Measure of departure from marginal point-symmetry for two-way contingency tables

Kouji Yamamoto, Kouji Tahata, Motoko Suzuki, Sadao Tomizawa


For two-way contingency tables, Tomizawa (1985) considered the point-symmetry and marginal point-symmetry models, and Tomizawa, Yamamoto and Tahata (2007) proposed a measure to represent the degree of departure from point-symmetry. 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.

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A. AGRESTI (1984), Analysis of ordinal categorical data, Wiley, New York.

Y. M. M. BISHOP, S. E. FIENBERG, P. W. HOLLAND (1975), Discrete multivariate analysis: theory and practice, The MIT Press, Cambridge, Massachusetts.

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.

K. HASHIMOTO (1999), Gendai nihon no kaikyuu kouzou (Class structure in modern Japan: theory, method and quantitative analysis), Toshindo Press, Tokyo (in Japanese).

S. KULLBACK, R. A. LEIBLER (1951), On information and sufficiency, “Annals of Mathematical Statistics”, 22, pp. 79-86.

G. P. PATIL, C. TAILLIE (1982), Diversity as a concept and its measurement, “Journal of the American Statistical Association”, 77, pp. 548-561.

T. R. C. READ, N. A. C. CRESSIE (1988), Goodness-of-fit statistics for discrete multivariate data, Springer, New York.

S. TOMIZAWA (1985), The decompositions for point-symmetry models in two-way contingency tables, “Biometrical Journal”, 27, pp. 895-905.

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.

K. D. WALL, G. A. LIENERT (1976), A test for point-symmetry in J-dimensional contingency-cubes, “Biometrical Journal”, 18, pp. 259-264.

DOI: 10.6092/issn.1973-2201/3620