Abstract
Loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables. In this paper we will show that there are cases in which these two techniques can be used complementary to each other. More specifically, we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices, each following a specific loglinear model. Therefore, in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models. A generalization of correspondence analysis, recently proposed by Escofier, will also be discussed. With this decomposition, which includes classical correspondence analysis as a special case, it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis. In this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Agresti, A. (1983). A survey of strategies for modeling cross-classifications having ordinal variables.Journal of the American Statistical Association, 78, pp. 184–198.
Andersen, E. B. (1980).Discrete statistical models with social science applications. Amsterdam: North Holland.
Benzécri, J. P. (1973).Analyse des données [The analysis of Data] (2 vols.). Paris: Dunod.
Benzécri, J. P. (1980).Pratique de l'analyse des données [The Practice of Data Analysis] (3 vols.). Paris: Dunod.
Bishop, Y. M., Y., Fienberg, S. E., & Holland, P. W. (1975).Discrete multivariate analysis: Theory and practice. Cambridge: MIT press.
Bonett, D. G., & Bentler, P. M. (1983). Goodness-of-fit procedures for the evaluation and selection of loglinear models.Psychological Bulletin, 93(1), pp. 149–166.
Daudin, J. J., & Trécourt, P. (1980). Analyse factorielle des correspondances et modèle log-lineaire: Comparaison des deux methodes sur un exemple [Correspondence analysis and the loglinear model: Comparison of two methods on an example].Revue de Statistique Appliquée, 28(1), 5–24.
de Leeuw, J. (1971).Canonical analysis of contingency tables (RB-002-71). Leiden: R.U.L./F.S.W., Department of Datatheory.
de Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients.Journal of Econometrics, 22, 113–128.
de Leeuw, J. (1984).Canonical analysis of categorical data. (Doctoral dissertation, University of Leiden, 1973). Leiden: DSWO-press.
Diekstra, R. F. W. (1981).Over suicide [about suicide]. Alphen a/d Rijn: Samson.
Dixon, W. J. (Ed.). (1981).BMDP statistical software 1981. Berkeley: University of California Press.
Douglas, J. D. (1973).The social meanings of suicide. Princeton: Princeton University Press.
Escofier, B. (1983). Analyse de la difference entre deux mesures sur le produit de deux mêmes ensembles [Analysis of the difference between two measures on the product of two identical sets].Cahiers de l'Analyse des Données, 3, 325–329.
Fienberg, S. E. (1980).The analysis of cross-classified categorical data (2nd ed.). Cambridge: MIT Press.
Fienberg, S. E., & Meyer, M. M. (1983). Loglinear models and categorical data analysis with psychometric and econometric applications.Journal of Econometrics, 22, 191–214.
Gifi, A. (1981).Non-linear multivariate analysis. Leiden: R.U.L./F.S.W., Department of Datatheory. (2nd. ed. in press, Leiden: DSWO press).
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.
Goodman, L. A. (1981a). Association models and the bivariate normal for contingency tables with ordered categories.Biometrika, 68, 347–355.
Goodman, L. A. (1981b). Association models and canonical correlation in the analysis of cross-classifications having ordered categories.Journal of the American Statistical Association, 76, 320–334.
Greenacre, M. J. (1984).Theory and applications of correspondence analysis. London: Academic Press.
Heiser, W. J., & Meulman, J. (1983). Analyzing rectangular tables by joint and constrained multi-dimensional scaling.Journal of Econometrics, 22, 139–167.
Heuer, J. (1979).Selbstmord bei Kinder und Jugendlichen [Suicide of children and youth]. Stuttgard, Ernst Klett Verlag.
Israëls, A. Z., & Sikkel, D. (1982).Correspondence analysis and comparisons with other techniques. Voorburg: Centraal Bureau voor Statistiek.
Jacobs, L. (1971)Adolescent suicide. New York: Wiley.
Kendall, M. G., & Stuart, A. (1973).The advanced theory of statistics, Vol. 2, 3rd. ed. London: Griffin.
Kroonenberg, P. M. (1983).Three-mode principal component analysis. Leiden: DSWO-press.
Leclerc, A. (1975). L'analyse des correspondances sur juxtaposition de tableaux de contingence [Correspondence analysis of concatenated contingency tables].Revue de Statistique Appliquée, 23(3), 5–16.
Nishisato, S. (1980).Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press.
Reynolds, H. T. (1977).The analysis of cross-classifications. Glencoe, IL: The Free Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van der Heijden, P.G.M., de Leeuw, J. Correspondence analysis used complementary to loglinear analysis. Psychometrika 50, 429–447 (1985). https://doi.org/10.1007/BF02296262
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02296262