Abstract
Correspondence analysis can be described as a technique which decomposes the departure from independence in a two-way contingency table. In this paper a form of correspondence analysis is proposed which decomposes the departure from the quasi-independence model. This form seems to be a good alternative to ordinary correspondence analysis in cases where the use of the latter is either impossible or not recommended, for example, in case of missing data or structural zeros. It is shown that Nora's reconstitution of order zero, a procedure well-known in the French literature, is formally identical to our correspondence analysis of incomplete tables. Therefore, reconstitution of order zero can also be interpreted as providing a decomposition of the residuals from the quasi-independence model. Furthermore, correspondence analysis of incomplete tables can be performed using existing programs for ordinary correspondence analysis.
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Baccini, A. (1984)Etude comparative des représentations graphiques en analyses factorielle des correspondances simples et multiples [A comparative study of graphical representations in simple and multiple correspondance analysis]. Toulouse: Laboratoire de Statistique et Probabilités. Université Paul Sabatier.
Benzécri, J. P. et al. (1973).L'analyse des données [Data analysis] (2 vols.). Paris: Dunod.
Benzécri, J. P. et al. (1980).Pratique de l'analyse des données [Practice of data analysis] (3 vols.). Paris: Dunod.
Bishop, Y. M. M., & Fienberg, S. E. (1969). Incomplete two-dimensional contingency tables.Biometrics, 25, 119–128.
Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975).Discrete multivariate analysis: Theory and practice. Cambridge: MIT-press.
Burtchy, B. (1984). Analyse factorielle des matrices d'échanges [Factor analysis of exchange matrices]. In E. Diday, M. Jambu, L. Lebart, J. Pagès, & R. Tomassone (Eds.),Data analysis and informatics, III. Amsterdam: North-Holland.
Caussinus, H. (1965). Contribution à l'analyse de la corrélation de deux caractères qualitatifs [Contribution to correlation analysis of two qualitative variables].Annales de la Faculté des Sciences de l'Université de Toulouse, 29, 77–182.
Caussinus, H., & de Falguerolles, A. (1986). Modèle de quasi-symétrie et analyse descriptive de tableaux carrés [The quasi-symmetrie model and the descriptive analysis of square tables].Publications du Laboratoire de Statistique et Probabilité, No. 02-86. Toulouse: Université Paul Sabatier
Daudin, J. J., & Trécourt, P. (1980). Analyse factorielle des correspondances et modéle log-lineaire: comparaison des deux méthodes sur un exemple [Correspondence analysis and the loglinear model: A comparison of the two methods using an example].Revue de Statistique Appliquée, 1, 5–24.
Deville, J.-C., & Malinvaud, E. (1983). Data analysis in official socio-economic statistics.Journal of the Royal Statistical Society, Series A, 146, 335–361.
Escofier, B. (1984). Analyse factorielle en reference á un modèle; application à l'analyse de tableaux d'échanges [Factorial analysis related to a model: Application to the analysis of exchange tables].Revue de Statistique Appliquée, 32(4), 25–36.
Foucart, T. (1985) Tableux symmeétriques et tableaux d'échanges [Symmetric tables and exchange tables].Revue de Statistique Appliquée, 33(2), 37–54.
Gabriel, K. R. (1971). The biplot-graphic display of matrices with application to principal component analysis.Biometrika, 58, 453–467.
Goodman, L. A. (1968). The analysis of cross-classified data: Independence, quasi-independence, and interactions in contingency tables with or without missing entries.Journal of American Statistical Association, 63, 1091–1131.
Goodman, L. A. (1985). The 1983 Henry L. Rietz memorial lecture. 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.
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.
Greenacre, M. J. (1984).Theory and applications of correspondence analysis. London: Academic Press.
Haberman, S. J. (1973). The analysis of residuals in cross-classified tables.Biometrics, 29, 205–220.
Israëls, A. Z., & Sikkel, D. (1982).Correspondence analysis and comparisons with other techniques. Voorburg: Centraal Bureau voor Statistiek.
Kendall, D. G., & Stuart, A. (1967).The advanced theory of statistics (Vol. 2, 2nd. ed.). London: Griffin.
Lauro, N. C., & Decarli, A. (1982). Correspondence analysis and log-linear models in multiway contingency tables study. Some remarks on experimental data.Metron (Rivista internazionale di statistica), 15(1,2), 213–234.
Mosteller, F. (1968). Association and estimation in contingency tables.Journal of American Statistical Association, 63, 1–28.
Nora, C. (1975).Une méthode de reconstitution et d'analyse de données incomplétes [A method for reconstitution and for the analysis of incomplete data]. Unpublished Thèse d'Etat, Université P. et M. Curie, Paris VI'.
Tenenhaus, M., & Young, F. W. (1985). An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis and other methods for quantifying categorical multivariate data.Psychometrika, 50, 91–119.
van der Heijden, P. G. M. (1987).Correspondence analysis of longitudinal categorical data. Leiden: D.S.W.O.-Press.
van der Heijden, P. G. M., & de Leeuw, J. (1985). Correspondence analysis used complementary to loglinear analysis.Psychometrika, 50, 429–447.
van der Heijden, P. G. M., & Worsley, K. J. (1988). Comment on “Correspondence analysis used complementary to loglinear analysis”.Psychometrika, 53, 287–291.
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de Leeuw, J., van der Heijden, P.G.M. Correspondence analysis of incomplete contingency tables. Psychometrika 53, 223–233 (1988). https://doi.org/10.1007/BF02294134
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DOI: https://doi.org/10.1007/BF02294134