An exact conditional approach is developed to test for certain forms of positive association between two ordinal variables (e.g. positive quadrant dependence, total positivity of order 2). The approach is based on the use of a test statistic measuring the goodness-of-fit of the model formulated according to the type of positive association of interest. The nuisance parameters, corresponding to the marginal distributions of the two variables, are eliminated by conditioning the inference on the observed margins. This, in turn, allows to remove the uncertainty on the conclusion of the test, which typically arises in the unconditional context where the null distribution of the test statistic depends on such parameters. Since the multivariate generalized hypergeometric distribution, which results from conditioning, is normally intractable, Markov chain Monte Carlo methods are used to obtain maximum likelihood estimates of the parameters of the constrained model. The Pearson's chi-squared statistics is used as a test statistic; a p-value for this statistic is computed through simulation, when the data are sparse, or exploiting the asymptotic theory based on the chi-bar squared distribution. The extension of the present approach to deal with bivariate contingency tables, stratified according to one or more explanatory discrete variables, is also outlined. Finally, three applications based on real data are presented.