The Annals of Statistics

On the Application of Symmetric Dirichlet Distributions and Their Mixtures to Contingency Tables, Part II

J. F. Crook and I. J. Good

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Abstract

This paper is a continuation of a paper in the Annals of Statistics (1976), 4 1159-1189 where, among other things, a Bayesian approach to testing independence in contingency tables was developed. Our first purpose now, after allowing for an improvement in the previous theory (which also has repercussions on earlier work on the multinomial), is to give extensive numerical results for two-dimensional tables, both sparse and nonsparse. We deal with the statistics $X^2, \Lambda$ (the likelihood-ratio statistic), a slight transformation $G$ of the Type II likelihood ratio, and the number of repeats within cells. The latter has approximately a Poisson distribution for sparse tables. Some of the "asymptotic" distributions are surprisingly good down to exceedingly small tail-area probabilities, as in the previous "mixed Dirichlet" approach to multinomial distributions (J. Roy. Statist. Soc. B. 1967, 29 399-431; J. Amer. Statist. Assoc. 1974, 69 711-720). The approach leads to a quantitative measure of the amount of evidence concerning independence provided by the marginal totals, and this amount is found to be small when neither the row totals nor the column totals are very "rough" and the two sets of totals are not both very flat. For Model 3 (all margins fixed), the relationship is examined between the Bayes factor against independence and its tail-area probability.

Article information

Source
Ann. Statist., Volume 8, Number 6 (1980), 1198-1218.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345194

Digital Object Identifier
doi:10.1214/aos/1176345194

Mathematical Reviews number (MathSciNet)
MR594638

Zentralblatt MATH identifier
0463.62052

JSTOR
links.jstor.org

Subjects
Primary: 62F15: Bayesian inference
Secondary: 62G10: Hypothesis testing 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

Keywords
Contingency tables multidimensional contingency tables mixtures of Dirichlet distributions Bayes factor Bayes/non-Bayes synthesis enumeration of arrays independence information in marginas total type II likelihood ratio number of repeats in a contingency table Fisher's "exact test" multinomial significance log-Cauchy hyperprior hierarchical Bayes

Citation

Crook, J. F.; Good, I. J. On the Application of Symmetric Dirichlet Distributions and Their Mixtures to Contingency Tables, Part II. Ann. Statist. 8 (1980), no. 6, 1198--1218. doi:10.1214/aos/1176345194. https://projecteuclid.org/euclid.aos/1176345194


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See also

  • Part I: I. J. Good. On the Application of Symmetric Dirichlet Distributions and their Mixtures to Contingency Tables. Ann. Statist., Volume 4, Number 6 (1976), 1159--1189.

Corrections

  • See Correction: J. F. Crook, I. J. Good. Corrections to "On the Application of Symmetric Dirichlet Distributions and Their Mixtures to Contingency Tables, Part II". Ann. Statist., Volume 9, Number 5 (1981), 1133--1133.