Abstract
In this article, we review, consolidate and extend a theory for sufficient dimension reduction in regression settings. This theory provides a powerful context for the construction, characterization and interpretation of low-dimensional displays of the data, and allows us to turn graphics into a consistent and theoretically motivated methodological body. In this spirit, we propose an iterative graphical procedure for estimating the meta-parameter which lies at the core of sufficient dimension reduction; namely, the central dimension-reduction subspace.
Similar content being viewed by others
References
Basu, D. and Pereira, C. A. B. (1983). Conditional independence in statistics, Sankhyā Ser. A, 45, 324–337.
Box, G. E. P. and Draper, N. (1987). Empirical Model-Building and Response Surfaces, Wiley, New York.
Carrol, R. J. and Li, K. C. (1995). Binary regressors in dimension reduction models: A new look at treatment comparison, Statist. Sinica, 5, 667–688.
Cheng, C. S. and Li, K. C. (1995). A study of the method of principal Hessian direction for analysis of designed experiments, Statist. Sinica, 5, 617–640.
Chiaromonte, F. (1997). A reduction paradigm for multivariate laws, L1 Statistical Procedures and Related Topics, (ed. Y. Dodge), lecture notes-monograph series, IMS Lecture Notes Monogr. Ser., 229–240, Hayward, California.
Chiaromonte, F. (1998). On multivariate structures and exhaustive reductions, Computing Science and Statistics, (ed. S. Weisberg), 30, 204–213, The Interface Foundation of North America, Fairfax Station, Virginia.
Chiaromonte, F. (2001). Structures and exhaustive reductions: A general framework for the simplification of multivariate data (submitted).
Cook, R. D. (1994a). On the interpretation of regression plots, J. Amer. Statist. Assoc., 89, 177–189.
Cook, R. D. (1994b). Using dimension-reduction subspaces to identify important inputs in models of physical systems, Proceedings of the Section on Physical and Engineering Sciences, 18–25, American Statistical Association, Alexandria, Virginia.
Cook, R. D. (1996a). Graphics for regressions with a binary response, J. Amer. Statist. Assoc., 91, 983–992.
Cook, R. D. (1996b). Added-variable plots and curvature in linear regression, Technometrics, 38, 275–278.
Cook, R. D. (1998a). Regression Graphics, Wiley, New York.
Cook, R. D. (1998b). Principal Hessian directions revisited, J. Amer. Statist. Assoc., 93, 84–100.
Cook, R. D. and Lee, H. (1999). Dimension reduction in binary response regression, J. Amer. Statist. Assoc., 97, 1187–1200.
Cook, R. D. and Weisberg, S. (1991). Discussion of “Sliced inverse regression for dimension reduction”, J. Amer. Statist. Assoc., 86, 316–342.
Cook, R. D. and Weisberg, S. (1994). An Introduction to Regression Graphics, Wiley, New York.
Cook, R. D. and Weisberg, S. (1999). Graphics in statistical analysis: Is the medium the message?, Amer. Statist., 53, 29–37.
Cook, R. D. and Wetzel, N. (1993). Exploring regression structure with graphics (with discussion), Test, 2, 33–100.
Cox, D. R. and Snell, E. J. (1968). A general definition of residuals, J. Roy. Statist. Soc. Ser. B, 30, 248–275.
Dawid, A. P. (1979). Conditional independence in statistical theory, J. Roy. Statist. Soc. Ser. B, 41, 1–31.
Dawid, A. P. (1980). Conditional independence for statistical operations, Ann. Statist., 8, 598–617.
Flury, B. and Riedwyl, H. (1998). Multivariate Statistics: A Practical Approach, Chapman and Hall, London.
Fujikoshi, Y. (1982). A test for additional information in canonical correlation analysis, Ann. Inst. Statist. Math., 34, 523–530.
Ibrahimy, A. and Cook, R. D. (1995). Regression design for one-dimensional subspaces, MODA4—Advances in Model-Oriented Data Analysis (eds. C. P. Kitsas and W. G. Muller), 86, 125–132, Physica, Heidelberg.
Li, K. C. (1991). Sliced inverse regression for dimension reduction (with discussion), J. Amer. Statist. Assoc., 86, 316–342.
Li, K. C. (1992). On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma, J. Amer. Statist. Assoc., 87, 1025–1039.
Li, K. C. and Duan, N. (1989). Regression analysis under link violation, Ann. Statist., 17, 1009–1952.
Li, K. C., Aragon Y. and Thomos-Agan C. (1995). Analyzing multivariate outcome data: SIR and a non linear theory of Hotelling's most predictable variates, (to appear in J. Amer. Statist. Assoc.).
McKay, R. J. (1977). Variable selection in multivariate regression: An application of simultaneous test procedures, J. Roy. Statist. Soc. Ser. B., 39, 371–380.
Swayne, D. F., Cook, D. and Buja, A. (1998). XGobi: Interactive dynamic data visualization in the X window system, J. Comput. Graph. Statist., 7, 113–130.
Author information
Authors and Affiliations
About this article
Cite this article
Chiaromonte, F., Cook, R.D. Sufficient Dimension Reduction and Graphics in Regression. Annals of the Institute of Statistical Mathematics 54, 768–795 (2002). https://doi.org/10.1023/A:1022411301790
Issue Date:
DOI: https://doi.org/10.1023/A:1022411301790