A probabilistic model for nonsymmetric correspondence analysis and prediction in contingency tables

  • Roberta Siciliano
  • Ab Mooijart
  • Peter G. M. van der Heijden


Nonsymmetric correspondence analysis is a model meant for the analysis of the dependence in a two-way continengy table, and is an alternative to correspondence analysis. Correspondence analysis is based on the decomposition of Pearson's Ф2-index

Nonsymmetric correspondence analysis is based on the decomposition of Goodman-Kruskal's τ-index for predicatablity.

In this paper, we approach nonsymmetric correspondence analysis as a statistical model based on a probability distribution. We provide algorithms for the maximum likelihood and the least-squares estimation with linear constraints upon model parameters.

The nonsymmetric correspondence analysis model has many properties that can be useful for prediction analysis in contingency tables. Predictability measures are introduced to identify the categories of the response variable that can be best predicted, as well as the categories of the explanatory variable having the highest predictability power. We describe the interpretation of model parameters in two examples. In the end, we discuss the relations of nonsymmetric correspondence analysis with other reduced-rank models.


Contingency Table Correspondence Analysis American Statistical Association Latent Class Analysis Latent Class Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Agresti, A. (1984).Analysis of ordinals categorical data. New York: Wiley.Google Scholar
  2. Agresti, A. (1990).Categorical Data Analysis. New York: Wiley.zbMATHGoogle Scholar
  3. Böckenholt, U. andBöckenholt, I. (1990). Canonical analysis of contingency table with linear constraints.Psychometrika, 55, 633–639.CrossRefGoogle Scholar
  4. Breiger, R. L. (1981) The social class structure of occupational mobility.American Journal of Sociology, 87, 578–611.CrossRefGoogle Scholar
  5. Clogg, C. C. (1982). Some models for the analysis of association in multiway cross-classifications having ordered categories.Journal of the American Statistical Associiation, 79, 762–771.CrossRefMathSciNetGoogle Scholar
  6. D'Ambra, L. andLauro, N. C. (1989). Nonsymmetrical analysis of three-way contingency tables.Multiway data analysis (eds. R. Coppi and S. Bolasco), 301–315. Amsterdam: North Holland.Google Scholar
  7. D'Ambra, L. andLauro, N. C. (1992) Non symmetrical exploratory data analysis.Statistica Applicata.Italian Journal of Applied Statistics 4, 4, Napoli: Curto.Google Scholar
  8. de Leeuw, J. andvan der Heiden, P. G. M. (1991) Reduced-rank models for contigency tables.Biometrika, 78, 239–242.CrossRefGoogle Scholar
  9. Escoufier, Y. (1988). Beyond correspondence analysis.Classification and related methods of data analysis (ed. H. H. Bock) 505–514. Amsterdam: North Holland.Google Scholar
  10. Gabriel, K. R. (1971). The biplot-graphic display of matrices with applications to principal component analysis.Biometrika, 58, 453–467.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Gilula, Z. (1986). Grouping and association in contingency table: an exploratory canonical correlation approach.Journal of the American Statistical Association, 81, 773–779.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Gilula, Z. andHaberman, S. J. (1986). Canonical analysis of contingency tables by maximum likelihood.Journal of the American Statistical Association, 81, 780–788.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Gilula, Z. andHaberman, S. J. (1988). The analysis of multivariate contingency tables by restricted canonical and restricted association models.Journal of the American Statistical Association, 83, 760–771.zbMATHCrossRefMathSciNetGoogle Scholar
  14. Gilula, Z andKrieger, A. M. (1989). Collapsed two-way contingency tables and the chi-square reduction principle.Journal of the Royal Statistical Society, Series B, 51, 425–433.zbMATHMathSciNetGoogle Scholar
  15. Gini, C. (1912) Variabilità e mutabilità. Contributi allo studio dele relazioni e delle distribuzioni statistiche.Studi Economico-Giuridici della Università di Cagliari.Google Scholar
  16. Good, I. J. (1969). Some applications of the singular value decomposition of a matrix.Technometrics 11, 823–831.zbMATHCrossRefGoogle Scholar
  17. Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories.Journal of the American Statistical Association, 70, 755–768.CrossRefGoogle Scholar
