The Annals of Statistics

Proportional hazards models with continuous marks

Yanqing Sun, Peter B. Gilbert, and Ian W. McKeague

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For time-to-event data with finitely many competing risks, the proportional hazards model has been a popular tool for relating the cause-specific outcomes to covariates [Prentice et al. Biometrics 34 (1978) 541–554]. This article studies an extension of this approach to allow a continuum of competing risks, in which the cause of failure is replaced by a continuous mark only observed at the failure time. We develop inference for the proportional hazards model in which the regression parameters depend nonparametrically on the mark and the baseline hazard depends nonparametrically on both time and mark. This work is motivated by the need to assess HIV vaccine efficacy, while taking into account the genetic divergence of infecting HIV viruses in trial participants from the HIV strain that is contained in the vaccine, and adjusting for covariate effects. Mark-specific vaccine efficacy is expressed in terms of one of the regression functions in the mark-specific proportional hazards model. The new approach is evaluated in simulations and applied to the first HIV vaccine efficacy trial.

Article information

Ann. Statist., Volume 37, Number 1 (2009), 394-426.

First available in Project Euclid: 16 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62N01: Censored data models
Secondary: 62N02: Estimation 62N03: Testing 62G20: Asymptotic properties

Competing risks distribution-free confidence bands and tests failure time data genetic data HIV vaccine trial pointwise and simultaneous confidence bands semiparametric model survival analysis


Sun, Yanqing; Gilbert, Peter B.; McKeague, Ian W. Proportional hazards models with continuous marks. Ann. Statist. 37 (2009), no. 1, 394--426. doi:10.1214/07-AOS554.

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  • [1] Aalen, O. O. and Johansen, S. (1978). An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scand. J. Statist. 5 141–150.
  • [2] Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.
  • [3] Cai, Z. and Sun, Y. (2003). Local linear estimation for time-dependent coefficients in Cox’s regression models. Scand. J. Statist. 30 93–111.
  • [4] Flynn, N. M., Forthal, D. N., Harro, C. D., Judson, F. N., Mayer, K. H., Para, M. F. and The rgp120 HIV Vaccine Study Group (2005). Placebo-controlled phase 3 trial of a recombinant glycoprotein 120 vaccine to prevent HIV-1 infection. J. Infectious Diseases 191 654–665.
  • [5] Gilbert, P. B., McKeague, I. W. and Sun, Y. (2008). The two-sample problem for failure rates depending on a continuous mark: An application to vaccine efficacy. Biostatistics 9 263–276.
  • [6] Graham, B. S. (2002). Clinical trials of HIV vaccines. Annual Review of Medicine 53 207–221.
  • [7] Huang, Y. and Louis, T. A. (1998). Nonparametric estimation of the joint distribution of survival time and mark variables. Biometrika 85 785–798.
  • [8] Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data. Wiley, New York.
  • [9] Lin, D. Y., Wei, L. J. and Ying, Z. (1993). Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 80 557–572.
  • [10] Martinussen, T. and Scheike, T. H. (2006). Dynamic Regression Models for Survival Data. Springer, New York.
  • [11] Nabel, G. J. (2001). Challenges and opportunities for development of an AIDS vaccine. Nature 410 1002–1007.
  • [12] Nickle, D. C., Heath, L., Jensen, M. A., Gilbert, P. B., Kosakovsky Pond, S. L. K. and Mullins, J. I. (2005). Amino acid substitution matrices for HIV-1 subtype B. Technical report, Univ. Washington.
  • [13] Olschewski, M. and Schumacher, M. (1990). Statistical analysis of quality of life data in cancer clinical trials. Statistics in Medicine 9 749–763.
  • [14] Prentice, R. L., Kalbfleisch, J. D., Peterson, A. V., Fluornoy, N., Farewell, V. T. and Breslow, N. E. (1978). The analysis of failure times in the presence of competing risks. Biometrics 34 541–554.
  • [15] Schumacher, M. (1984). Two-sample tests of Cramér–von Mises and Kolmogorov–Smirnov type for randomly censored data. Internat. Statist. Rev. 52 263–281.
  • [16] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • [17] Sun, Y. and Wu, H. (2005). Semiparametric time-varying coefficients regression model for longitudinal data. Scand. J. Statist. 32 21–47.
  • [18] Tsiatis, A. A. (1975). A nonidentifiability aspect of the problem of competing risks. Proc. Natl. Acad. Sci. USA 72 20–22.
  • [19] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.
  • [20] UNAIDS (Joint United Nations Programme for HIV/AIDS) (2004). AIDS Epidemic Update 2004.
  • [21] Wyatt, R., Kwong, P. D., Desjardins, E., Sweet, R. W., Robinson, J., Hendrickson, W. A. and Sodroski, J. G. (1998). The antigenic structure of the HIV gp120 envelope glycoprotein. Nature 393 705–711.