Quasi-Independence in Ordinal Triangular Contingency Tables
本文重新解释了有序三角列联表中准独立性的含义，将其与有序关联度量联系起来，并提出了一个利用有序信息的大样本检验方法，适用于更窄的备择假设。
Abstract The quasi-independence model is most often used to analyze incomplete contingency tables. Like independence in complete tables, the structurally nonzero probabilities under this model are multiplicative (in a two-way classification, πij = α,βj for some positive parameters αi and βj ). The only interpretation of quasi-independence known in the literature is that it is a form of independence, conditional on any complete (rectangular) subtable (Bishop, Fienberg, and Holland 1975). Unfortunately, this interpretation does not provide a clear understanding of quasi-independence in terms of association. Restricting attention to ordinal triangular tables, an alternative and more explicit interpretation is given in this article through some commonly used coefficients of ordinal association. A large-sample test for quasi-independence in ordinal triangular tables is also formulated. The likelihood ratio dependence condition, introduced by Lehmann (1966), plays a fundamental role in studying the dependence structures of bivariate probability models (Agresti 1984). This article's interpretation of quasi-independence is given with respect to a family of such models. Specifically, it is shown that quasi-independence minimizes (maximizes) ordinal association in upper-right (left) or lower-left (right) ordinal triangular tables in the class of positive (negative) likelihood ratio dependence models with a fixed set of marginals. The tools used include a probability inequality for a pair of positively likelihood ratio dependent variables (X, Y) that satisfy Pr(X ≤ Y) = 1. This interpretation is analogous to that of independence, and is often helpful in understanding the appropriate alternative hypothesis against which the null hypothesis of quasi-independence has to be tested. For instance, one might expect negative likelihood ratio dependence for the models in the fingerprint data analyzed by Waite (1915) and Goodman (1968). Therefore, the appropriate alternative hypothesis here should correspond to a smaller degree of association than that corresponding to quasi-independence. Similarly, for the data on disability rating of stroke patients analyzed by Bishop and Fienberg (1969), one could envision positive likelihood ratio dependence for the underlying models, in which case the degree of association corresponding to any alternative should be higher than that corresponding to quasi-independence. The proposed test for quasi-independence, unlike the usual chi-squared tests considered by Goodman (1968) and Bishop and Fienberg (1969), exploits the ordinal nature of row and column variables, and has the advantage of being applicable to narrower alternatives. In addition, unlike independence, quasi-independence does not minimize (maximize) ordinal association within the larger class of positive (negative) quadrant dependence models.