Model selection for contingency tables with algebraic statistics

Asymmetry models for square contingency tables: exact tests via algebraic statistics

“…Conducting algebraic tests for each model yields a large computational effort. Krampe and Kuhnt (2009) discuss how to reduce this effort in the case of graphical models. The general results can be extended to the analysis of symmetry structures, for completeness, we include the quasi symmetry model treated in the algebraic framework by Rapallo (2005).…”

“…Normally, 'exact' simulated p-values using the Diaconis-Sturmfels procedure require the simulation of an individual Markov chain for each model of interest as well as a separate generation of the Gröbner basis. Following the concept of Krampe and Kuhnt (2009), we showed that it suffices to generate a Markov chain according to the complete symmetry model. For all other considered models, 'exact' simulated p-values can then be calculated by selecting those data sets from the Markov chain with the same value of the sufficient statistic as the observed data set.…”

“…Krampe and Kuhnt (2009) show how the inherent hierarchy of graphical models can be used to reduce the computational effort and let the simulation study be based on a single Markov chain generation for all models. In this paper we show how the same idea can be followed up for symmetry models since they exhibit a hierarchy as well.…”

“…Model DS g 1 : x 24 x 31 − x 13 x 42 g 2 : x 32 x 34 − x 23 x 43 g 3 : x 21 x 34 − x 12 x 43 g 4 : x 21 x 23 − x 12 x 32 Model OQS g 1 : x 12 x 32 − x 21 x 23 ; g 2 : x 2 23 x 31 − x 13 x 2 32 ; g 3 : x 12 x 23 x 31 − x 13 x 21 x 32 ; g 4 : x 3 34 x 41 − x14 x 3 43 ; g 5 : x 23 x 41 x 2 34 − x 14 x 32 x 2 43 ; g 6 : x 12 x 23 x 34 x 41 − x 14 x 21 x 32 x 43 ; g 7 : x 2 12 x 34 x 41 − x 14 x 2 21 x 43 ; g 8 : ) S ⊂ OQS ⊂ DS: If δ i−j = δ i−j model DS reduces to model OQS, if further α = 1 we get model S (iii) S ⊂ OQS ⊂ QS: If α 1 = · · · α I = α then model QS becomes model OQS and further model S if α = 1Statements (i) to (iii) now follow from Theorem 1.5.1,Krampe and Kuhnt (2009).…”

A short history of algebraic statistics

“…These include the Bowker test for matched pair data [41], rater agreement test [59], calibration of Fourier analysis of ranking data [23], in model selection [42], sequential sampling in multi-way contingency tables with given constraints [21], for quasi-independence [39] and weaken independence [18], to mention a few. The computation of Markov bases in special cases has received much attention e.g.…”