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The association in two-way ordinal contingency tables through global odds ratios

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Abstract

Hirotsu’s statistic is a suitable measure for studying the association between two variables on an ordinal scale. For visualizing the nature of the association, such a statistic can be decomposed by performing doubly ordered cumulative correspondence analysis. An alternative measure for describing the association between two ordered variables could be global odds ratios. In this paper we consider a generalization of the doubly ordered cumulative correspondence analysis in order to represent the global odds ratios in the two-dimensional plot.

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References

  1. Armstrong, B.G., Sloan, M.: Ordinal regression models for epidemiologic data. Am. J. Epidemiol. 129(1), 191–204 (1989)

    Article  Google Scholar 

  2. Agresti, A.: An Introduction to Categorical Data Analysis. Wiley, Hoboken (1996)

    MATH  Google Scholar 

  3. Agresti, A.: Analysis of Ordinal Categorical Data, 2nd edn. Wiley, New York (2010)

    Book  MATH  Google Scholar 

  4. Agresti, A., Coull, B.A.: The analysis of contingency tables under inequality constraints. J. Stat. Plan. Inference 107, 45–73 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aitchison, J., Greenacre, M.J.: Biplots of compositional data. Appl. Stat. 51, 375–392 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Barlow, R., Bartholomew, D., Bremner, J., Brunk, H.: Statistical Inference Under Order Restrictions. Wiley, New York (1972)

    MATH  Google Scholar 

  7. Becker, M.P., Clogg, C.C.: Analysis of sets of two-way contingency tables using association models. J. Am. Stat. Assoc. 84, 142 151 (1989)

    MathSciNet  Google Scholar 

  8. Beh, E.J.: Partitioning Pearson’s Chi-squared statistic for singly ordered two-way contingency tables. Aust. N. Zeal. J. Stat. 43(3), 327–33 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Beh, E.J., D’Ambra, L., Simonetti, B.: Cumulative correspondence analysis for ordered categorical data using Taguchi’s statistic. Commun. Stat. Theory Methods 40, 1620–1632 (2011)

    Article  MATH  Google Scholar 

  10. Beh, E.J., Lombardo, R.: Correspondence Analysis: Theory, Practice and New Strategies. Wiley, New York (2014)

    MATH  Google Scholar 

  11. Cochran, W.G.: The combination of estimates from different experiments. Biometrics 10, 101–129 (1954)

    Article  Google Scholar 

  12. Coehn, J.: The cost of dichotomization. Appl. Psychol. Meas. 7, 249–253 (1983)

    Article  Google Scholar 

  13. Cohen, J., Cohen, P.: Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences. Erlbaum, Hillsdale (1983)

    Google Scholar 

  14. Camminatiello, I., D’Ambra, A., Sarnacchiaro, P.: The association in a two-way contingency table through log odds ratio analysis: the case of Sarno river pollution. Springerplus 3, 384–492 (2014)

    Article  Google Scholar 

  15. D’Ambra, A.: Cumulative correspondence analysis using orthogonal polynomials. Commun. Stat. Theory Methods 46(6), 2942–2954 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. D’Ambra, A., Amenta, P.: Correspondence analysis with linear constraints of ordinal crossclassifications. J. Classif. 28, 70–92 (2011)

    Article  Google Scholar 

  17. D’Ambra, L., Amenta, P., D’Ambra, A.: Decomposition of cumulative chi-squared statistics, with some new tools for their interpretation. Stat. Methods Appl. 27(2), 297–318 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. D’Ambra, A., Amenta, P., Beh, E.J.: Confidence regions and other tools for an extension of correspondence analysis based on cumulative frequencies. Adv. Stat. Anal. (2020). https://doi.org/10.1007/s10182-020-00382-5

    Article  MATH  Google Scholar 

  19. D’Ambra, L., Beh, E.J., Amenta, P.: Catanova for two-way contingency tables with ordinal variables using orthogonal polynomials. Commun. Stat. Theory Methods 34(8), 1755–1969 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. D’Ambra, A., Crisci, A., D’Ambra, L.: Association models in the weighted log ratio analysis for rates. Aust. N. Zeal. J. Stat. 59(2), 169–185 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. De Rooij, M., Anderson, C.J.: Visualizing, summarizing, and comparing odds ratio structures. Methodology 3, 139–148 (2007)

    Article  Google Scholar 

  23. De Rooij, M., Heiser, W.J.: Graphical representations and odds ratios in a distance-association model for the analysis of cross-classified data. Psychometrika 70, 99–123 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Eshima, N., Tubata, M., Tsujitani, M.: Property of the RC(M) association model and a summary measure of association in the contingency table. J. Jpn. Statist. Soc. 31(1), 15–26 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Farrington, D.P., Loeber, R.: Some benefits of dichotomization in psychiatric and criminological research. Crim. Behav. Ment. Health 10, 102–122 (2000)

