Given a two-way contingency table in which the rows and columns both define ordinal variables, the parameters are estimated by maximizing the likelihood function under the restrictions that all local log-odds parameters are non-negative. An exact analytic form of the rational solution to this problem is obtained. When such a rational solution exists, the likelihood ratio for testing the hypothesis of independence against that of positive association is obtained explicitly. Under the null hypothesis, an asymptotic equivalence of this ratio to a chi-squared statistic is established and its distribution is obtained. The method is illustrated on a data set.