Decomposition of Independence Using the Logit Uniform Association Model and Equality of Concordance and Discordance for Two-Way Classifications

For two-way contingency tables with ordered categories, the present paper gives a theorem that the independence model holds if and only if the logit uniform association model holds and equality of concordance and discordance for all pairs of adjacent rows and all dichotomous collapsing of the columns holds. Using the theorem, we analyze the cross-classification of duodenal ulcer patients according to operation and dumping severity.

KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Tahata, K. , Miyamoto, N. and Tomizawa, S. (2015) Decomposition of Independence Using the Logit Uniform Association Model and Equality of Concordance and Discordance for Two-Way Classifications. Open Journal of Statistics, 5, 514-518. doi: 10.4236/ojs.2015.56054.

 [1] Goodman, L.A. (1979) Simple Models for the Analysis of Association in Cross-Classifications Having Ordered Categories. Journal of the American Statistical Association, 74, 537-552. http://dx.doi.org/10.1080/01621459.1979.10481650 [2] Agresti, A. (1984) Analysis of Ordinal Categorical Data. Wiley, New York. [3] Tomizawa, S., Miyamoto, N. and Sakurai, M. (2008) Decomposition of Independence Model and Separability of Its Test Statistic for Two-Way Contingency Tables with Ordered Categories. Advances and Applications in Statistics, 8, 209-218. [4] Kendall, M.G. (1945) The Treatment of Ties in Ranking Problems. Biometrika, 33, 239-251.http://dx.doi.org/10.1093/biomet/33.3.239 [5] Tahata, K., Miyamoto, N. and Tomizawa, S. (2008) Decomposition of Independence Using Pearson, Kendall and Spearman’s Correlations and Association Model for Two-Way Classifications. Far East Journal of Theoretical Statistics, 25, 273-283. [6] Stuart, A. (1963) Calculation of Spearman’s Rho for Ordered Two-Way Classifications. The American Statistician, 17, 23-24. [7] Kendall, M. and Gibbons, J.D. (1990) Rank Correlation Methods. 5th Edition, Edward Arnold, London. [8] Tahata, K. and Tomizawa, S. (2014) Symmetry and Asymmetry Models and Decompositions of Models for Contingency Tables. SUT Journal of Mathematics, 50, 131-165. [9] Grizzle, J.E., Starmer, C.F. and Koch, G.G. (1969) Analysis of Categorical Data by Linear Models. Biometrics, 25, 489-504. http://dx.doi.org/10.2307/2528901 [10] Tomizawa, S. (1992) More Parsimonious Linear-by-Linear Association Model in the Analysis of Cross-Classifications Having Ordered Categories. Biometrical Journal, 34, 129-140. http://dx.doi.org/10.1002/bimj.4710340202