procesos de Poisson. Ph.D. thesis, Universitat Politécnica de Catalunya.

Pawlowsky-Glahn V, Egozcue JJ (2001). Geometric approach to statistical analysis on the simplex. Stochastic Environmental Research and Risk Assessment (SERRA), 15(5), 384-398.

Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015). Modeling and analysis of compositional data. Statistics in practice. John Wiley & Sons, Chichester UK. ISBN

272 pp.




DOI: http://dx.doi.org/10.17713/ajs.v45i4.136

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@Matthias Templ (using Open Journal Systems) -- see previous editions at http://www.stat.tugraz.at/AJS/Editions.html