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In contingency table analysis, one is interested in testing whether a modelof interest (e.g., the independent or symmetry model) holds usinggoodness-of-fit tests. When the null hypothesis where the model is true isrejected, the interest turns to the degree to which the probability structureof the contingency table deviates from the model. Many indexes have beenstudied to measure the degree of the departure, such as the Yule coefficientand Cram\'er coefficient for the independence model, and Tomizawa's symmetryindex for the symmetry model. The inference of these indexes is performed usingsample proportions, which are estimates of cell probabilities, but it iswell-known that the bias and mean square error (MSE) values become largewithout a sufficient number of samples. To address the problem, this studyproposes a new estimator for indexes using Bayesian estimators of cellprobabilities. Assuming the Dirichlet distribution for the prior of cellprobabilities, we asymptotically evaluate the value of MSE when plugging theposterior means of cell probabilities into the index, and propose an estimatorof the index using the Dirichlet hyperparameter that minimizes the value.Numerical experiments show that when the number of samples per cell is small,the proposed method has smaller values of bias and MSE than other methods ofcorrecting estimation accuracy. We also show that the values of bias and MSEare smaller than those obtained by using the uniform and Jeffreys priors.

Improving the accuracy of estimating indexes in contingency tables using Bayesian estimators