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Graphical Methods for Hierarchical Log-Linear Models

  • Hong, Chong-Sun (Department of Statistics, Sungkyunkwan University) ;
  • Lee, Ui-Ki (Research Institute of Applied Statistics, Sungkyunkwan University)
  • Published : 2006.12.31

Abstract

Most graphical methods for categorical data can describe the structure of data and represent a measure of association among categorical variables. Among them the polyhedron plot represents sequential relationships among hierarchical log-linear models for a multidimensional contingency table. This kind of plot could be explored to describe the differences among sequential models. In this paper we suggest graphical methods, containing all the information, that reflect the relationship among all log-linear models in a certain hierarchical structure. We use the ideas of a correlation diagram.

References

  1. Barrowman, N.J. and Myers, R.A. (2000). Still more spawner-recruitment curves: the hockey stick and its generalizations. Canadian Journal of Fisheries and Aquatic Sciences, Vol. 57, 665-676 https://doi.org/10.1139/cjfas-57-4-665
  2. Barrowman, N.J. and Myers, R.A. (2003). Raindrop plots: a new way to display collections of likelihoods and distributions. American Statistician, Vol. 57, 268-274 https://doi.org/10.1198/0003130032369
  3. Becker, R.A., Cleveland, W.S., and Shyu, M.J. (1996). The visual design and control of trellis display. Journal of Computational and Graphical Statistics. Vol. 6(2), 123-155
  4. Bishop, Y.M.M., Fienberg, S.E., and Holland P.W. (1975). Discrete Multivariate Analysis, MIT Press
  5. Christensen, R. (1990). Log-Linear Models and Logistic Regression, Springer, New York
  6. Cleveland, W.S. and McGill, R. (1984), Graphical perception: theory, experimentation, and application to the development of graphical methods. Journal of American Statistical Association, Vol. 79(387), 531-554 https://doi.org/10.2307/2288400
  7. Cohen, A. (1980). On the graphical display of the significant components in a two-way contingency table. Communications in Statistics-Theory and Methods, Vol. 9, 1025-1041 https://doi.org/10.1080/03610928008827940
  8. Corsten, L.C.A. and Gabriel, K.R. (1976). Graphical exploration in comparing variance matrice. Biometrics, Vol. 32, 851-863 https://doi.org/10.2307/2529269
  9. Darroch, J.N., Lauritzen, S.L., and Speed, T.P. (1980). Markov-fields and log-linear models for contingency tables. Annals of Statistics, Vol. 8, 522-39 https://doi.org/10.1214/aos/1176345006
  10. Doi, M., Nakamura, T., and Yamamoto, E. (2001). Conservative tendency of the crude odds ratio. Journal of Japan Statistical Society, Vol. 31(1), 53-65 https://doi.org/10.14490/jjss1995.31.53
  11. Fienberg, S.E. (1968). The Estimation of Exponential Probabilities in Two-way Contingency Tables, Ph. D. Thesis, Department of Statistics, Harvard University
  12. Fienberg, S.E. and Gilbert, J.P. (1970). The geometry of a $2{\times}2$ contingency tables. Journal of American Statistical Association, Vol. 65, 694-701 https://doi.org/10.2307/2284580
  13. Fienberg, S.E. (1975). Perspective canada as a social report. Social Indicators Research, Vol. 2, 154-174
  14. Fienberg, S.E. (1983). The Analysis of Cross-Classified Categorical Data, MIT Press
  15. Friendly, M. (1991). The SAS System for Statistical Graphics, SAS Institute Inc
  16. Friendly, M. (1992). Mosaic displays for log-linear models, proceedings of the statistical graphics section. American Statistical Association, 61-68
  17. Friendly, M. (1994a). Mosaic displays for multi-way contingency tables. Journal of American Statistical Association, Vol. 89, 190-200 https://doi.org/10.2307/2291215
  18. Friendly, M. (1994b). SAS/IML graphics for fourfold displays. Observations, Vol. 3(4), 47-56
  19. Gabriel, K.R. (1971). The biplot graphical display of matrices with applications to principal component analysis. Biometrika, Vol. 58, 453-467 https://doi.org/10.1093/biomet/58.3.453
  20. Gay, D.M. (1983). Algorithm 611. subroutines for unconstrained minimization using a model / trust-region approach. ACM Transactions on Mathematical Software, Vol. 9, 503-524 https://doi.org/10.1145/356056.356066
  21. Gay, D.M. (1984). A trust region approach to linearly constrained optimization, Numerical Analysis, Proceedings, Dundee 1983, ed. F.A. Lootsma, Berlin: Springer, 171-189
  22. Hartigan, J.A. and Kleiner, B. (1981). Mosaic for contingency tables. Com -puter Science and Statistics, Proceedings of the 13th Symposium on the Interface, ED. W. F. Eddy, New York: Springer-Verlag, 268-273
  23. Hartigan, J.A. and Kleiner, B. (1984). A mosaic of the television ratings. American Statisticians, Vol. 38, 32-35 https://doi.org/10.2307/2683556
  24. Hong, C.S., Choi. H.J., and Oh, M.G. (1999). Geometric descriptions for hierarchical log-linear models, InterStat, September, 1999. InterStat on the Internet
  25. Li, X., Buechner, J.M., Tarwater, P.M., and Munoz, A. (2003). A diamond -shaped equiponderant graphical display of the effects of two categorical predictors on continuous outcomes. American Statisticians, Vol. 57, 193-199 https://doi.org/10.1198/0003130031883
  26. Ries, P.N. and Smith, H. (1963). The use of chi-square for preference testing In multidimensional problems. Chemical Engineering Progress, Vol. 59, 39-43
  27. Trosset. M.W. (2005). Visualizing correlation. Journal of Computational and Graphical Statistics, Vol. 14, 1-19 https://doi.org/10.1198/106186005X27004
  28. Tufte, E.R. (1983). The Visual Display of Quantitative Information, Graphics Press. Cheshire, Connecticut
  29. Tukey, J.W. (1977). Exploratory Data Analysis, Addison-Wesley Publishing Company
  30. Wilkinson, L. (1999). The Grammar of Graphics, Springer, New York
  31. Yamamoto, E. and Doi, M. (2001) Noncollapsibility of common odds ratios without/with confounding. Bulletin of The 53rd Session of the International Statistical Institute, Book 3, 39-40