Dimensionality Reduction Methods for Contingency Tables with Ordinal Variables

  • Luigi D’AmbraEmail author
  • Pietro Amenta
  • Antonello D’Ambra
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 227)


Several extensions of correspondence analysis have been introduced in literature coping with the possible ordinal structure of the variables. They usually obtain a graphical representation of the interdependence between the rows and columns of a contingency table, by using several tools for the dimensionality reduction of the involved spaces. These tools are able to enrich the interpretation of the graphical planes, providing also additional information, with respect to the usual singular value decomposition. The main aim of this paper is to suggest an unified theoretical framework of several methods of correspondence analysis coping with ordinal variables.


Ordinal variables Single and double cumulative correspondence analysis Orthogonal polynomials Generalized singular value decomposition 


  1. 1.
    Beh, E.J.: Simple correspondence analysis of ordinal cross-classifications using orthogonal polynomials. Biom. J. 39, 589–613 (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beh, E.J., D’Ambra, L., Simonetti, B.: Correspondence analysis of cumulative frequencies using a decomposition of Taguchi‘s statistic. Commun. Stat. Theory Methods 40, 1620–1632 (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Beh, E.J., Lombardo, R.: Correspondence Analysis: Theory, Practice and New Strategies. Wiley (2014)Google Scholar
  4. 4.
    Cuadras, C.M., Cuadras, D.: A unified approach for the multivariate analysis of contingency tables. Open J. Stat. 5, 223–232 (2015)CrossRefGoogle Scholar
  5. 5.
    D’Ambra, A.: Cumulative correspondence analysis using orthogonal polynomials. Commun. Stat. Theory Methods (to appear)Google Scholar
  6. 6.
    D’Ambra, L., Beh, E.J., Amenta, P.: CATANOVA for two-way contingency tables with an ordinal response using orthogonal polynomials. Commun. Stat. Theory Methods 34, 1755–1769 (2005)CrossRefzbMATHGoogle Scholar
  7. 7.
    D’Ambra, L., Beh, E.J., Camminatiello, I.: Cumulative correspondence analysis of two-way ordinal contingency tables. Commun. Stat. Theory Methods 43(6), 1099–1113 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D’Ambra, L., Lauro, N.: Non symmetrical analysis of three-way contingency tables. In: Coppi, R., Bolasco, S. (eds.) Multiway Data Analysis, pp. 301–315. Elsevier Science Publishers B.V, Amsterdam (1989)Google Scholar
  9. 9.
    Emerson, P.L.: Numerical construction of orthogonal polynomials from a general recurrence formula. Biometrics 24, 696–701 (1968)CrossRefGoogle Scholar
  10. 10.
    Goodman, L.A.: A single general method for the analysis of cross-classified data: reconciliation and synthesis of some methods of Pearson, Yule, and Fisher, and also some methods of correspondence analysis and association analysis. J. Amer. Stat. Assoc. 91, 408–428 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goodman, L.A., Kruskal, W.H.: Measures of association for cross-classifications. J. Amer. Stat. Assoc. 49, 732–764 (1954)zbMATHGoogle Scholar
  12. 12.
    Hirotsu, C.: Cumulative chi-squared statistic as a tool for testing goodness of fit. Biometrika 73, 165–173 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Light, R., Margolin, B.: An analysis of variance for categorical data. J. Amer. Stat. Assoc. 66(335), 534–544 (1971)Google Scholar
  14. 14.
    Lombardo, R., Beh, E.J., D’Ambra, L.: Non-symmetric correspondence analysis with ordinal variables. Comput. Stat. Data Anal. 52, 566–577 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Mood, A.M.: On the asymptotic efficiency of certain non-parametric two-sample tests. Ann. Math. Stat. 25, 514–522 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nair, V.N.: Chi-squared type tests for ordered alternatives in contingency tables. J. Amer. Stat. Assoc. 82, 283–291 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rayner, J.C.W., Best, D.J.: Smooth extensions of Pearson’s product moment correlation and Spearman’s rho. Stat. Probab. Lett. 30, 171–177 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Satterthwaite, F.E.: An approximate distribution of estimates of variance components. Biom. Bull. 2(6), 110–114 (1946)CrossRefGoogle Scholar
  19. 19.
    Taguchi, G.: Statistical Analysis. Maruzen, Tokyo (1966)Google Scholar
  20. 20.
    Taguchi, G.: A new statistical analysis for clinical data, the accumulating analysis, in contrast with the chi-square test. Saishin Igaku 29, 806–813 (1974)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Luigi D’Ambra
    • 1
    Email author
  • Pietro Amenta
    • 2
  • Antonello D’Ambra
    • 3
  1. 1.Department of Economics, Management and InstitutionsUniversity “Federico II” of NaplesNaplesItaly
  2. 2.Department of Law, Economics, Management and Quantitative MethodsUniversity of SannioBeneventoItaly
  3. 3.Department of EconomicsUniversity of Campania “L.Vanvitelli”CapuaItaly

Personalised recommendations