Properties of the $ Q_t $-test which was proposed in Jayasekara and Yanagawa (1995) for detecting non-linear differences of two populations in an ordinal categorical table are further studied in this paper under different asymptotic framework. It is assumed in this paper that each marginal sum relative to the gland total tends to constant which is away from zero when the gland total tends to infinitive. The asymptotic distributions of the test statistics are obtained under null and contigious alternatives. It is shown that single test statistic, or combinatin of several test statistics have high powers for detecting various patterns of non-linear responses.