    Article  Google Scholar 

  26. Gilula, Z., Krieger, A.M., Ritov, Y.: Ordinal association in contingency tables: some interpretive aspects. J. Am. Stat. Assoc. 83(402), 540–545 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Greenacre, M.J.: Power transformations in correspondence analysis. Comput. Stat. Data Anal. 53, 3107–3116 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hirotsu, C.: Cumulative Chi-squared statistic as a tool for testing goodness of fit. Biometrika 73, 165–173 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hirotsu, C.: Modelling and analysing the generalized interaction. In: Proceedings Third IEEE Conference: Control Applications, vol. 2, pp. 1283–1288. ISBN-10: 0780318722 (1994)

  30. Kateri, M.: Contingency Table Analysis: Methods and Implementation Using R. Birkhauser, Basel (2014)

    Book  MATH  Google Scholar 

  31. Kruskal, W.H., Wallis, W.A.: Use of ranks in one-criterion variance analysis. J. Am. Stat. Assoc. 47, 583–621 (1952)

    Article  MATH  Google Scholar 

  32. Landis, J.R., Heyman, E.R., Koch, G.G.: Average partial association in three-way contingency tables: a review and discussion of alternative tests. Int. Stat. Rev. 46, 237–254 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lang, J.N., Iannario, M.: Improved tests of independence in singly-ordered two-way contingency tables. Comput. Stat. Data Anal. 63, 339–351 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lebart, L., Warwick, K.M., Morineau, A.: Multivariate Descriptive Statistical Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  35. Lombardo, R., Beh, E.J., D’Ambra, A.: Studying the dependence between ordinal-nominal categorical variables via orthogonal polynomials. J. Appl. Stat. 38(10), 2119–2132 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mantel, N.: Chi-square tests with one degree of freedom: extensions of the Mantel–Haenszel procedure. J. Am. Stat. Assoc. 58, 690–700 (1963)

    MathSciNet  MATH  Google Scholar 

  37. Mantel, N., Haenszel, W.: Statistical aspects of the analysis of data from retrospective studies of disease. J. Natl Cancer Inst. 22(4), 719–748 (1959)

    Google Scholar 

  38. Nair, V.N.: Chi-squared-type test for ordered alternative in contingency tables. J. Am. Stat. Assoc. 82(397), 283–291 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pawitan, Y.: Change-point problem. In: Armitage, p., Colton, T. (eds.) Encyclopedia of Biostatistics, 2nd edn. John Wiley & Sons (2005). https://doi.org/10.1002/0470011815.b2a12011

  40. Rayner, J.C.W., Best, D.J.: Analysis of singly ordered two-way contingency tables. J. Appl. Math. Decis. Sci. 4, 83–98 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sarnacchiaro, P., D’ambra, A.: Explorative data analysis and Catanova for ordinal variables: an integrated approach. J. Appl. Stat. 34(9), 1035–1050 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sarnacchiaro, P., D’Ambra, A.: Cumulative correspondence analysis to improve the public train transport. Electron. J. Appl. Stat. Anal. 2, 15–24 (2011)

    Google Scholar 

  43. Sarnacchiaro, P., D’Ambra, L., Camminatiello, I.: Measures of association and visualization of log-odds ratio for a two-way contingency table. Aust. N. Zeal. J. Stat. 57, 363–376 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. 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 

  45. Williamson, J.M., Kim, K.M., Lipsitz, S.R.: Analyzing bivariate ordinal data using a global odds ratio. J. Am. Stat. Assoc. 90, 1432–1437 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Ida Camminatiello

    Present address: University of Campania L. Vanvitelli Corso Gran Priorato di Malta, 81043, Capua, CE, Italy

Authors and Affiliations

  1. University of Campania L. Vanvitelli Corso Gran Priorato di Malta, 81043, Capua, CE, Italy

    Antonello D’Ambra

  2. University of Napoli Federico II Monte Sant’Angelo, Via Cintia, 21, 80126, Naples, Italy

    Luigi D’Ambra

Authors
  1. Ida Camminatiello
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  2. Antonello D’Ambra
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  3. Luigi D’Ambra
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Correspondence to Ida Camminatiello.

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Camminatiello, I., D’Ambra, A. & D’Ambra, L. The association in two-way ordinal contingency tables through global odds ratios. METRON 80, 9–22 (2022). https://doi.org/10.1007/s40300-021-00224-7

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