Chi-Square, Partition of

A chi-square statistic suitable for testing a primary hypothesis can be partitioned into components such that each component gives a test for a corresponding secondary hypothesis. Some partitionings are exact and some are approximate. The theory is based on the Fisher–Cochran theorem about decomposing quadratic functions of normal variables. The history of this technique is surveyed. Applications are described for contingency tables, with the main focus on the two-way table and likelihood-ratio statistics. Brief mention is also made of partitioning into non-chi-square components, such as a decomposition that forms the basis of correspondence analysis.