A conditional Bayesian approach for testing independence in two-way contingency tables


  • Z. SABERI Department of Statistics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran
  • M. GANJALI Department of Statistics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran


Bayes Factor, bayesian P-value, hierarchical Bayes, important sampling, noncentral hypergeometric distribution


Bayesian methods for exact small-sample analysis with categorical data in   contingency tables are considered. Point null hypotheses versus two-sided hypothesis are tested concerning log odds ratios in these tables with fixed row margins. The conditional distribution of the sufficient statistics for interesting parameters conditional on sufficient statistics of other nuisance parameters in the model is obtained and used to eliminate the effect of nuisance parameters. This distribution is Fisher's multivariate noncentral hypergeometric distribution. Three Bayesian approaches, hierarchical Bayes, empirical Bayes, and noninformative Bayes are considered and compared by simulation studies.


Agresti, A. 2002 . Categorical Data Analysis, Wiley, New York.

Agresti, A. & Hitchcock, D. B. 2005. Bayesian inference for categorical data analysis, Statistical Methods & Applications, 14 : 297--330.

Altham, P. M. E. 1969 . Exact Bayesian analysis of a 2 <$>times <$> 2 contingency table, and Fisher's exact significance test. Journal of the Royal Statistical Society, Series B., Methodological, 31 : 261-269.

Altham, P. M. E. 1971 . The analysis of matched proportions. Biometrika 58 : 561-576.

Casella, G. & Moreno, E. 2009. Assessing robustness of intrinsic tests of independence in two-way contingency tables. JASA, 104 (487): 1261-1271.

Cochran, W. G. 1954. Some methods of strengthening common <$>chi ^{2} <$> tests. Biometrics, 10 : 417-451.

Cressie, N. & Read, T. R. C. 1984 . Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B, 46 : 440-464.

Graubard, B. I., & Korn, E. I. 1987 . Choice of column scores for testing independence in ordered <$>2times k<$> tables. Biometrics, 43 : 471-476.

Jeffreys, H. 1961 . Theory of Probability. Oxford University Press.

Kass, R. & Raftery, A. 1995 . Bayes factors. Journal of the American Statistical Association, 90 :773-795.

Leonard, T. 1972. Bayesian methods for binomial data. Biometrika, 59 : 581-589.

McCullagh, P. & Nelder, J. A. 1989. Generalized Linear Models, 2nd edition. London: Chapman & Hall.

Pardo, L. 2006. Statistical Inference Based on Divergence Measures. Chapman & Hall/CRC.

Robert, C.P. & Casella, G. 2004. Monte Carlo Statistical Methods, 2nd edition, New York: Springer-Verlag.

Srinivasan, R. 2002. Importance sampling - Applications in communications and detection, Springer-Verlag, Berlin.