arXiv subjects

Wataru Urasaki

Publications and source records attributed to Wataru Urasaki.

6 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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arXiv2026

A family of divergence-based correlation measures for contingency tables under bivariate normality

We propose a family of association measures for two-way contingency tables whose latent distribution can be assumed to be bivariate normal. When this assumption holds, the power-divergence measuring departure from independence can be approximated in closed form as a function of the latent correlation coefficient. By inverting this relationship, we obtain a family of measures $ρ_{(λ)}$, indexed by a scalar parameter $-1 \leq λ\leq 1$, that directly approximates the latent correlation. Special cases include the informational measure of correlation proposed by Linfoot (1957) at $λ= 0$ and Pearson's contingency coefficient $C$ at $λ= 1$. Additionally, we derive asymptotic distributions via the delta method and construct two families of confidence intervals. Simulation studies confirm that the proposed measures approximate the true latent correlation more faithfully than conventional divergence-based measures, and that they successfully distinguish between weak and moderate associations where existing measures tend to give indistinguishable values. Compared with the polychoric correlation coefficient, the proposed measures are computed several thousand times faster and remain numerically stable even when the latent correlation is close to one.

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arXiv2024

A measure of departure from symmetry via the Fisher-Rao distance for contingency tables

A measure of asymmetry is a quantification method that allows for the comparison of categorical evaluations before and after treatment effects or among different target populations, irrespective of sample size. We focus on square contingency tables that summarize survey results between two time points or cohorts, represented by the same categorical variables. We propose a measure to evaluate the degree of departure from a symmetry model using cosine similarity. This proposal is based on the Fisher-Rao distance, allowing asymmetry to be interpreted as a geodesic distance between two distributions. Various measures of asymmetry have been proposed, but visualizing the relationship of these quantification methods on a two-dimensional plane demonstrates that the proposed measure provides the geometrically simplest and most natural quantification. Moreover, the visualized figure indicates that the proposed method for measuring departures from symmetry is less affected by very few cells with extreme asymmetry. A simulation study shows that for square contingency tables with an underlying asymmetry model, our method can directly extract and quantify only the asymmetric structure of the model, and can more sensitively detect departures from symmetry than divergence-type measures.

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arXiv2024

Visualization for departures from symmetry with the power-divergence-type measure in two-way contingency tables

When the row and column variables consist of the same category in a two-way contingency table, it is specifically called a square contingency table. Since it is clear that the square contingency tables have an association structure, a primary objective is to examine symmetric relationships and transitions between variables. While various models and measures have been proposed to analyze these structures understanding changes between two variables in behavior at two-time points or cohorts, it is also necessary to require a detailed investigation of individual categories and their interrelationships, such as shifts in brand preferences. This paper proposes a novel approach to correspondence analysis (CA) for evaluating departures from symmetry in square contingency tables with nominal categories, using a power-divergence-type measure. The approach ensures that well-known divergences can also be visualized and, regardless of the divergence used, the CA plot consists of two principal axes with equal contribution rates. Additionally, the scaling is independent of sample size, making it well-suited for comparing departures from symmetry across multiple contingency tables. Confidence regions are also constructed to enhance the accuracy of the CA plot.

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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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