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


  • Kouji Yamamoto Osaka University Hospital
  • Kouji Tahata Science University of Tokio
  • Motoko Suzuki Science University of Tokio
  • Sadao Tomizawa Science University of Tokio



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.


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.




How to Cite

Yamamoto, K., Tahata, K., Suzuki, M., & Tomizawa, S. (2011). Measure of departure from marginal point-symmetry for two-way contingency tables. Statistica, 71(3), 367-380.