Log-normal Distribution Type Symmetry Model for Square Contingency Tables with Ordered Categories


  • Kiyotaka Iki




For the analysis of square contingency tables with the same row and column ordinal classications, this article proposes a new model which indicates that the log-ratios of symmetric cell probabilities are proportional to the difference between log-row category and log-column category. The proposed model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate log-normal distribution. Also, this article gives the decomposition of the symmetry model using the proposed model with the orthogonality of test statistics. Examples are given. The simulation studies based on bivariate log-normal distribution are given.


Agresti, A. (1983). A simple diagonals-parameter symmetry and quasi-symmetry model. Statistics and Probability Letters, 1, 313-316.

Agresti, A. (2013). Categorical Data Analysis, 3rd ed. New Jersey: Wiley.

Bishop, Y. M. M., Fienberg, S. E., and Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. Cambridge: The MIT Press.

Bhapkar, V. P. and Darroch, J. N. (1990). Marginal symmetry and quasi symmetry of general order. Journal of Multivariate Analysis, 34, 173-184.

Bowker, A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association, 43, 572-574.

Darroch, J. N. and Ratcliff, D. (1972). Generalized iterative scaling for log-linear models. Annals of Mathematical Statistics, 43, 1470-1480.

Lang, J. B. (1996). On the partitioning of goodness-of-fit statistics for multivariate categorical response models. Journal of the American Statistical Association, 91, 1017-1023.

Lang, J. B. and Agresti, A. (1994). Simultaneously modeling joint and marginal distributions of multivariate categorical responses. Journal of the American Statistical Association, 89, 625-632.

Tomizawa, S. (1984). Three kinds of decompositions for the conditional symmetry model in a square contingency table. Journal of the Japan Statistical Society, 14, 35-42.

Tomizawa, S. (1985). Analysis of data in square contingency tables with ordered categories using the conditional symmetry model and its decomposed models. Environmental Health Perspectives, 63, 235-239.



How to Cite

Iki, K. (2018). Log-normal Distribution Type Symmetry Model for Square Contingency Tables with Ordered Categories. Austrian Journal of Statistics, 47(3), 39-48. https://doi.org/10.17713/ajs.v47i3.701