Partitions of Pearson’s Chi-square statistic for frequency tables: a comprehensive account

Sébastien Loisel, Yoshio Takane*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)
120 Downloads (Pure)


Pearson’s Chi-square statistic for frequency tables depends on what is hypothesized as the expected frequencies. Its partitions also depend on the hypothesis. Lancaster (J R Stat Soc B 13:242–249, 1951) proposed ANOVA-like partitions of Pearson’s statistic under several representative hypotheses about the expected frequencies. His expositions were, however, not entirely clear. In this paper, we clarify his method of derivations, and extend it to more general situations. A comparison is made with analogous decompositions of the log likelihood ratio statistic associated with log-linear analysis of contingency tables.

Original languageEnglish
Pages (from-to)1429–1452
Number of pages24
JournalComputational Statistics
Issue number4
Early online date4 Sept 2015
Publication statusPublished - Dec 2016


  • ANOVA-like partitions
  • Helmert-like contrasts
  • Likelihood ratio (LR) statistic
  • Metric matrix
  • One-way tables
  • Orthogonal transformations
  • Three-way tables
  • Two-way tables

ASJC Scopus subject areas

  • Statistics and Probability
  • Computational Mathematics
  • Statistics, Probability and Uncertainty


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