On Quasi-Independence and Quasi-Dependence in Contingency Tables, with Special Reference to Ordinal Triangular Contingency Tables

Abstract

In the usual statistical analysis of the entries (frequencies) in a contingency table having I rows and J columns (I ≥ 2, J ≥ 2), an important role is often played by the usual null hypothesis of independence (i.e., the hypothesis that the row variable and the column variable are statistically independent of each other). In the situation considered in this article, the entries in some of the I × J cells of the contingency table are omitted from the analysis (because these entries are either missing, unreliable, void, or restricted in certain ways); and in this situation, the usual null hypothesis of independence cannot be applied in a meaningful way. The concept of “quasi-independence” was introduced for this situation; and an important role can often be played in this situation by the null hypothesis of quasi-independence (a generalization of the usual null hypothesis of independence). For this situation, we introduce also the concept of likelihood-ratio “quasi-dependence” (a generalization of the usual concept of likelihood-ratio dependence in the I × J table), and we consider the null hypothesis of null quasi-dependence and the alternative hypotheses of positive quasi-dependence and negative quasi-dependence. The relationship between quasi-independence and null quasi-dependence is considered, and methods are introduced for testing these null hypotheses against the aforementioned alternative hypotheses. Special attention is given to the triangular contingency table; and some of the results presented for this special case can also be applied more generally to the I × J contingency table. With the approach presented in this article, we are able to simplify and gain further insight into some formulas appearing in the earlier literature on the quasi-independence model applied to contingency tables. This approach has certain other advantages as well. For example, with this approach we are able to introduce various tests of the quasi-independence model against alternative hypotheses of positive or negative quasi-dependence, and these tests will in some cases be more powerful than other tests proposed in the earlier literature.

Key Words:

• Independence and quasi-independence in contingency tables
• Likelihood-ratio dependence and likelihood-ratio quasi-dependence
• Maximum-likelihood estimation
• Null quasi-dependence
• positive quasi-dependence, and negative quasi-dependence
• Tests of null quasi-dependence and tests of quasi-independence
• Triangular contingency tables