Asymmetry Index on Marginal Homogeneity for Square Contingency Tables with Ordered Categories

Affiliation(s)

Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan.

Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba, Japan.

ABSTRACT

For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homogeneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed measures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed.

For square contingency tables with ordered categories, the present paper considers two kinds of weak marginal homogeneity and gives measures to represent the degree of departure from weak marginal homogeneity. The proposed measures lie between –1 to 1. When the marginal cumulative logistic model or the extended marginal homogeneity model holds, the proposed measures represent the degree of departure from marginal homogeneity. Using these measures, three kinds of unaided distance vision data are analyzed.

KEYWORDS

Marginal Homogeneity; Marginal Cumulative Logistic Model; Measure; Square Contingency Table

Marginal Homogeneity; Marginal Cumulative Logistic Model; Measure; Square Contingency Table

Cite this paper

K. Tahata, K. Kawasaki and S. Tomizawa, "Asymmetry Index on Marginal Homogeneity for Square Contingency Tables with Ordered Categories,"*Open Journal of Statistics*, Vol. 2 No. 2, 2012, pp. 198-203. doi: 10.4236/ojs.2012.22023.

K. Tahata, K. Kawasaki and S. Tomizawa, "Asymmetry Index on Marginal Homogeneity for Square Contingency Tables with Ordered Categories,"

References

[1] A. Stuart, “A Test for Homogeneity of the Marginal Distributions in a Two-Way Classification,” Biometrika, Vol. 42, No. 3-4, 1955, pp. 412-416. doi:10.1093/biomet/42.3-4.412

[2] A. Agresti, “Analysis of Ordinal Categorical Data,” John Wiley, New York, 1984.

[3] S. Tomizawa, “Three Kinds of Decompositions for the Conditional Symmetry Model in a Square Contingency Table,” Journal of the Japan Statistical Society, Vol. 14, No. 1, 1984, pp. 35-42.

[4] S. Tomizawa, “A Generalization of the Marginal Homogeneity Model for Square Contingency Tables with Ordered Categories,” Journal of Educational and Behavioral Statistics, Vol. 20, 1995, pp. 349-360.

[5] K. Tahata and S. Tomizawa, “Generalized Marginal Homogeneity Model and Its Relation to Marginal Equimoments for Square Contingency Tables with Ordered Categories,” Advances in Data Analysis and Classification, Vol. 2, No. 3, 2008, pp. 295-311. doi:10.1007/s11634-008-0028-1

[6] K. Tahata, T. Iwashita and S. Tomizawa, “Measure of Departure from Symmetry of Cumulative Marginal Probabilities for Square Contingency Tables with Ordered Categories,” SUT Journal of Mathematics, Vol. 42, 2006, pp. 7-29.

[7] K. Tahata, T. Iwashita and S. Tomizawa, “Measure of Departure from Conditional Marginal Homogeneity for Square Contingency Tables with Ordered Categories,” Statistics, Vol. 42, No. 5, 2008, pp. 453-466. doi:10.1080/02331880802190521

[8] Y. M. M. Bishop, S. E. Fienberg and P. W. Holland, “Discrete Multivariate Analysis: Theory and Practice,” The MIT Press, Cambridge, 1975.

[9] A. Stuart, “The Estimation and Comparison of Strengths of Association in Contingency Tables,” Biometrika, Vol. 40, 1953, pp. 105-110. doi:10.2307/2333101

[10] K. Yamamoto, K. Tahata, N. Miyamoto and S. Tomizawa, “Comparison between Several Square Tables Data Using Models of Symmetry and Asymmetry,” Quantum Probability and White Noise Analysis: Quantum Bio-Informatics, Vol. 21, 2007, pp. 337-349.

[11] P. McCullagh, “A Class of Parametric Models for the Analysis of Square Contingency Tables with Ordered Categories,” Biometrika, Vol. 65, No. 2, 1978, pp. 413-418. doi:10.1093/biomet/65.2.413

[1] A. Stuart, “A Test for Homogeneity of the Marginal Distributions in a Two-Way Classification,” Biometrika, Vol. 42, No. 3-4, 1955, pp. 412-416. doi:10.1093/biomet/42.3-4.412

[2] A. Agresti, “Analysis of Ordinal Categorical Data,” John Wiley, New York, 1984.

[3] S. Tomizawa, “Three Kinds of Decompositions for the Conditional Symmetry Model in a Square Contingency Table,” Journal of the Japan Statistical Society, Vol. 14, No. 1, 1984, pp. 35-42.

[4] S. Tomizawa, “A Generalization of the Marginal Homogeneity Model for Square Contingency Tables with Ordered Categories,” Journal of Educational and Behavioral Statistics, Vol. 20, 1995, pp. 349-360.

[5] K. Tahata and S. Tomizawa, “Generalized Marginal Homogeneity Model and Its Relation to Marginal Equimoments for Square Contingency Tables with Ordered Categories,” Advances in Data Analysis and Classification, Vol. 2, No. 3, 2008, pp. 295-311. doi:10.1007/s11634-008-0028-1

[6] K. Tahata, T. Iwashita and S. Tomizawa, “Measure of Departure from Symmetry of Cumulative Marginal Probabilities for Square Contingency Tables with Ordered Categories,” SUT Journal of Mathematics, Vol. 42, 2006, pp. 7-29.

[7] K. Tahata, T. Iwashita and S. Tomizawa, “Measure of Departure from Conditional Marginal Homogeneity for Square Contingency Tables with Ordered Categories,” Statistics, Vol. 42, No. 5, 2008, pp. 453-466. doi:10.1080/02331880802190521

[8] Y. M. M. Bishop, S. E. Fienberg and P. W. Holland, “Discrete Multivariate Analysis: Theory and Practice,” The MIT Press, Cambridge, 1975.

[9] A. Stuart, “The Estimation and Comparison of Strengths of Association in Contingency Tables,” Biometrika, Vol. 40, 1953, pp. 105-110. doi:10.2307/2333101

[10] K. Yamamoto, K. Tahata, N. Miyamoto and S. Tomizawa, “Comparison between Several Square Tables Data Using Models of Symmetry and Asymmetry,” Quantum Probability and White Noise Analysis: Quantum Bio-Informatics, Vol. 21, 2007, pp. 337-349.

[11] P. McCullagh, “A Class of Parametric Models for the Analysis of Square Contingency Tables with Ordered Categories,” Biometrika, Vol. 65, No. 2, 1978, pp. 413-418. doi:10.1093/biomet/65.2.413