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A copula model for bivariate hybrid censored survival data with application to the MACS study

Lifetime Data Analysis volume 16, pages 231–249 (2010)Cite this article

Abstract

A copula model for bivariate survival data with hybrid censoring is proposed to study the association between survival time of individuals infected with HIV and persistence time of infection with an additional virus. Survival with HIV is right censored and the persistence time of the additional virus is subject to interval censoring case 1. A pseudo-likelihood method is developed to study the association between the two event times under such hybrid censoring. Asymptotic consistency and normality of the pseudo-likelihood estimator are established based on empirical process theory. Simulation studies indicate good performance of the estimator with moderate sample size. The method is applied to a motivating HIV study which investigates the effect of GB virus type C (GBV-C) co-infection on survival time of HIV infected individuals.

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References

  • Birk M, Lindback S, Lidman C (2002) No influence of GB virus C replication on the prognosis in a cohort of HIV-1-infected patients. AIDS 16: 2482–2485

    Article  Google Scholar 

  • Ding AA, Wang W (2004) Testing independence for bivariate current status data. J Am Stat Assoc 99: 145–155

    MATH  Article  MathSciNet  Google Scholar 

  • Fleming TR, Harrington DP (1991) Counting process and survival analysis. John wiley & Sons, New York

    Google Scholar 

  • Groeneboom P, Wellner JA (1992) Information bounds and nonparametric maximum likelihood estimation. Birkhauser, Boston

    MATH  Google Scholar 

  • Hougaard P (1989) Fitting a multivariate failure time distribution. IEEE Trans Reliab 38: 444–448

    MATH  Article  Google Scholar 

  • Hsu L, Prentice RL (1996) A generalisation of the mantel-haenszel test to bivariate failure time data. Biometrika 4: 905–911

    Article  MathSciNet  Google Scholar 

  • Huang J, Wellner JA (1995) Asymptotic normality of the npmle of linear functionals for interval censored data, case 1. Stat Neerl 49: 153–163

    MATH  Article  MathSciNet  Google Scholar 

  • Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley-Interscience, New York

    MATH  Google Scholar 

  • Liang KE, Self SG, Bandeen-Rocche K, Zeger S (1995) Some recent developments for regression analysis of multivariate failure time data. Lifetime Data Anal 1: 403–415

    MATH  Article  Google Scholar 

  • Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer-Verlag, New York

    MATH  Google Scholar 

  • Oakes D (1982) A concordance test for independence in the presence of censoring. Biometrics 38: 451–455

    MATH  Article  Google Scholar 

  • Oakes D (1989) Bivariate survival models induced by frailties. J Am Stat Assoc 84: 487–493

    MATH  Article  MathSciNet  Google Scholar 

  • Oakes D (2000) Survival analysis. J Am Stat Assoc 95: 282–285

    Article  Google Scholar 

  • Pepe MS (1991) Inference for events with dependent risks in multiple endpoint studies. J Am Stat Assoc 86: 770–778

    MATH  Article  MathSciNet  Google Scholar 

  • Shih JH, Louis T (1995) Inference on the association parameter in copula models for bivariate survival data. Biometrics 51: 1384–1399

    MATH  Article  MathSciNet  Google Scholar 

  • Shih JH, Louis TA (1996) Tests of independence for bivariate survival data. Biometrics 4: 1440–1449

    Article  MathSciNet  Google Scholar 

  • Tillmann H, Heiken H, Knapik-Botor A, Heringlake S, Ockenga J et al (2001) Infection with GB virus C and reduced mortality among HIV-infected patients. N Engl J Med 345: 715–724

    Article  Google Scholar 

  • Toyoda H, Fukuda Y et al (1998) Effect of GB virus C/hepatitis G virus coinfection on the course of HIV infection in hemophilia patients in Japan. J Acquir Immune Defic Syndr Hum Retrovirol 17: 209–213

    Google Scholar 

  • van der Vaart AW (1998) Asymptotic statistics. Cambridge Univ. Press, Cambridge

    MATH  Google Scholar 

  • van der Vaart AW, Wellner JA (1996) Weak convergence and empirical processes with application to statistics. Springer-Verlag, New York

    Google Scholar 

  • von Mises R (1947) On the asymptotic distribution of differentiable statistical functions. Ann Math Statist 18: 309–348

    MATH  Article  Google Scholar 

  • Wang W, Ding AA (2000) On assessing the association for bivariate current status data. Biometrika 87: 879–893

    MATH  Article  MathSciNet  Google Scholar 

  • Williams C, Klinzman D, Yamashita T, Xiang J et al (2004) Persistent GB virus C infection and survival in HIV-infected men. N Engl J Med 350: 981–990

    Article  Google Scholar 

  • Xiang J, Wunschmann W, Diekema D, Klinzman D, Patrick K et al (2001) Effect of coinfection with GB virus C on survival among patients with HIV infection. N Engl J Med 345: 707–714

    Article  Google Scholar 

  • Zhang W, Chaloner K, Tillmann HS, Williams CF, Stapleton JT (2006) Effect of early and late GBV-C viremia on survival of HIV-infected individuals: a meta-analysis. HIV Med 7: 173–180

    Article  Google Scholar 

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Affiliations

  1. Division of Biostatistics, Edwards Lifesciences, One Edwards Way, Irvine, CA, 92612, USA

    Suhong Zhang

  2. Department of Biostatistics, University of Iowa, C22 GH, 200 Hawkins Drive, Iowa City, IA, 52242, USA

    Ying Zhang & Kathryn Chaloner

  3. Department of Internal Medicine, University of Iowa and Iowa City VA Medical Center, SW54-15 GH, 200 Hawkins Drive, Iowa City, IA, 52242, USA

    Jack T. Stapleton

Authors
  1. Suhong Zhang
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  2. Ying Zhang
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  3. Kathryn Chaloner
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  4. Jack T. Stapleton
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Correspondence to Ying Zhang.

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Zhang, S., Zhang, Y., Chaloner, K. et al. A copula model for bivariate hybrid censored survival data with application to the MACS study. Lifetime Data Anal 16, 231–249 (2010). https://doi.org/10.1007/s10985-009-9139-z

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  • DOI: https://doi.org/10.1007/s10985-009-9139-z

Keywords

  • Association measure
  • Bivariate survival model
  • Copula
  • Current status data
  • Kendall’s τ
  • Right censored data
  • Empirical process