## The Annals of Statistics

### Dimension reduction for censored regression data

#### Abstract

Without parametric assumptions, high-dimensional regression analysis is already complex. This is made even harder when data are subject to censoring. In this article, we seek ways of reducing the dimensionality of the regressor before applying nonparametric smoothing techniques. If the censoring time is independent of the lifetime, then the method of sliced inverse regression can be applied directly. Otherwise, modification is needed to adjust for the censoring bias. A key identity leading to the bias correction is derived and the root-$n$ consistency of the modified estimate is established. Patterns of censoring can also be studied under a similar dimension reduction framework. Some simulation results and an applica-tion to a real data set are reported.

#### Article information

Source
Ann. Statist., Volume 27, Number 1 (1999), 1-23.

Dates
First available in Project Euclid: 5 April 2002

https://projecteuclid.org/euclid.aos/1018031098

Digital Object Identifier
doi:10.1214/aos/1018031098

Mathematical Reviews number (MathSciNet)
MR1701098

Zentralblatt MATH identifier
0932.62050

Subjects
Primary: 65G05 62J20: Diagnostics

#### Citation

Li, Ker-Chau; Wang, Jane-Ling; Chen, Chun-Houh. Dimension reduction for censored regression data. Ann. Statist. 27 (1999), no. 1, 1--23. doi:10.1214/aos/1018031098. https://projecteuclid.org/euclid.aos/1018031098

#### References

• BERAN, R. 1981. Nonparametric regression with randomly censored survival data. Technical report, Univ. California, Berkeley. Z. BRILLINGER. 1991. Comment on Sliced inverse regression for dimension reduction,'' by K. C. Li. J. Amer. Statist. Assoc. 86 333. Z.
• CARROLL, R. J. and LI, K. C. 1992. Measurement error regression with unknown link: dimension reduction and data visualization. J. Amer. Statist. Assoc. 87 1040 1050. Z.
• CARROLL, R. J. and LI, K. C. 1995. Binary regressors in dimension reduction models: a new look at treatment comparisons. Statist. Sinica 5 667 688. Z.
• CHEN, C. H. and LI, K. C. 1998. Can SIR be as popular as multiple linear regression? Statist. Sinica 8 289 316. Z.
• COOK, R. D. 1994. On the interpretation of regression plots. J. Amer. Statist. Assoc. 89 177 189. Z.
• COOK, R. D. and NACHTSHEIM, C. J. 1994. Re-weighting to achieve elliptically contoured covariates in regression. J. Amer. Statist. Assoc. 89 592 599. Z.
• COOK, R. D. and WEISBERG, S. 1991. Comment on Sliced inverse regression for dimension reduction,'' by K. C. Li. J. Amer. Statist. Assoc. 86 328 332. Z.
• COOK, R. D. and WEISBERG, S. 1994. An Introduction to Regression Graphics. Wiley, New York. Z.
• DABROWSKA, D. M. 1987. Non-parametric regression with censored survival time data. Scand. J. Statist. 14 181 197. Z.
• DABROWSKA, D. M. 1992. Variable bandwidth conditional Kaplan Meier estimate. Scand. J. Statist. 19 351 361. Z.
• DIACONIS, P. and FREEDMAN, D. 1984. Asymptotics of graphical projection pursuit. Ann. Statist. 12 793 815. Z.
• DOKSUM, K. A. 1987. An extension of partial likelihood methods for proportional hazard models to general transformation models. Ann. Statist. 15 325 345. Z.
• DOKSUM, K. A. and GASKO, M. 1990. On a correspondence between models in binary regression analysis and in survival analysis. Internat. Statist. Rev. 58 243 252. Z.
• DUAN, N. and LI, K. C. 1991. Slicing regression: a link-free regression method. Ann. Statist. 19 505 530. Z.
• FAN, J. and GIJBELS, I. 1994. Censored regression: local linear approximations and their applications, J. Amer. Statist. Assoc. 89 560 570. Z.
• FLEMING, T. R. and HARRINGTON, D. P. 1991. Counting Processes and Survival Analysis. Wiley, New York. Z.
• HALL, P. and LI, K. C. 1993. On almost linearity of low dimensional projection from high dimensional data. Ann. Statist. 21 867 889. Z.
• HSING, T. and CARROLL, R. J. 1992. An asymptotic theory for sliced inverse regression. Ann. Statist. 20 1040 1061. Z. Z.
• HUBER, P. 1985. Projection pursuit with discussion. Ann. Statist. 13 435 526. Z. Z.
• LI, K. C. 1991. Sliced inverse regression for dimension reduction with discussion. J. Amer. Statist. Assoc. 86 316 342. Z.
• LI, K. C. 1992. Uncertainty analysis for mathematical models with SIR. In Probability and Z. Statistics Z. P. Jiang, S. H. Yan, P. Cheng and R. Wu, eds. 138 162. World Scientific Press, Singapore.
• LI, K. C. 1997. Nonlinear confounding in high dimensional regression. Ann. Statist. 25 577 612. Z.
• MCCULLAGH, P. and NELDER, J. A. 1989. Generalized Linear Models, 2nd. ed. Chapman and Hall, London. Z.
• SCHOTT, J. R. 1994. Determining the dimensionality in sliced inverse regression. J. Amer. Statist. Assoc. 89 141 148. Z.
• ZHU, L. X. and NG, K. W. 1995. Asymptotics of sliced inverse regression. Statist. Sinica 5 727 736.
• LOS ANGELES, CALIFORNIA DAVIS, CALIFORNIA E-MAIL: kcli@math.ucla.edu