Abstract
Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akaike, H. (1974). A new look at the statistical model identification.IEEE Transactions on Automatic Control, 19, 716–723.
Andersen, E. B. (1980).Discrete statistical models with social science applications. Amsterdam: North-Holland.
Anderson, J. A. (1982). Logistic discrimination. In P. R. Krishnaiah, & L. N. Kanal (Eds.),Handbook of statistics 2. Amsterdam: North Holland.
Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975).Discrete multivariate analysis: Theory and practice. Cambridge, MA: The MIT Press.
Bradley, R. A., & Terry, M. E. (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons.Biometrika, 39, 324–345.
Carroll, J. D. (1973). Categorical conjoint measurement. In P. E. Green & Y. Wind, (Eds.),Multiattribute decisions in marketing: A measurement approach. Hinsdale IL: Dryden Press.
Carroll, J. D., & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via anN-way generalization of “Eckart-Young” decomposition.Psychometrika, 35, 238–319.
Caussinus, H. (1965). Contribution a l'analyse statistique tableau de correlation [Contribution to Statistical Analysis of Correlation Table].Annals of the Faculty of Science University, Toulouse,29, 77–182.
Caussinus, H. (1986). Discussion of paper by L. A. Goodman.International Statistical Review, 54, 274–278.
Colonius, H: (1981). A new interpretation of stochastic test models.Psychometrika, 46, 223–225.
Coomb, C. H. (1964).A theory of data. New York: Wiley.
Cramer, H. (1946).Mathematical methods of statistics. Princeton, NJ: Princeton University Press.
de Leeuw, J. (1984). Beyond homogeneity analysis (RR-84-08). Leiden: University of Leiden, Department of Data Theory.
Efron, B. (1986). Double exponential families and their use in generalized linear regression.Journal of the American Statistical Association, 81, 709–721.
Escoufier, Y. (1987, June). In the neighborhood of correspondence analysis. Paper presented at the first IFCS Conference, Aachen.
Fisher, R. A. (1940). The precision of discriminant functions.Annals of Eugenics, 10, 422–429.
Fisher, R. A. (1948).Statistical methods for research workers (10th ed.). London: Oliver and Boyd.
Gilula, Z., & Haberman, S. J. (1986). Canonical analysis of contingency tables by maximum likelihood.Journal of the American Statistical Association, 81, 780–788.
Goodman, L. A. (1981). Association models and canonical correlation in the analysis of cross-classifications having ordered categories.Journal of the American Statistical Association, 76, 320–334.
Goodman, L. A. (1985). New methods for the analysis of two-way contingency tables: An alternative to Diaconis and Efron.The Annals of Statistics, 13, 887–893.
Goodman, L. A. (1986). Some useful extensions of the usual correspondence analysis and the usual log-linear models approach in the analysis of contingency tables.International Statistical Review, 54, 243–309.
Green, D. M., & Swets, J. A. (1966).Signal detection theory and psychophysics. New York: Wiley.
Greenacre M. (1984).Theory and applications of correspondence analysis. London: Academic Press.
Guilford, J. P. (1954).Psychometric methods (2nd ed.) New York: McGraw Hill.
Hayashi, C. (1952). On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view.Annals of the Institute of Statistical Mathematics, 2, 69–98.
Heiser, W. J. (1981),Unfolding analysis of proximity data. Unpublished Doctoral Dissertation, University of Leiden.
Heiser, W. J. (1987). Joint ordination of species and sites: the unfolding technique. In P. Legendre, & L. Legendre (Eds.),Developments in numerical ecology (pp. 189–221). Berlin: Springer.
Ihm, P., & van Groenewoud, H. (1975). A multivariate ordering of vegetation data based on gaussian type gradiant response curves.Journal of Ecology, 63, 767–778.
Ihm, P., & van Groenewoud, H. (1984). Correspondence analysis and Gaussian ordination.COMPST AT Lectures, 3, 5–60.
Johnson, P. O. (1950). The quantification of qualitative data in discriminant analysis.Journal of the American Statistical Association, 45, 65–76.
Jorgensen, B. (in press). Exponential dispersion models.Journal of the Royal Statistical Society, Series B.
