Combination of Neighboring Cells in Contingency Tables
Abstract In the application of the χ 2-test for independence to contingency tables in case the expected frequencies in some of the cases are small, a commonly recommended procedure is to coalesce two or more rows and/or columns. Based on Cramer's modified χ 2 minimum method, this paper develops means of combining individual cells instead of whole rows or columns. Specific results are given for a pair of cells in the same row (or column), a block of cells, two pairs which form a 2×2 block, and two row pairs whose columns do not overlap. Results generally are simple if the sets of cells combined form a block or blocks which do not overlap. The complications that arise in the case of partial overlapping are illustrated for two row pairs which have a single column in common. The estimates of the probability of an item falling in each of the three columns on the null hypothesis are obtained from a quadratic equation which, however, always leads to a unique set of consistent estimates.