This article first illuminates possible discrepancies between normal and logistic models in the bivariate dichotomous case of qualitative response analyses. Then a Cox-type test statistic for separate families of hypotheses is proposed for the purpose of comparing the two models. An asymptotic distribution of the new test statistic is derived, and also the consistency of the test is shown. Before applying the Cox-type tests to actual data, Berkson's minimum chi-square estimators of the two models are explained in Section 3. In Section 4, logistic, normal, and linear models are worked out for a set of economic data, and the Cox-type tests are applied to compare them.