Krippendorff, K. (1986).Information theory: Structural models for qualitative data. Beverley Hills, CA: Sage Publications. (Sage University Paper Series on Quantitative Applications in the Social Sciences.)
Landwehr, J., Pregibon, D., & Shoemaker, A. (1984). Graphical methods for assessing logistic regression models.Journal of the American Statistical Association, 79, 61–71. (With discussions)
Luce, R. D. (1959).Individual choice behavior: A theoretical analysis. New York: Wiley.
Maxwell, A. E. (1961). Canonical variate analysis when the variables are dichotomous.Educational and Psychological Measurement, 21, 259–271.
McCullagh, P., & Nelder, J. A. (1983).Generalized linear models. London: Chapman and Hall.
McGill, W. J. (1954). Multivariate information transmission.Psychometrika, 19, 97–116.
Nishisato, S. (1980).Analysis of categorical data: dual scaling and its applications. Toronto: University of Toronto Press.
Nishisato, S. (1982).Shitsu teki deta no suryoka [Quantification of qualitative data]. Tokyo: Asakura Shoten. (In Japanese)
Pregibon, D. (1981). Logistic regression diagnostics.The Annals of Statistics, 9, 705–724.
Ramsay, J. O. (1978). Confidence regions for multidimensional scaling analysis.Psychometrika, 43, 145–160.
Rubin, D. B. (1984). Assessing the fit of logistic regressions using the implied discriminant analysis. (Comment on Landwahr,et al.'s, paper).Journal of the American Statistical Association, 79, 79–80.
Snee, R. (1974) Graphical display of two-way contingency tables.American Statistician, 38, 9–12.
Srole, L., Michael, T. S., Opler, S. T., and Rennie, T. A. C. (1962).Mental health in metropolis: The midtown Manhattan study. New York: McGraw Hill.
Strauss, D. (1981). Choice by features: An extension of Luce's model to account for similarities.British Journal of Mathematical and Statistical Psychology, 34, 50–61.
Takane, Y. (1980). Maximum likelihood estimation in the generalized case of Thurstone's model of comparative judgment.Japanese Psychological Research, 22, 188–196.
Takane, Y., Bozdogan, H., & Shibayama, T. (1987). (in press). Ideal point discriminant analysis,Psychometrika.
Takane, Y., & Shibayama, T. (1984). Multiple discriminant analysis for predictor variables measured at various scale levels.Proceedings of the 12th Annual Meeting of the Behaviormetric Society of Japan, 99–100.
Takane, Y., & Shibayama, T. (1986). Comparisons of models for stimulus recognition data. In J. de Leeuw, W. Heiser, J. Meulman, and F. Critchley (Eds.),Multidimensional data analysis (pp. 119–148). Leiden: DSWO Press. (With discussion)
ter Braak, C. J. F. (1986). Canonical correspondence analysis: A new eigenvector technique for multivariate direct gradient analysis.Ecology, 67, 1167–1179.
ter Braak, C. J. F. (in press, June). Partial canonical correspondence analysis. In H. H. Bock (Ed.),Proceedings of the first IFCS conference, Aachen.
Thurstone, L. L. (1929). Theory of attitude measurement.Psychological Review, 36, 222–241.
Torgerson, W. S. (1958).Theory and methods of scaling. New York: Wiley.
van der Heijden, P. G. M., & de Leeuw, J. (1985). Correspondence analysis used complementary to logilinear analysis.Psychometrika, 50, 429–447.
Yanai, H. (1987, June). Partial correspondence analysis. Paper presented at the 1987 Annual Meeting of the North American Classification Society, Montreal.
Author information
Authors and Affiliations
Additional information
Presented as the Presidential Address to the Psychometric Society's Annual and European Meetings, June, 1987. Preparation of this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Chikio Hayashi of University of the Air in Japan for providing the ISM data, and to Jim Ramsay and Ivo Molenaar for their helpful comments on an earlier draft of this paper.
Rights and permissions
About this article
Cite this article
Takane, Y. Analysis of contingency tables by ideal point discriminant analysis. Psychometrika 52, 493–513 (1987). https://doi.org/10.1007/BF02294815
Issue Date:
DOI: https://doi.org/10.1007/BF02294815