An algebraic theory of contrasts for Neyman’s modified chi-square statistic

Sébastien Loisel, Yoshio Takane*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


The use of Pearson’s chi-square goodness of fit statistic is well established in the analysis of contingency tables. This statistic measures the overall discrepancy between observed and expected proportions, thereby indicating the degree of empirical validity of the hypothesis under which the expected proportions are derived. A theory of contrasts that capture parts of Pearson’s global index of fit has been developed, allowing to test specific aspects of the overall discrepancy. This theory, however, has one clear disadvantage when used in multiple comparison settings. In Pearson’s statistic, an hypothesized mean structure (expected proportions) has a direct bearing on its variance-covariance structure, so that once the former is rejected, the latter is also rejected. This is OK if only a single omnibus test is performed, but is problematic in multiple comparison procedures because the tests performed earlier may denounce the variance-covariance structure assumed in later tests. For this reason, Goodman used Neyman’s modified chi-square statistic for his post-hoc tests in contingency tables. In this statistic, mean and variance-covariance structures are separate entities (i.e., they can be specified independently), overcoming the weakness of Pearson’s statistic. This paper develops a theory of contrasts specifically tailored to Neyman’s statistic. Goodman’s method, however, is not entirely trouble free. A simple cure is proposed, making Goodman’s procedure theoretically more rigorous and applicable to wider situations.

Original languageEnglish
Pages (from-to)335-360
Number of pages26
Issue number1
Early online date13 Sept 2022
Publication statusPublished - Jan 2023


  • Additive partitions
  • Neyman’s modified chi-square statistic
  • Orthogonal contrasts
  • Pearson’s chi-square statistic
  • Three-way and higher-order contingency tables
  • Two-way contingency tables
  • v (a contrast vectorthat captures the maximum amount of variability in Neymans statistic)

ASJC Scopus subject areas

  • Analysis
  • Experimental and Cognitive Psychology
  • Clinical Psychology
  • Applied Mathematics


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