  18. Goodman, L. A. (1981) Criteria for determining whether certain categories in a cross-classification table should be combined with special reference to occupational categories in occupational mobility tables.American Journal of Sociology, 87, 612–650.CrossRefGoogle Scholar
  19. Goodman, L. A. (1985) The analysis of cross-classified data having ordered and/or unordered categories: association modes, correlation models and asymmetry models for contingency tables with or without missing entries.The Annals of Statistics, 13, 10–69.zbMATHMathSciNetGoogle Scholar
  20. Goodman, L. A. (1986) Some useful extensions to the usual correspondence analysis approach and the usual loglinear approach in the analysis of contingency tables (with comments).International Statistical Review, 54, 243–309.zbMATHMathSciNetCrossRefGoogle Scholar
  21. Goodman, L. A. (1987) New methods for analyzing the intrinsic character of qualitative variables using cross-classified data.American Journal of Sociology, 93, 529–583.CrossRefGoogle Scholar
  22. Goodman, L. A. (1991) Models, measures, and graphical displays in the analysis of contingency tables (with discussion).Journal of the American Statistical Association, 86, 1085–1138.zbMATHCrossRefMathSciNetGoogle Scholar
  23. Goodman, L. A. andKruskal, W. H. (1954) Measures of association for cross-classifications.Journal of the American Statistical Association, 49, 732–764.zbMATHCrossRefGoogle Scholar
  24. Guttman, L. (1971) Measurement as structural theory.Psychometrika, 36, 329–347.CrossRefGoogle Scholar
  25. Greenacre, M. J. (1984)Theory and applications of corespondence analysis. New York: Academic Press.Google Scholar
  26. Hildebrand, D. K., Laing, J. D. andRosenthal, H. (1977).Prediction analysis of cross-classifications. New York: Wiley.zbMATHGoogle Scholar
  27. Kendall, M. G. andStuart, A. (1979).The advanced theory of statistics, Vol. 2 (4th Ed.), New York: Hafner.zbMATHGoogle Scholar
  28. Lauro, N. C. andD'Ambra, L. (1984) L'analyse non symmetrique des correspondances.Data analysis and informatics III (eds. E. Diday et al.), 433–446. Amsterdam: North Holland.Google Scholar
  29. Lauro, N. C. andSiciliano, R. (1989) Exploratory methods and modelling for contingency tables analysis: an integrated approach.Statistica Applicata. Italian Journal of Applied Statiscs, 1, 5–32.Google Scholar
  30. Light, R. J. andMargolin, B. H. (1971) An analysis of variance for categorical data.Journal of the American Statistical Association, 66, 534–544.zbMATHCrossRefMathSciNetGoogle Scholar
  31. Margolin, B. H. andLight, R. J. (1974) An analysis of variance for categorical data, II: small sample comparisons with chi square and other competitors.Journal of the American Statistical Association, 69, 755–764.zbMATHCrossRefMathSciNetGoogle Scholar
  32. Nishisato, S. (1980)Analysis of categorical data: dual scaling and its applications. Toronto: University of Toronto Press.zbMATHGoogle Scholar
  33. Siciliano R. (1992). Reduced-rank models for dependence analysis of contingency tables.Statistica applicata Italian Journal of Applied Statistics, 4, 4, Napoli, Curto.Google Scholar
  34. Siciliano R. andMooijaart, A. (1991)An algorithm for maximum likelihood estimation of generalized reduced-rank models for contingency table analysis. Technical Report-PRM 04-91, University of Leiden.Google Scholar
  35. Siciliano, R., Lauro N. C. andMooijaart, A. (1990) Exploratory approach and maximum likelihood estimation of models for nonsymmetrical analysis of two-way multiple contingency tables.Compstat ′90, 157–162.Google Scholar
  36. Siciliano, R. Mooijaart, A. andvan der Heijden, P. G. M. (1990)Non-symmetric correspondence analysis by maximum likelihood. Technical Report-PRM 05-90, University of Leiden.Google Scholar
  37. Takane, Y., Yanai, H. andMayekawa, S. (1991) Relationships among several methods of linearly constrained correspondence analysis.Psychometrika, in press.Google Scholar
  38. van der Heijden, P. G. M., Mooijaart, A. andde Leeuw, J. (1989) Latent budget analysis.Statistical Modelling (eds. A. de Carli et al.). Berlin: Springer Verlag.Google Scholar
  39. Wasserman, S. andFaust, K. (1989) Canonical analysis of the composition and structure of social networks.Sociological methodology 1989 (ed. C. C. Clogg), 1–42. London, Basil Blackwell.Google Scholar

Copyright information

© Società Italiana di Statistica 1993

Authors and Affiliations

  • Roberta Siciliano
    • 3
  • Ab Mooijart
    • 1
  • Peter G. M. van der Heijden
    • 2
  1. 1.Leiden UniversityThe Netherlands
  2. 2.University of UtrechtThe Netherlands
  3. 3.Dipartimento di Matematica e StatisticaUniversità di Napoli Federico IINapoliItaly

Personalised recommendations