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Familywise decompositions of Pearson’s chi-square statistic in the analysis of contingency tables

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  • Rosaria Lombardo

    (University of Campania “Luigi Vanvitelli”)

  • Yoshio Takane

    (University of Victoria)

  • Eric J. Beh

    (University of Newcastle)

Abstract

Pearson’s chi-square statistic is well established for testing goodness-of-fit of various hypotheses about observed frequency distributions in contingency tables. A general formula for ANOVA-like decompositions of Pearson’s statistic is given under the independence assumption along with their extensions to higher-order tables. Mathematically, it makes the terms in the partitions and orthogonality among them obvious. Practically, it enables simultaneous analyses of marginal and joint probabilities in contingency tables under a variety of hypotheses about the marginal probabilities. Specifically, this framework accommodates the specification of theoretically driven probabilities as well as the well known cases in which the marginal probabilities are fixed or estimated from the data. The former allows tests of prescribed marginal probabilities, while the latter allows tests of the associations among variables after eliminating the marginal effects. Mixtures of these two cases are also permitted. Examples are given to illustrate the tests.

Suggested Citation

  • Rosaria Lombardo & Yoshio Takane & Eric J. Beh, 2020. "Familywise decompositions of Pearson’s chi-square statistic in the analysis of contingency tables," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(3), pages 629-649, September.
    • Handle: RePEc:spr:advdac:v:14:y:2020:i:3:d:10.1007_s11634-019-00374-7
    DOI: 10.1007/s11634-019-00374-7
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    References listed on IDEAS

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    1. Sébastien Loisel & Yoshio Takane, 2016. "Partitions of Pearson’s Chi-square statistic for frequency tables: a comprehensive account," Computational Statistics, Springer, vol. 31(4), pages 1429-1452, December.
    2. Peter Heijden & Jan Leeuw, 1985. "Correspondence analysis used complementary to loglinear analysis," Psychometrika, Springer;The Psychometric Society, vol. 50(4), pages 429-447, December.
    3. André Carlier & Pieter Kroonenberg, 1996. "Decompositions and biplots in three-way correspondence analysis," Psychometrika, Springer;The Psychometric Society, vol. 61(2), pages 355-373, June.
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    1. Rosaria Lombardo & Eric J. Beh & Francesco Prattichizzo & Giuseppe Lucisano & Antonio Nicolucci & Björn Eliasson & Hanne Krage Carlsen & Rosalba La Grotta & Valeria Pellegrini & Antonio Ceriello, 2024. "Testing and Visualization of Associations in Three-Way Contingency Tables: A Study of the Gender Gap in Patients with Type 1 Diabetes and Cardiovascular Complications," Mathematics, MDPI, vol. 12(14), pages 1-13, July.

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