A Note on the Additive Partitioning of Chi-Square in Contingency Tables

The exact partition of the total chi-square statistic in a r X s contingency table in such a way as to afford comparisons among frequencies achieved by pooling adjacent rows and columns has been treated by Irwin 1949, Lancaster 1949, 1950, and Kimball 1954. The statistics obtained by such partitionings may be used to test orthogonal contrasts of a particular type, each with one degree of freedom. In general, however, an experimenter may be interested in testing other orthogonal contrasts, many of which may be handled by the approach suggested by Irwin and Lancaster, or more simply by short-cut formulas such as those given by Kimball. The purpose of this paper is to present short-cut formulas for handling a broader class of orthogonal contrasts which may be tested with one or more degrees of freedom. The component chi-square values computed by these formulas will add exactly to the total chisquare as computed in the usual way. The most conspicuous omissions will be those contrasts whose analogues in the analysis of variance involve the use of orthogonal polynomials for separating out linear, quadratic, cubic, etc., sums of squares (Yates 1948, Cochran 1954).