Positive association in a two-way contingency table:likelihood ratio tests

Given a two-way contingency table in which the rows and columns both define ordinal variables, there are many ways in which the informal idea of positive association between those variables might be defined. This paper considers a variety of definitions expressed as inequality constraints on cross-product ratios. Logical relationships between the definitions are explored. Each definition can serve as a composite alternative against which the null hypothesis of no association may be tested. For a broad class of such alternatives a decomposition of the log-likelihood gives both an explicit likelihood ratio statistic and its asymptotic null hypothesis distribution. Results are derived for multinomial sampling and for fully conditional sampling with row and column totals fixed.