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Exploratory Analysis of Contingency Tables by Loglinear Formulation and Generalizations of Correspondence Analysis

Published online by Cambridge University Press:  01 January 2025

Vartan Choulakian
Vartan Choulakian*
Affiliation:
Université de Moncton, Moncton, New Brunswick
*
Requests for reprints should be sent to Vartan Choulakian, Département de mathématiques, physique et informatique, Université de Moncton, Moncton, N. B., E1A 3E9 CANADA.
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Abstract

Goodman's (1979, 1981, 1985) loglinear formulation for bi-way contingency tables is extended to tables with or without missing cells and is used for exploratory purposes. A similar formulation is done for three-way tables and generalizations of correspondence analysis are deduced. A generalized version of Goodman's algorithm, based on Newton's elementary unidimensional method is used to estimate the scores in all cases.

Keywords

exploratory data analysiscontingency tablesmissing cellsloglinear formulationinteractionsgeneralized correspondence analysisNewton's algorithmmutual information
Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 2 , June 1988 , pp. 235 - 250
Copyright
Copyright © 1988 The Psychometric Society

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Footnotes

This research was partially supported by National Science and Engineering Research Council of Canada, Grant No. A8724. The author is grateful to the reviewers and the editor for helpful comments.

References

Bartlett, M. S. (1935). Contingency table interactions. Journal of Royal Statistical Society, Supplement 2, 248–252.CrossRefGoogle Scholar
Bahadur, R. R. (1961). A representation of the joint distribution of responses ton dichotomous items. In Solomon, H. (Eds.), Studies in item analysis and prediction (pp. 158–168). Stanford, CA: Stanford University Press.Google Scholar
Bener, A. (1982). Decomposition des interactions dans dans une correspondance multiple [Decomposition of interactions in a multi-way table]. Les Cahiers de l'Analyse des Données, 7, 25–32.Google Scholar
Benzécri, J. P. (1973). L'Analyse des Données: Tome 2, L'Analyse des Correspondances [Data Analysis: Vol. II, Correspondence analysis], Paris: Dunod.Google Scholar
Bishop, Y. M. M., Fienberg, S. E., Holland, P. W. (1975). Discrete Multivariate Analysis, Cambridge, MA: MIT Press.Google Scholar
Carroll, J. D., Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via anN-way generalization of “Eckart-Young” decomposition. Psychometrika, 35, 283–320.CrossRefGoogle Scholar
Caussinus, H. (1965). Contribution à l'analyse de la corrélation de deux caractères qualitatifs [Contribution to the correlation analysis of two qualitative variables]. Annales de la Faculté des sciences de Toulouse, 29, 77–182.Google Scholar
Ĉencov, N. N. (1982). Statistical decision rules and optimal inference, Vol. 53, Providence, RI: American Mathematical Society.Google Scholar
Choulakian, V. (1986) Analysis of multiway categorical tables by simple correspondence analysis. Manuscript submitted for publication.Google Scholar
Csiszar, I. (1975). I-Divergence geometry of probability distributions and minimization problems. The Annals of Probability, 3(1), 146–158.CrossRefGoogle Scholar
Darroch, J. N. (1974). Multiplicative and additive interaction in contingency tables. Biometrika, 61, 207–214.CrossRefGoogle Scholar
Daudin, J. J., Trécourt, P. (1980). Analyse Factorielle de correspondance et modèle log-linéaire: comparaison de deux méthodes sur exemple [Correspondence analysis and loglinear model: Comparison of both methods on an example]. Revue de Statistique Appliquée, 28, 5–25.Google Scholar
de Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients. Journal of Econometrics, 22, 113–117.CrossRefGoogle Scholar
Foucart, T. (1984). Analyse Factorielle de tableaux multiples [Factorial analysis of multiway tables], Paris: Masson.Google Scholar
Goodman, L. (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.Google Scholar
Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of American Statistical Association, 74, 537–552.CrossRefGoogle Scholar
Goodman, L. A. (1981). Association models and canonical correlation in the analysis of cross-classifications having ordered categories. Journal of American Statistical Association, 76, 320–334.Google 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. Annals of Statistics, 13(1), 10–69.CrossRefGoogle Scholar
Greenacre, M. J. (1984). Theory and applications of correspondence analysis, London: Academic Press.Google Scholar
Harshman, R. A., Ladefoged, P., Goldstein, L. (1977). Factor analysis of tongue shapes. Journal of Acoustical Society of America, 62, 693–707.CrossRefGoogle ScholarPubMed
Holland, P. W., Thayer, D. T. (1983). Using modern statistical methods to improve the presentation of the annual summaries of GRE candidate background data, Princeton, NJ: Educational Testing Service.CrossRefGoogle Scholar
Kruskal, J. B. (1983). Multilinear methods. Proceedings of Symposia in Applied Mathematics, 28, 75–103.CrossRefGoogle Scholar
Ku, H. H., Kullback, S. (1968). Interaction in multidimensional contingency tables: An information theoretic approach. Journal Research National Bureau of Standards, 728, 159–199.Google Scholar
Lancaster, H. O. (1951). Complex contingency tables treated by the partition ofx 2. Journal of Royal Statistical Society, Series B, 13, 242–249.CrossRefGoogle Scholar
Lancaster, H. O. (1969). The chi-squared distribution, New York: Wiley.Google Scholar
Lancaster, H. O. (1975). The multiplicative definition of interaction. Australian Journal of Statistics, 13, 36–44.CrossRefGoogle Scholar
Lancaster, H. O. (1975). Joint probability distributions in the Meixner classes. Journal of Royal Statistical Society, Series B, 37, 434–443.CrossRefGoogle Scholar
Lauro, N. C., Decarli, A. (1982). Correspondence analysis and log-linear models in multi-way contingency tables study: Some remarks on experimental data. Metron, 40, 21–234.Google Scholar
Lebart, L., Morineau, A., Warwick, K. M. (1984). Multivariate descriptive analysis: Correspondence analysis and related techniques for large matrices, New York: Wiley-Interscience.Google Scholar
Leclerc, A., Chevalier, A., Luce, D., Blanc, M. (1985). Analyses des correspondances et modèle logistique: Possibilités et intérêts d'approches complémentaires [Correspondence analysis and logistic model: Possibilities for complementary approaches]. Revue de Statistique Appliquée, 33(1), 25–40 Google Scholar
Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications, Toronto: University of Toronto Press.CrossRefGoogle Scholar
Roy, S. N., Kastenbaum, M. A. (1956). On the hypothesis of “no interaction” in a multi-way contingency table. Annals of Mathematical Statistics, 27, 749–757.CrossRefGoogle Scholar
Spring, D. (1985). On the second derivative test for constrained local maxima. The American Mathematical Monthly, 92, 631–643.CrossRefGoogle Scholar
Tenenhaus, M., Young, F. (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.CrossRefGoogle Scholar
van der Heijden, P. G. M., de Leeuw, J. (1985). Correspondence analysis used complementary to loglinear analysis. Psychometrika, 50, 429–447.CrossRefGoogle Scholar

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Vartan Choulakian