Hostname: page-component-5f745c7db-nzk4m Total loading time: 0 Render date: 2025-01-06T05:52:14.914Z Has data issue: true hasContentIssue false
  • English
  • Français

Correspondence Analysis and Association Models Constrained by a Conditional Independence Graph

Published online by Cambridge University Press:  01 January 2025

Antoine de Falguerolles ,
Said Jmel  and
Joe Whittaker
Antoine de Falguerolles
Affiliation:
Universitë Paul Sabatier, Toulouse
Said Jmel
Affiliation:
Université Nancy II, Nancy
Joe Whittaker*
Affiliation:
Lancaster University
*
Requests for reprints should be sent to Joe Whittaker, Mathematics and Statistics Department, Lancaster University, LAI 4YF UNITED KINGDOM. E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the Goodman association model (RC) and of correspondence analysis (CA) is considered.

Estimation of the row and column scores is presented in this context by developing a unified framework that includes both models. Incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables, depending on the topology of the graph. An example of doubly additive scores in the analysis of a Burt subtable is given.

Keywords

RC-association modelcollapsibilityconditional independencecorrespondence analysisgraphical modelindependence graphlog-linear modelmultiway contingency tablescore additivity
Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 2 , June 1995 , pp. 161 - 180
Copyright
Copyright © 1995 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Thanks are due to anonymous referees who substantially improved the original draft of this paper.

