Measure of departure from marginal point-symmetry for two-way contingency tables. (English) Zbl 1453.62526
Summary: For two-way contingency tables, S. Tomizawa [Biom. J. 27, 895–905 (1985; Zbl 0579.62037)] considered the point-symmetry and marginal point-symmetry models, and S. Tomizawa et al. proposed a measure to represent the degree of departure from point-symmetry [“An entropy measure of departure from point-symmetry for two-way contingency tables”, Symmetry Cult. Sci. 18, 279–297 (2007)]. The present paper proposes a measure to represent the degree of departure from marginal pointsymmetry for two-way tables. The proposed measure is expressed by using Cressie-Read power-divergence or Patil-Taillie diversity index. This measure would be useful for comparing the degrees of departure from marginal point-symmetry in several tables. The relationship between the degree of departure from marginal point-symmetry and the measure is shown when it is reasonable to assume underlying bivariate normal distribution. Examples are shown.
MSC:
| 62H17 | Contingency tables |
Citations:
Zbl 0579.62037References:
| [1] | A. AGRESTI (1984), Analysis of ordinal categorical data, Wiley, New York. · Zbl 0647.62052 |
| [2] | Y. M. M. BISHOP, S. E. FIENBERG, P. W. HOLLAND (1975), Discrete multivariate analysis: theory and practice, The MIT Press, Cambridge, Massachusetts. · Zbl 0332.62039 |
| [3] | N. A. C. CRESSIE, T. R. C. READ (1984), Multinomial goodness-of-fit tests, “Journal of the Royal Statistical Society”, series B, 46, pp. 440-464. · Zbl 0571.62017 |
| [4] | K. HASHIMOTO (1999), Gendai nihon no kaikyuu kouzou (Class structure in modern Japan: theory, method and quantitative analysis), Toshindo Press, Tokyo (in Japanese). |
| [5] | S. KULLBACK, R. A. LEIBLER (1951), On information and sufficiency, “Annals of Mathematical Statistics”, 22, pp. 79-86. · Zbl 0042.38403 |
| [6] | G. P. PATIL, C. TAILLIE (1982), Diversity as a concept and its measurement, “Journal of the American Statistical Association”, 77, pp. 548-561. · Zbl 0511.62113 |
| [7] | T. R. C. READ, N. A. C. CRESSIE (1988), Goodness-of-fit statistics for discrete multivariate data, Springer, New York. · Zbl 0663.62065 |
| [8] | S. TOMIZAWA (1985), The decompositions for point-symmetry models in two-way contingency tables, “Biometrical Journal”, 27, pp. 895-905. · Zbl 0579.62037 |
| [9] | S. TOMIZAWA, K. YAMAMOTO, K. TAHATA (2007), An entropy measure of departure from pointsymmetry for two-way contingency tables, “Symmetry: Culture and Science”, 18, pp. 279-297. |
| [10] | K. D. WALL, G. A. LIENERT (1976), A test for point-symmetry in J-dimensional contingency-cubes, “Biometrical Journal”, 18, pp. 259-264. · Zbl 0335.62030 |
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