Abstract
For three-way contingency tables with two ordinal and one nominal variables, this paper shows that (i) the conditional independence model holds if and only if conditional covariance between ordinal variables at each level of the nominal variable equals zero and the heterogeneous linear-by-linear association model holds, (ii) the conditional independence model holds if and only if an average conditional covariance between ordinal variables across levels of the nominal variable equals zero and the homogeneous linear-by-linear association model holds, and (iii) the likelihood ratio test statistic for the conditional independence model is asymptotically equivalent to the sum of the likelihood ratio test statistic for the decomposed models.