The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Collapsibility Using Raindrop Plot

RAINDROP PLOT을 이용한 차원축소

  • Hong C. S. (Department of Statistics, Sungkyunkwan University) ;
  • Kim B. J. (Department of Statistics, Sungkyunkwan University) ;
  • Park J. Y. (Department of Statistics, Sungkyunkwan University)
  • 홍종선 (성균관대학교 경제학부 통계학전공) ;
  • 김범준 (성균관대학교 통계학과) ;
  • 박지용 (성균관대학교 통계학과)
  • Published : 2005.07.01
Download PDF
⟨ Previous Next ⟩

Abstract

For categorical data analysis, the collapsibility were explained with the odds ratio (cross-product ratio). When these theories with these odds ratios are applied to real $2{\times}2{\times}K$ contingency tables, it is impossible to decide whether data are collapsible. Among graphical methods to represent odds ratios, Contour plot which is developed by Doi, Nakamura and Yamamoto (2001) could explain the structure of these data, but cannot decide on the collapsibility. In this paper, by using the Raindrop plot proposed by Barrowman and Myers (2003), we suggest an alternative method which can not only explain the structure of data, but also decide on the collapsibility.

범주형 자료분석에서 차원축소(collapsibility)는 오즈비로 설명되었다. 실제의 $2{\times}2{\times}K$ 분할표 자료를 이 이론에 적응시켰을 때 오즈비의 값으로 차원축소가 가능한지의 여부를 판단하기는 어렵다. 오즈비를 시각적으로 표현하는 방법 중에서 Doi, Nakamura와 Yamamoto(2001)가 제안한 Contour plot을 통해서 분할표 자료를 설명하는 것은 가능하지만 차원축소의 가능성을 결정하기에는 한계가 있다. 본 연구에서는 오즈비의 신뢰구간을 시각적으로 표현할 수 있는 방법으로 Barrowman과 Myers(2003)가 제안한 Raindrop plot을 이용하여 $2{\times}2{\times}K$ 분할표 자료를 설명할 수 있으며 동시에 차원축소의 가능성을 판단할 수 있는 방법을 제안하고자 한다.

Keywords

References

  1. Agresti, A. (1984). Analysis of Ordinary Categorical Data, John Wiley and Sons
  2. Agresti, A. (1990). Categorical Data Analysis, John Wiley and Sons
  3. 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, 57, 665-676 https://doi.org/10.1139/cjfas-57-4-665
  4. Barrowman, N. J. and Myers, R. A. (2003). Raindrop plots: A new way to display collections of likelyhoods and distributions, The American Statistician, 57, 268-274 https://doi.org/10.1198/0003130032369
  5. Bishop, Yvonne M. M., Fienberg, Steve E., and Holland, Paul W. (1975). Discrete Multivariate Analysis, Cambridge, Massachusetts: MIT Press
  6. Christensen, Ronaldo. (1990). Log-Linear Models, New York: Springer-Verlag
  7. Cohen, A. (1980). On the graphical display of the significant components in a two-way contigency table, Communications in Statistics - Theory and Methods, A9, 1025-1041
  8. Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics, Chapman Hall
  9. Darroch, J. N., Lauritzen, S. L., and Speed, T. P. (1980). Markov fields and log-linear interaction models for contingency tables, Annals of Statistics, 57, 552-539
  10. Doi, M., Nakamura, T., and Yamamoto, E. (2001). Conservative tendency of the crude odds ratio, Journal of Japan Statistical Society, 1, 1-19
  11. Ducharme, G. R. and Lepage, Y. (1986). Testing collapsibility in contingency tables, Journal of the Royal Statistical Society, B, 48, 197-205
  12. Efron, B. (1996). Empirical Bayes methods for combining likelihoods, Journal of the American Statistical Association, 91, 538-565 https://doi.org/10.2307/2291646
  13. Fienberg, S. E. and Gilbert, J. P. (1970). The geometry of 2 x 2 contingency tables, Journal of the American Statistical Association, 65, 694-701 https://doi.org/10.2307/2284580
  14. Fienberg, S. E. (1975). Perspective Canada as a social report, Social Indicators Research, 2, 154-174
  15. Fienberg, S. E. (1980). The Analysis of Cross-Classified Data, 2nd ed, The MIT press
  16. Friendly, M. (1992). Mosaic displays for log-linear models, Proceedings of the Statistical Graphics Section, the American Statistical Association, 61-68
  17. Friendly, M. (1994). Mosaic displays for multi-way contigency tables, Journal of the American Statistical Association, 89, 190-200 https://doi.org/10.2307/2291215
  18. Hartigan, J. A. and Kleiner, B. (1981). Mosaic for contingency tables, Computer Science and Statistics: Proceedings of the 13th Symposium on the Interface, ED. W. F. Eddy, New York: Spring-Verlag, 268-273
  19. Hartigan, J. A. and Kleiner, B. (1984). A Mosaic of the television ratings, The American Statstician, 38, 32-35 https://doi.org/10.2307/2683556
  20. Hong, C. S. and Lee, J. C. (2000). Ring chart II for multidimensional categorical data analysing using conditional ring charts, Korean Journal of Applied Statistics, 13, 163-178
  21. Hyndman, R. J. (1996). Computing and graphing highest density regions, The American Statistician, 50, 120-126 https://doi.org/10.2307/2684423
  22. Jeong, D. B., Hong, C. S., and Yoon, S. H. (2003). Empirical comparisons of disparity measures for partial association models in three dimensional contingency tables, The Korean Communications in Statistics, 10, 135-144 https://doi.org/10.5351/CKSS.2003.10.1.135
  23. Oh, M, G., Hong, C. S., and Lee, J. C. (1999). Ring chart for categorical data, Korean Journal of Applied Statistics, 12, 225-240
  24. Tukey, J. W. (1977). Exploratory Data Analysis, Addison-Wesley Publishing Company
  25. 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