The Annals of Applied Statistics

Doubly robust estimation of optimal treatment regimes for survival data—with application to an HIV/AIDS study

Runchao Jiang, Wenbin Lu, Rui Song, Michael G. Hudgens, and Sonia Naprvavnik

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In many biomedical settings, assigning every patient the same treatment may not be optimal due to patient heterogeneity. Individualized treatment regimes have the potential to dramatically improve clinical outcomes. When the primary outcome is censored survival time, a main interest is to find optimal treatment regimes that maximize the survival probability of patients. Since the survival curve is a function of time, it is important to balance short-term and long-term benefit when assigning treatments. In this paper, we propose a doubly robust approach to estimate optimal treatment regimes that optimize a user specified function of the survival curve, including the restricted mean survival time and the median survival time. The empirical and asymptotic properties of the proposed method are investigated. The proposed method is applied to a data set from an ongoing HIV/AIDS clinical observational study conducted by the University of North Carolina (UNC) Center of AIDS Research (CFAR), and shows the proposed methods significantly improve the restricted mean time of the initial treatment duration. Finally, the proposed methods are extended to multi-stage studies.

Article information

Ann. Appl. Stat., Volume 11, Number 3 (2017), 1763-1786.

Received: February 2016
Revised: March 2017
First available in Project Euclid: 5 October 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Doubly robust estimation median survival time optimal treatment regimen restricted mean survival time


Jiang, Runchao; Lu, Wenbin; Song, Rui; Hudgens, Michael G.; Naprvavnik, Sonia. Doubly robust estimation of optimal treatment regimes for survival data—with application to an HIV/AIDS study. Ann. Appl. Stat. 11 (2017), no. 3, 1763--1786. doi:10.1214/17-AOAS1057.

Export citation


  • Bai, X., Tsiatis, A. A. and O’Brien, S. M. (2013). Doubly-robust estimators of treatment-specific survival distributions in observational studies with stratified sampling. Biometrics 69 830–839.
  • Chen, P.-Y. and Tsiatis, A. A. (2001). Causal inference on the difference of the restricted mean lifetime between two groups. Biometrics 57 1030–1038.
  • Cox, D. R. (1972). Regression models and life-tables. J. Roy. Statist. Soc. Ser. B 34 187–220.
  • Dombrowski, J. C., Kitahata, M. M., Rompaey, S. E. V., Crane, H. M., Mugavero, M. J., Eron, J. J., Boswell, S. L., Rodriguez, B., Mathews, W. C., Martin, J. N., Moore, R. D. and Golden, M. R. (2013). High levels of antiretroviral use and viral suppression among persons in HIV care in the United States, 2010. J. Acquir. Immune Defic. Syndr. 63 299–306.
  • Gill, R. D., Keiding, N. and Andersen, P. K. (1997). Statistical Models Based on Counting Processes. Springer, New York.
  • Goldberg, Y. and Kosorok, M. R. (2012). Q-learning with censored data. Ann. Statist. 40 529–560.
  • Gunthard, H. F., Aberg, J. A., Eron, J. J., Hoy, J. F., Telenti, A., Benson, C. A., Burger, D. M., Cahn, P., Gallant, J. E., Glesby, M. J. Reiss, P. Saag, M. S. Thomas, D. L. Jacobsen, D. M. and Volberding, P. A. (2014). Antiretroviral treatment of adult HIV infection: 2014 recommendations of the International Antiviral Society-USA panel. JAMA J. Am. Med. Assoc. 312 410–425.
  • Howe, C. J., Cole, S. R., Napravnik, S. and Eron, J. J.Jr. (2010). Enrollment, retention, and visit attendance in the University of North Carolina Center for AIDS Research Clinical Cohort, 2001–2007. AIDS Res. Hum. Retrovir. 26 875–881.
  • Irwin, J. O. (1949). The standard error of an estimate of expectation of life, with special reference to expectation of tumourless life in experiments with mice. J. Hyg. 47 188–189.
  • Jiang, R., Lu, W., Song, R., and Davidian, M. (2016). On estimation of optimal treatment regimes for maximizing $t$-year survival probability. J. R. Stat. Soc. Ser. B. Stat. Methodol. To appear.
  • Jiang, R., Lu, W., Song, R., Hudgens, M. G. and Naprvavnik, S. (2017). Supplement to “Doubly robust estimation of optimal treatment regimes for survival data—with application to an HIV/AIDS study.” DOI:10.1214/17-AOAS1057SUPP.
  • Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457–481.
  • Mebane, W. R. Jr. and Sekhon, J. S. (2011). Genetic optimization using derivatives: The rgenoud package for R. J. Stat. Softw. 42 1–26.
  • Murphy, S. A. (2003). Optimal dynamic treatment regimes. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 331–366.
  • Murphy, S. A. (2005). A generalization error for Q-learning. J. Mach. Learn. Res. 6 1073–1097.
  • Panel on Antiretroviral Guidelines for Adults and Adolescents (2016). Guidelines for the use of antiretroviral agents in HIV-1-infected adults and adolescents, Department of Health and Human Services. Available at:
  • Robins, J. M. (2004). Optimal structural nested models for optimal sequential decisions. In Proceedings of the Second Seattle Symposium in Biostatistics. Lect. Notes Stat. 179 189–326. Springer, New York.
  • Tian, L., Alizadeh, A. A., Gentles, A. J. and Tibshirani, R. (2014). A simple method for estimating interactions between a treatment and a large number of covariates. J. Amer. Statist. Assoc. 109 1517–1532.
  • Watkins, C. J. C. H. and Dayan, P. (1992). Q-learning. Mach. Learn. 8 279–292.
  • Willig, J. H., Abroms, S., Westfall, A. O., Routman, J., Adusumilli, S., Varshney, M., Allison, J., Chatham, A., Raper, J. L., Kaslow, R. A., Saag, M. S. and Mugavero, M. J. (2008). Increased regimen durability in the era of once daily fixed-dose combination antiretroviral therapy. AIDS 22 1951–1960.
  • Zhao, Y., Kosorok, M. R. and Zeng, D. (2009). Reinforcement learning design for cancer clinical trials. Stat. Med. 28 3294–3315.
  • Zhao, Y. Q., Zeng, D., Laber, E. B., Song, R., Yuan, M. and Kosorok, M. R. (2015). Doubly robust learning for estimating individualized treatment with censored data. Biometrika 102 151–168.
  • Zucker, D. M. (1998). Restricted mean life with covariates: Modification and extension of a useful survival analysis method. J. Amer. Statist. Assoc. 93 702–709.

Supplemental materials

  • Supplement to “Doubly robust estimation of optimal treatment regimes for survival data—with application to an HIV/AIDS study”. It contains regularity conditions referenced in Theorems 1 and 2, and additional simulation results referenced in Sections 4 and 6.