ON QUASI-INDEPENDENCE AND QUASI-DEPENDENCE IN CONTINGENCY-TABLES, WITH SPECIAL REFERENCE TO ORDINAL TRIANGULAR CONTINGENCY-TABLES

In the usual statistical analysis of the entries (frequencies) in a co ntingency table having I rows and J columns (I greater than or equal t o 2, J greater than or equal to 2), an important role is often played by the usual null hypothesis of independence (i.e., the hypothesis tha t the row variable and the column variable are statistically independe nt of each other). In the situation considered in this article, the en tries in some of the I X J cells of the contingency table are omitted from the analysis (because these entries are either missing, unreliabl e, void, or restricted in certain ways); and in this situation, the us ual null hypothesis of independence cannot be applied in a meaningful way. The concept of ''quasi-independence'' was introduced for this sit uation; and an important role can often be played in this situation by the null hypothesis of quasi-independence (a generalization of the us ual null hypothesis of independence). For this situation, we introduce also the concept of likelihood-ratio ''quasi-dependence'' (a generali zation of the usual concept of likelihood-ratio dependence in the I X J table), and we consider the null hypothesis of null quasi-dependence and the alternative hypotheses of positive quasi-dependence and negat ive quasi-dependence. The relationship between quasi-independence and null quasi-dependence is considered, and methods are introduced for te sting these null hypotheses against the aforementioned alternative hyp otheses. Special attention is given to the triangular contingency tabl e; and some of the results presented for this special case can also be applied more generally to the I x J contingency table. With the appro ach presented in this article, we are able to simplify and gain furthe r insight into some formulas appearing in the earlier literature on th e quasi-independence model applied to contingency tables. This approac h has certain other advantages as well. For example, with this approac h we are able to introduce various tests of the quasi-independence mod el against alternative hypotheses of positive or negative quasi-depend ence, and these tests will in some cases be more powerful than other t ests proposed in the earlier literature.