The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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Approximating Exact Test of Mutual Independence in Multiway Contingency Tables via Stochastic Approximation Monte Carlo

  • Received : 2012.08.16
  • Accepted : 2012.10.09
  • Published : 2012.10.31
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Abstract

Monte Carlo methods have been used in exact inference for contingency tables for a long time; however, they suffer from ergodicity and the ability to achieve a desired proportion of valid tables. In this paper, we apply the stochastic approximation Monte Carlo(SAMC; Liang et al., 2007) algorithm, as an adaptive Markov chain Monte Carlo, to the exact test of mutual independence in a multiway contingency table. The performance of SAMC has been investigated on real datasets compared to with existing Markov chain Monte Carlo methods. The numerical results are in favor of the new method in terms of the quality of estimates.

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References

  1. Agresti, A. (1992). A survey of exact inference for contingency tables, Statistical Science, 7, 131-153. https://doi.org/10.1214/ss/1177011454
  2. Agresti, A. (1999). Exact inference for categorical data: Recent advances and continuing controversies, Statistics in Medicine, 18, 2191-2207. https://doi.org/10.1002/(SICI)1097-0258(19990915/30)18:17/18<2191::AID-SIM249>3.0.CO;2-M
  3. Agresti, A. (2002). Categorical Data Analysis, 2nd edition, Wiley.
  4. Andrieu, C., Moulines, E. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions, SIAM Journal on Control and Optimization, 44, 283-312. https://doi.org/10.1137/S0363012902417267
  5. Beh, E. J. and Davy, P. J. (1997). Multiple correspondence analysis of ordinal multi-way contingency tables using orthogonal polynomials, In preparation.
  6. Beh, E. J. and Davy, P. J. (1998). Partitioning Pearson's chi-squared statistic for a completely ordered three-way contingency table, Australian and New Zealand Journal of Statistics, 40, 465-477. https://doi.org/10.1111/1467-842X.00050
  7. Booth, J. G. and Butler, R. W. (1999). An importance sampling algorithm for exact conditional test in log-linear models, Biometrika, 86, 321-332. https://doi.org/10.1093/biomet/86.2.321
  8. Caffo, B. S. and Booth, J. G. (2001). A Markov chain Monte Carlo algorithm for approximating exact conditional probabilities, Journal of Computational and Graphical Statistics, 10, 730-745. https://doi.org/10.1198/106186001317243421
  9. Chen, H. F. (2002). Stochastic Approximation and Its Applications, Kluwer Academic Publishers, Dordrecht.
  10. Deloera, J. A. and Onn, S. (2006). Markov basis of three-way tables are arbitrarily complicated, Journal of Symbolic Computation, 41, 173-181. https://doi.org/10.1016/j.jsc.2005.04.010
  11. Diaconis, P. and Sturmfels, B. (1998). Algebraic algorithms for sampling from conditional distributions, The Annals of Statistics, 26, 363-397. https://doi.org/10.1214/aos/1030563990
  12. Dobra, A. (2003). Markov bases for decomposable graphical models, Bernoulli, 9, 1093-1108. https://doi.org/10.3150/bj/1072215202
  13. Gastwirth, J. L. (1988). Statistical Reasoning in Law and Public Policy 1, Academic, San Diego.
  14. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 97-109. https://doi.org/10.1093/biomet/57.1.97
  15. Liang, F. (2009). On the use of stochastic approximation Monte Carlo for Monte Carlo integration, Statistics & Probability Letters, 79, 581-587. https://doi.org/10.1016/j.spl.2008.10.007
  16. Liang, F., Liu, C. and Carroll, R. (2007). Stochastic approximation in Monte Carlo computation, Journal of American Statistical Association, 102, 477, 305-320. https://doi.org/10.1198/016214506000001202
  17. McCullagh, P. (1986). The conditional distribution of goodness-of-fit statistics for discrete data, Journal of the American Statistical Association, 81, 104-107. https://doi.org/10.1080/01621459.1986.10478244
  18. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines, Journal of Chemical Physics, 21, 1087-1091. https://doi.org/10.1063/1.1699114
  19. Paul, S. and Deng, D. (2000). Goodness of fit of generalized linear models to sparse data, Journal of the Royal Statistical Society, Series B, 62, 323-333. https://doi.org/10.1111/1467-9868.00234
  20. Robbins, H. and Monro, S. (1951). A stochastic approximation method, Annals of Mathematical Statistics, 22, 400-407. https://doi.org/10.1214/aoms/1177729586
  21. Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms, Biometrika, 83, 95-110. https://doi.org/10.1093/biomet/83.1.95