Positive Association Among Three Binary Variables and Cross-Product Ratios

We show that, when the three-way association level among the three binary variables, X, U 1 and U 2 is fixed, D P e pr(X e 1¦U 1 e 1) - pr(X e 1¦U 1 e 0) increases as the cross-product ratio of U 1 and U 2 increases under the assumption that X is positively associated with U 1 and U 2 . We then discuss some implications of this property. Copyright 2007, Oxford University Press.

[1]  Sung-Ho Kim,et al.  Stochastic ordering and robustness in classification from a Bayesian network , 2005, Decis. Support Syst..

[2]  D. Rubin.  Causal Inference Using Potential Outcomes , 2005 .

[3]  Helena Chmura Kraemer,et al.  Reconsidering the odds ratio as a measure of 2×2 association in a population , 2004, Statistics in medicine.

[4]  B. Junker,et al.  A characterization of monotone unidimensional latent variable models , 1997 .

[5]  R. Mislevy.  Evidence and inference in educational assessment , 1994 .

[6]  K. Tatsuoka.  Toward an Integration of Item-Response Theory and Cognitive Error Diagnosis. , 1987 .

[7]  P. Rosenbaum,et al.  Conditional Association and Unidimensionality in Monotone Latent Variable Models , 1985 .

[8]  Dorothy T. Thayer,et al.  Differential Item Performance and the Mantel-Haenszel Procedure. , 1986 .

[9]  P. Holland.  Statistics and Causal Inference , 1985 .

[10]  L. A. Goodman,et al.  Measures of association for cross classifications , 1979 .

[11]  Stephen E. Fienberg,et al.  The analysis of cross-classified categorical data , 1980 .

[12]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[13]  L. A. Goodman,et al.  Measures of Association for Cross Classifications, IV: Simplification of Asymptotic Variances , 1972 .

[14]  L. A. Goodman,et al.  Measures of Association for Cross Classifications III: Approximate Sampling Theory , 1963 .

[15]  L. A. Goodman,et al.  Measures of Association for Cross Classifications. II: Further Discussion and References , 1959 .

[16]  Jerome Cornfield,et al.  A Statistical Problem Arising from Retrospective Studies , 1956 .

[17]  E. H. Simpson,et al.  The Interpretation of Interaction in Contingency Tables , 1951 .

[18]  G. Yule.  NOTES ON THE THEORY OF ASSOCIATION OF ATTRIBUTES IN STATISTICS , 1903 .