# A hybrid Newton-type method for censored survival data using double weights in linear models

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## Abstract

As an alternative to the Cox model, the rank-based estimating method for censored survival data has been studied extensively since it was proposed by Tsiatis [Tsiatis AA (1990) Ann Stat 18:354–372] among others. Due to the discontinuity feature of the estimating function, a significant amount of work in the literature has been focused on numerical issues. In this article, we consider the computational aspects of a family of doubly weighted rank-based estimating functions. This family is rich enough to include both estimating functions of Tsiatis (1990) for the randomly observed data and of Nan et al. [Nan B, Yu M, Kalbfleisch JD (2006) Biometrika (to appear)] for the case-cohort data as special examples. The latter belongs to the biased sampling problems. We show that the doubly weighted rank-based discontinuous estimating functions are monotone, a property established for the randomly observed data in the literature, when the generalized Gehan-type weights are used. Though the estimating problem can be formulated to a linear programming problem as that for the randomly observed data, due to its easily uncontrollable large scale even for a moderate sample size, we instead propose a Newton-type iterated method to search for an approximate solution of the (system of) discontinuous monotone estimating equation(s). Simulation results provide a good demonstration of the proposed method. We also apply our method to a real data example.

## Keywords

Censored linear regression Double weights Two-stage design Case-cohort design Hybrid Newton-type method Generalized Gehan-type weights Monotone estimating function Linear programming## Notes

### Acknowledgment

We thank Drs. Howard M. Sandler and Jeremy M. G. Taylor for providing the prostate cancer data.

## References

- Borgan Ø, Langholz B, Samuelsen SO, Goldstein L, Pogoda J (2000) Exposure stratified case-cohort designs. Lifetime Data Anal 6:39–58MATHMathSciNetCrossRefGoogle Scholar
- Breslow NE, McNeney B, Wellner JA (2003) Large sample theory for semiparametric regression models with two-phase, outcome dependent sampling. Ann Stat 31:1110–1139MATHMathSciNetCrossRefGoogle Scholar
- Fygenson M, Ritov Y (1994) Monotone estimating equations for censored data. Ann Stat 22:732–746MATHMathSciNetGoogle Scholar
- Horvitz DG, Thompson DJ (1952) A generalization of sampling without replacement from a finite universe. J Am Stat Assoc 47:663–685MATHMathSciNetCrossRefGoogle Scholar
- Huang Y (2002) Calibration regression of censored lifetime medical cost. J Am Stat Assoc 97:318–327MATHCrossRefGoogle Scholar
- Jin Z, Lin DY, Wei LJ, Ying Z (2003) Rank-based inference for the accelerated failure time model. Biometrika 90:341–353MATHMathSciNetCrossRefGoogle Scholar
- Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. John Wiley & Sons, Inc., New JerseyMATHGoogle Scholar
- Lin DY, Geyer CJ (1992) Computational methods for semiparametric linear regression with censored data. J Comput Graph Stat 1:77–90CrossRefGoogle Scholar
- Lin DY, Wei LJ, Ying Z (1998) Accelerated failure time models for counting processes. Biometrika 85:605–618MATHMathSciNetCrossRefGoogle Scholar
- Nan B, Yu M, Kalbfleisch JD (2006) Censored linear regression for case-cohort studies. Biometrika (to appear)Google Scholar
- Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, Inc., New YorkMATHGoogle Scholar
- Prentice RL (1986) A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika 73:1–11MATHMathSciNetCrossRefGoogle Scholar
- Ritov Y (1990) Estimation in a linear regression model with censored data. Ann Stat 18:303–328MATHMathSciNetGoogle Scholar
- Self SG, Prentice RL (1988) Asymptotic distribution theory and efficiency results for case-cohort studies. Ann Stat 16:64–81MATHMathSciNetGoogle Scholar
- Stewart GW (1996) Afternotes on numerical analysis. SIAMGoogle Scholar
- Taylor JMG, Yu M, Sandler HM (2005) Individualized predictions of disease progression following radiation therapy for prostate cancer. J Clin Oncol 23(4):816–825CrossRefGoogle Scholar
- Tsiatis AA (1990) Estimating regression parameters using linear rank tests for censored data. Ann Stat 18:354–372MATHMathSciNetGoogle Scholar
- Ying Z (1993) A large sample study of rank estimation for censored regression data. Ann Stat 21:76–99MATHMathSciNetGoogle Scholar