Skip to main content

Advertisement

SpringerLink
Log in

Joint modeling of longitudinal and survival data with the Cox model and two-phase sampling

  • Published:
Lifetime Data Analysis Aims and scope Submit manuscript

Abstract

A common objective of cohort studies and clinical trials is to assess time-varying longitudinal continuous biomarkers as correlates of the instantaneous hazard of a study endpoint. We consider the setting where the biomarkers are measured in a designed sub-sample (i.e., case-cohort or two-phase sampling design), as is normative for prevention trials. We address this problem via joint models, with underlying biomarker trajectories characterized by a random effects model and their relationship with instantaneous risk characterized by a Cox model. For estimation and inference we extend the conditional score method of Tsiatis and Davidian (Biometrika 88(2):447–458, 2001) to accommodate the two-phase biomarker sampling design using augmented inverse probability weighting with nonparametric kernel regression. We present theoretical properties of the proposed estimators and finite-sample properties derived through simulations, and illustrate the methods with application to the AIDS Clinical Trials Group 175 antiretroviral therapy trial. We discuss how the methods are useful for evaluating a Prentice surrogate endpoint, mediation, and for generating hypotheses about biological mechanisms of treatment efficacy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Cao W, Tsiatis A, Davidian M (2009) Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data. Biometrika 96(3):723–734

    Article  MathSciNet  MATH  Google Scholar 

  • Dafni U, Tsiatis A (1998) Evaluating surrogate markers of clinical outcome when measured with error. Biometrics 54(4):1445–1462

    Article  MATH  Google Scholar 

  • DeGruttola V, Tu X (1994) Modeling progression of CD4-lymphocyte count and its relationship to survival time. Biometrics 50(4):1003–1014

    Article  MATH  Google Scholar 

  • Gilbert P, Yu X, Rotnitzky A (2014) Optimal auxiliary-covariate-based two-phase sampling design for semiparametric efficient estimation of a mean or mean difference, with application to clinical trials. Stat Med 33(6):901–917

    Article  MathSciNet  Google Scholar 

  • Guo X, Carlin B (2004) Separate and joint modeling of longitudinal and event time data using standard computer packages. Am Stat 58(1):16–24

    Article  MathSciNet  Google Scholar 

  • Hall P, Racine J, Li Q (2004) Cross-validation and the estimation of conditional probability densities. J Am Stat Assoc 99(468):1015–1026

    Article  MathSciNet  MATH  Google Scholar 

  • Hammer S, Katzenstein D, Hughes M, Gundacker H, Schooley R, Haubrich R, Henry W, Lederman M, Phair J, Niu M, Hirsch M, Merigan T (1996) A trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults with CD4 cell counts from 200 to 500 per cubic millimeter. N Engl J Med 335(15):1081–1090

    Article  Google Scholar 

  • Haneuse S, Saegusa T, Lumley T (2011) osDesign: an R package for the analysis, evaluation, and design of two-phase and case-control studies. J Stat Softw 43(11):1–29

    Article  Google Scholar 

  • Haynes B, Gilbert P, McElrath M, Zolla-Pazner S, Tomaras G, Alam S, Evans D, Montefiori D, Karnasuta C, Sutthent R, Liao H, DeVico A, Williams Lewis C GK, Pinter A, Fong Y, Janes H, DeCamp A, Huang Y, Rao M, Billings E, Karasavvas N, Robb M, Ngauy V, de Souza M, Paris R, Ferrari G, Bailer R, Soderberg K, Andrews C, Berman P, Frahm N, De Rosa S, Alpert M, Yates N, Shen X, Koup R, Pitisuttithum P, Kaewkungwal J, Nitayaphan S, Rerks-Ngarm S, Michael N, Kim J (2012) Immune-correlates analysis of an HIV-1 vaccine efficacy trial. N Engl J Med 366(14):1275–1286

    Article  Google Scholar 

  • Henderson R, Diggle P, Dobson A (2000) Joint modelling of longitudinal measurements and event time data. Biostatistics 1(4):465–480

    Article  MATH  Google Scholar 

  • Kobayashi F, Kuroki M (2014) A new proportion measure of the treatment effect captured by candidate surrogate endpoints. Stat Med 33(19):3338–3353