References

Agresti, A. (1990). Categorical data analysis, New York: Wiley.Google Scholar
Andersen, E. B. (1990). The statistical analysis of categorical data, Berlin: Springer Verlag.CrossRefGoogle Scholar
Asmussen, S., Edwards, D. (1983). Collapsibility and response variables in contingency tables. Biometrika, 70, 567–578.CrossRefGoogle Scholar
Baccini, A., Caussinus, H., de Falguerolles, A. (1991). RC models in exploratory data analysis (discussion contribution to Goodman's paper). Journal of the American Statistical Association, 86, 1115–1117.Google Scholar
Baccini, A., Caussinus, H., de Falguerolles, A. (1993). Analysing dependence in large contingency tables: Dimensionality and patterns in scatter-plots. In Cuadras, C. M., Rao, C. R. (Eds.), Multivariate analysis: Future directions 2 (pp. 245–263). Amsterdam: North-Holland.Google Scholar
Baccini, A., Khoudraji, A. (1992). A least squares procedure for estimating the parameters in an association model. Computational Statistics, 7, 287–300.Google Scholar
Becker, M. P. (1990). Maximum likelihood estimation of the RC(M) association model. Applied Statistics, 39, 152–167.CrossRefGoogle Scholar
Becker, M. P., Clogg, C. C. (1989). Analysis of sets two-way contingency tables using association models. Journal of the American Statistical Association, 84, 142–151.CrossRefGoogle Scholar
Benzécri, J. P. (1973). L'analyse des correspondances: Vol. 2. L'analyse des données [Correspondence analysis. Vol. 2. Data analysis], Paris: Dunod.Google Scholar
Bishop, Y. M., Fienberg, S., Holland, P. (1975). Discrete multivariate analysis, Cambridge, MA: M.I.T. Press.Google Scholar
Böckenholt, U., Böckenholt, I. (1990). Canonical analysis of contingency tables with linear constraints. Psychometrika, 55, 633–639.CrossRefGoogle Scholar
Caussinus, H., de Falguerolles, A. (1987). Tableaux carrés: modélisation et méthodes factorielles [Squares Tables: Models and methods]. Revue de Statistique Appliquée, 35, 35–52.Google Scholar
Darroch, J. N., Lauritzen, S. L., Speed, T. P. (1980). Markov field and log-linear interaction models for contingency tables. The Annals of Statistics, 8, 522–539.CrossRefGoogle Scholar
de Falguerolles, A., Francis, B. (1992). Algorithmic approaches for fitting bilinear models. In Dodge, Y., Whittaker, J. (Eds.), Computational Statistics I: Proceedings of Compstat92 (pp. 77–82). Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
de Leeuw, J., van der Heijden, P. G. M. (1991). Reduced rank models for contingency tables. Biometrika, 78, 229–232.CrossRefGoogle Scholar
Escoufier, Y. (1988). Beyond correspondence analysis. In Bock, H. H. (Eds.), Classification and related methods of data analysis, Amsterdam: North-Holland.Google Scholar
Escoufier, Y., Junca, S. (1986). Least-squares approximation of frequencies and their logarithms (discussion contribution to Goodman's paper). International Statistical Review, 54, 279–283.CrossRefGoogle Scholar
Gabriel, K. R. (1971). The bi-plot graphic display of matrices with application to principal component analysis. Biometrika, 58, 453–467.CrossRefGoogle Scholar
Gilula, Z. (1986). Grouping and association in contingency tables: An exploratory canonical correlation approach. Journal of the American Statistical Association, 81, 773–779.CrossRefGoogle Scholar
Gilula, Z., Haberman, S. J. (1986). Canonical analysis of contingency tables by maximum likelihood. Journal of the American Statistical Association, 81, 780–788.CrossRefGoogle Scholar
Gilula, Z., Haberman, 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.CrossRefGoogle Scholar
Glonek, G. F. V., Darroch, J. N., Speed, T. P. (1988). On the existence of maximum likelihood estimators for hierarchical log-linear models. Scandinavian Journal of Statistics, 15, 187–193.Google Scholar
Goodman, L. A. (1970). The multivariate analysis of qualitative data: Interaction among multiple classifications. Journal of the American Statistical Association, 65, 226–256.CrossRefGoogle Scholar
Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74, 537–552.CrossRefGoogle Scholar
Goodman, L. A. (1981). Criteria for determining whether certain categories in cross-classification tables should be combined. American Journal of Sociology, 87, 612–650.CrossRefGoogle Scholar
Goodman, L. A. (1985). The analysis of cross-classified data having ordered and/or unordered categories: Association models, correlation models, and asymmetry models for contingency tables with or without missing entries. The Annals of Statistics, 13, 10–69.CrossRefGoogle Scholar
Goodman, L. A. (1986). Some useful extensions of the usual correspondence analysis approach and the usual log-linear models approach in the analysis of contingency tables. International Statistical Review, 54, 243–309.CrossRefGoogle Scholar
Goodman, L. A. (1991). Measures, models and graphical displays in the analysis of cross-classified data. Journal of the American Statistical Association, 86, 1085–1110.CrossRefGoogle Scholar
Greenacre, M. J. (1984). Theory and application of correspondence analysis, London: Academic Press.Google Scholar
Haberman, S. J. (1979). Analysis of qualitative data: Volume 2, New developments, New York: Academic Press.Google Scholar
Israëls, A. (1987). Eigenvalue techniques for qualitative data, Leiden, The Netherlands: DSWO Press.Google Scholar
Leclerc, A. (1975). L'analyse des correspondances sur juxtaposition de tableaux de contingence [Correspondence analysis on juxaposed contingency tables]. Revue de Statistique Appliquée, 23, 5–16.Google Scholar
Siciliano, S., Mooijaart, A., van der Heijden, P. G. M. (1990). Non-symmetric correspondence analysis by maximum likelihood (PRM 05-90, Leiden Psychological Report), Leiden: University of Leiden.Google Scholar
Takane, Y., Yanai, H., Mayekawa, S. (1991). Relationships among several methods of linearly constrained correspondence analysis. Psychometrika, 56, 667–684.CrossRefGoogle Scholar
Taylor, W. F. (1953). Distance functions and regular best asymptotically normal estimates. The Annals of Mathematical Statistics, 24, 85–92.CrossRefGoogle Scholar
van der Heijden, P. G. M. (1987). Correspondence analysis of longitudinal categorical data, Leiden, The Netherlands: DSWO Press.Google Scholar
van der Heijden, P. G. M., de Falguerolles, A., de Leeuw, J. (1989). A combined approach to contingency table analysis using correspondence and loglinear analysis (with discussion). Applied Statistics, 38, 249–292.CrossRefGoogle Scholar
Whittaker, J. (1990). Graphical models in applied multivariate statistics, Chichester: Wiley.Google Scholar