Partition of the Chi-Squared Statistic in a Contingency Table

Abstract

The Pearson statistic, a well-known goodness-of fit test in the analysis of contingency tables, gives little guidance as to why a null hypothesis is rejected. One approach to determine the source(s) of deviation from the null is the decomposition of a chi-squared statistic. This allows writing the statistic as the sum of independent chi-squared statistics. First, three major types of contingency tables and the usual chi-squared tests are reviewed. Three types of decompositions are presented and applied: one based on the partition of the contingency table into independent subtables; one derived from smooth models and one from the eigendecomposition of the central matrix defining the statistics. A comparison of some of the omnibus statistics decomposed above to a χ2(1)-distributed statistic shows that the omnibus statistics lack power compared to this statistic for testing hypothesis of equal success probabilities against monotonic trend in the success probabilities in a column-binomial contingency table.