arXiv subjects

Sadao Tomizawa

Publications and source records attributed to Sadao Tomizawa.

4 recordsLinked to original sources

arXiv2026

Association measures for two-way contingency tables based on multi-categorical proportional reduction in error

In two-way contingency tables under an asymmetric situation, where the row and column variables are defined as explanatory and response variables, respectively, quantifying the extent to which the explanatory variable contributes to predicting the response variable is important. One quantification method is the association measure, which indicates the degree of association in a range from $0$ to $1$. Among various measures that have been proposed, those based on proportional reduction in error (PRE) are particularly notable for their simplicity and intuitive interpretation. These measures, including Goodman-Kruskal's lambda proposed in 1954, are widely implemented in statistical software such as R and SAS and remain extensively used. However, a well-known limitation of PRE measures is their potential to return a value of $0$ despite no independence. This issue arises because the measures are constructed based solely on the maximum joint and marginal probabilities, failing to make full use of the information available in the contingency table. To address this problem, we propose an extension of PRE measures designed for the proportional reduction in error with multiple categories. The properties of the proposed measures are examined, and their utility is demonstrated through numerical experiments. The results suggest their potential as practical tools in applied statistics.

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arXiv2023

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

In contingency table analysis, one is interested in testing whether a model of interest (e.g., the independent or symmetry model) holds using goodness-of-fit tests. When the null hypothesis where the model is true is rejected, the interest turns to the degree to which the probability structure of the contingency table deviates from the model. Many indexes have been studied to measure the degree of the departure, such as the Yule coefficient and Cramér coefficient for the independence model, and Tomizawa's symmetry index for the symmetry model. The inference of these indexes is performed using sample proportions, which are estimates of cell probabilities, but it is well-known that the bias and mean square error (MSE) values become large without a sufficient number of samples. To address the problem, this study proposes a new estimator for indexes using Bayesian estimators of cell probabilities. Assuming the Dirichlet distribution for the prior of cell probabilities, we asymptotically evaluate the value of MSE when plugging the posterior means of cell probabilities into the index, and propose an estimator of 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 of correcting estimation accuracy. We also show that the values of bias and MSE are smaller than those obtained by using the uniform and Jeffreys priors.

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arXiv2023

Generalized Cramér's coefficient via $f$-divergence for contingency tables

Various measures in two-way contingency table analysis have been proposed to express the strength of association between row and column variables in contingency tables. Tomizawa et al. (2004) proposed more general measures, including Cramér's coefficient, using the power-divergence. In this paper, we propose measures using the $f$-divergence that has a wider class than the power-divergence. Unlike statistical hypothesis tests, these measures provide quantification of the association structure in contingency tables. The contribution of our study is proving that a measure applying a function that satisfies the condition of the $f$-divergence has desirable properties for measuring the strength of association in contingency tables. With this contribution, we can easily construct a new measure using a divergence that has essential properties for the analyst. For example, we conducted numerical experiments with a measure applying the $θ$-divergence. Furthermore, we can give further interpretation of the association between the row and column variables in the contingency table, which could not be obtained with the conventional one. We also show a relationship between our proposed measures and the correlation coefficient in the bivariate normal distribution of latent variables in the contingency tables.

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arXiv2023

Geometric Mean Type of Proportional Reduction in Variation Measure for Two-Way Contingency Tables

In a two-way contingency table analysis with explanatory and response variables, the analyst is interested in the independence of the two variables. However, if the test of independence does not show independence or clearly shows a relationship, the analyst is interested in the degree of their association. Various measures have been proposed to calculate the degree of their association, one of which is the proportional reduction in variation (PRV) measure which describes the PRV from the marginal distribution to the conditional distribution of the response. The conventional PRV measures can assess the association of the entire contingency table, but they can not accurately assess the association for each explanatory variable. In this paper, we propose a geometric mean type of PRV (geoPRV) measure that aims to sensitively capture the association of each explanatory variable to the response variable by using a geometric mean, and it enables analysis without underestimation when there is partial bias in cells of the contingency table. Furthermore, the geoPRV measure is constructed by using any functions that satisfy specific conditions, which has application advantages and makes it possible to express conventional PRV measures as geometric mean types in special cases.

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