    Article  MathSciNet  Google Scholar 

  • Lin D, Fleming T, DeGruttola V (1997) Estimating the proportion of treatment effect explained by a surrogate marker. Stat Med 16(13):1515–1527

    Article  Google Scholar 

  • Prentice R (1982) Covariate measurement errors and parameter estimation in a failure time regression model. Biometrika 69(2):331–342

    Article  MathSciNet  MATH  Google Scholar 

  • Prentice R (1986) A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika 73(1):1–11

    Article  MathSciNet  MATH  Google Scholar 

  • Prentice R (1989) Surrogate endpoints in clinical trials: definition and operational criteria. Stat Med 8(4):431–440

    Article  Google Scholar 

  • Qi L, Wang C, Prentice R (2005) Weighted estimators for proportional hazards regression with missing covariates. J Am Stat Assoc 100(472):1250–1263

    Article  MathSciNet  MATH  Google Scholar 

  • Rizopoulos D, Verbeke G, Lesaffre E (2009) Fully exponential Laplace approximations for the joint modelling of survival and longitudinal data. J R Stat Soc Ser B (Stat Methodol) 71(3):637–654

    Article  MathSciNet  MATH  Google Scholar 

  • Self S, Pawitan Y (1992) Modeling a marker of disease progression and onset of disease. In: Jewell N, Dietz K, Farewell V (eds) AIDS Epidemiol. Birkhäuser, Boston, pp 231–255

    Chapter  Google Scholar 

  • Song X, Davidian M, Tsiatis A (2002) An estimator for the proportional hazards model with multiple longitudinal covariates measured with error. Biostatistics 3(4):511–528

    Article  MATH  Google Scholar 

  • Tsiatis A, Davidian M (2001) A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error. Biometrika 88(2):447–458

    Article  MathSciNet  MATH  Google Scholar 

  • Tsiatis A, Davidian M (2004) Joint modeling of longitudinal and time-to-event data: an overview. Stat Sin 14:809–834

    MathSciNet  MATH  Google Scholar 

  • Tsiatis A, DeGruttola V, Wulfsohn M (1995) Modeling the relationship of survival to longitudinal data measured with error. Applications to survival and CD4 counts in patients with AIDS. J Am Stat Assoc 90(429):27–37

    Article  MATH  Google Scholar 

  • Wang C (2006) Corrected score estimator for joint modeling of longitudinal and failure time data. Stat Sin 16:235–253

    MathSciNet  MATH  Google Scholar 

  • Wang C, Xie S, Prentice R (2001) Recalibration based on an approximate relative risk estimator in Cox regression with missing covariates. Stat Sin 11:1081–1104

    MathSciNet  MATH  Google Scholar 

  • Wu L, Liu W, Yi G, Huang Y (2012) Analysis of longitudinal and survival data: joint modeling, inference methods, and issues. J Probab Stat. doi:10.1155/2012/640153

  • Wulfsohn M, Tsiatis A (1997) A joint model for survival and longitudinal data measured with error. Biometrics 53(1):330–339

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Institute Of Allergy And Infectious Diseases of the National Institutes of Health [Award Numbers R37AI054165, UM1AI068635], and by the Bill and Melinda Gates Foundation [Award Number OPP1110049]. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH or BMGF. The authors thank Ching-Yun Wang for statistical input, the AIDS Clinical Trials Group (ACTG) operations, laboratory, and statistical data management centers for generating and providing the ACTG 175 data, and the ACTG 175 study participants and investigators, in particular protocol chair Scott Hammer who reviewed and approved this work.

Author information

Authors and Affiliations

  1. Department of Biostatistics, University of Washington, Seattle, WA, 98195, USA

    Rong Fu & Peter B. Gilbert

  2. Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, Seattle, WA, 98109, USA

    Peter B. Gilbert

Authors
  1. Rong Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter B. Gilbert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rong Fu.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 449 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, R., Gilbert, P.B. Joint modeling of longitudinal and survival data with the Cox model and two-phase sampling. Lifetime Data Anal 23, 136–159 (2017). https://doi.org/10.1007/s10985-016-9364-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10985-016-9364-1

Keywords

Search

Navigation