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

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.