A Simple Interaction Model for Two-Dimensional Contingency Tables.

A model specifying independence within each of two disjoint groups of rows is presented as an alternative to complete independence in a two-dimensional contingency table with r rows and two columns. If the measure used to assess the improvement in fit is a member of a class of order-producing (OP) functions, then the partition that maximizes the measure can be located after checking only (r - 1) of the (2[sup r-1] - 1) possible partitions. The likelihood ratio statistic and corresponding Pearson goodness-of-fit statistics for independence in the collapsed table are two such OP functions. In the case of equal row totals the asymptotic distribution of the Pearson statistic is approximately that of 2(pi - 2)/pi times a chi-squared variable with r/(pi - 2) degrees of freedom. [ABSTRACT FROM AUTHOR]