Exact Conditional Testing of Certain Forms of Positive Association for Bivariate Ordinal Data

We describe an exact conditional approach to test for certain forms of positive association between two ordinal variables. The approach is based on maximizing a conditional version of the multinomial likelihood for the observed table given the row and column margins. This allows us to remove the uncertainty that typically arises in testing hypotheses on the association between two categorical variables due to the presence of nuisance parameters corresponding to the marginal distributions of the two variables. Conditional maximum likelihood estimates of the parameters are obtained through Markov chain Monte Carlo methods. The Pearson’s chi-squared is used as test statistic. A p-value for this statistic is computed by simulation, when data are sparse, or by exploiting the asymptotic theory.