Joint Modeling of Repeated Measures and Competing Failure Events in a Study of Chronic Kidney Disease

  • Wei Yang
  • Dawei Xie
  • Qiang Pan
  • Harold I. Feldman
  • Wensheng Guo
S.I. : Organ Failure and Transplantation


We are motivated by the chronic renal insufficiency cohort (CRIC) study to identify risk factors for renal progression in patients with chronic kidney diseases. The CRIC study collects two types of renal outcomes: glomerular filtration rate (GFR) estimated annually and end-stage renal disease (ESRD). A related outcome of interest is death which is a competing event for ESRD. A joint modeling approach is proposed to model a longitudinal outcome and two competing survival outcomes. We assume multivariate normality on the joint distribution of the longitudinal and survival outcomes. Specifically, a mixed effects model is fit on the longitudinal outcome and a linear model is fit on each survival outcome. The three models are linked together by having the random terms of the mixed effects model as covariates in the survival models. EM algorithm is used to estimate the model parameters, and the nonparametric bootstrap is used for variance estimation. A simulation study is designed to compare the proposed method with an approach that models the outcomes sequentially in two steps. We fit the proposed model to the CRIC data and show that the protein-to-creatinine ratio is strongly predictive of both estimated GFR and ESRD but not death.


Joint modeling Multivariate normality Informative dropout Competing risk Chronic kidney disease 


  1. 1.
    Anderson AH, Yang W, Hsu C-Y, Joffe MM, Leonard MB, Xie D, Chen J, Greene T, Jaar BG, Kao P et al (2012) Estimating gfr among participants in the chronic renal insufficiency cohort (cric) study. Am J Kidney Dis 60(2):250–261CrossRefGoogle Scholar
  2. 2.
    Andrinopoulou E-R, Rizopoulos D, Takkenberg JJM, Lesaffre E (2014) Joint modeling of two longitudinal outcomes and competing risk data. Stat Med 33(18):3167–3178MathSciNetCrossRefGoogle Scholar
  3. 3.
    Coresh J, Turin T, Matsushita K et al (2014) Decline in estimated glomerular filtration rate and subsequent risk of end-stage renal disease and mortality. JAMA 311(24):2518–2531Google Scholar
  4. 4.
    DeGruttola V, Tu XM (1994) Modelling progression of CD4-lymphocyte count and its relationship to survival time. Biometrics 50(4):1003–1014CrossRefMATHGoogle Scholar
  5. 5.
    Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the em algorithm. J R Stat Soc Ser B (methodological) 39:1–38Google Scholar
  6. 6.
    Deslandes E, Chevret S (2010) Joint modeling of multivariate longitudinal data and the dropout process in a competing risk setting: application to ICU data. BMC Med Res Methodol 10(1):69CrossRefGoogle Scholar
  7. 7.
    Diggle P, Kenward MG (1994) Informative drop-out in longitudinal data analysis. J R Stat Soc Ser C (Applied Statistics) 43(1):49–93MATHGoogle Scholar
  8. 8.
    Efron B, Tibshirani RJ (1994) An introduction to the bootstrap. CRC Press, Boca RatonGoogle Scholar
  9. 9.
    Elashoff RM, Li G, Li N (2007) An approach to joint analysis of longitudinal measurements and competing risks failure time data. Stat Med 26(14):2813–2835MathSciNetCrossRefGoogle Scholar
  10. 10.
    Feldman HI, Appel LJ, Chertow GM, Cifelli D, Cizman B, Daugirdas J, Fink JC, Franklin-Becker ED, Go AS, Hamm LL et al (2003) The chronic renal insufficiency cohort (cric) study: design and methods. J Am Soc Nephrol 14(suppl 2):S148–S153CrossRefGoogle Scholar
  11. 11.
    Fine JP, Gray RJ (1999) A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc 94(446):496–509MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Guo X, Carlin BP (2004) Separate and joint modeling of longitudinal and event time data using standard computer packages. Am Stat 58(1):16–24MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hillis SL (1995) Residual plots for the censored data linear regression model. Stat Med 14(18):2023–2036CrossRefGoogle Scholar
  14. 14.
    Hogan JW, Roy J, Korkontzelou C (2004) Handling drop-out in longitudinal studies. Stat Med 23(9):1455–1497CrossRefGoogle Scholar
  15. 15.
    Hsieh F, Tseng Y-K, Wang J-L (2006) Joint modeling of survival and longitudinal data: likelihood approach revisited. Biometrics 62(4):1037–1043MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Huang X, Li G, Elashoff RM, Pan J (2011) A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects. Lifetime Data Anal 17:80–100MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Klahr S, Levey AS, Beck GJ, Caggiula AW, Hunsicker L, Kusek JW, Striker G (1994) The effects of dietary protein restriction and blood-pressure control on the progression of chronic renal disease. N Engl J Med 330(13):877–884CrossRefGoogle Scholar
  18. 18.
    Koller MT, Raatz H, Steyerberg EW, Wolbers M (2012) Competing risks and the clinical community: irrelevance or ignorance? Stat Med 31(11–12):1089–1097MathSciNetCrossRefGoogle Scholar
  19. 19.
    Law M, Jackson D (2015) Residual plots for linear regression models with censored outcome data: a refined method for visualising residual uncertainty. Commun Stat Simul Comput. doi:10.1080/03610918.2015.1076470
  20. 20.
    Leung K-M, Elashoff RM, Afifi AA (1997) Censoring issues in survival analysis. Annu Rev Public Health 18(1):83–104CrossRefGoogle Scholar
  21. 21.
    Li L, Hu B, Greene T (2009) A semiparametric joint model for longitudinal and survival data with application to hemodialysis study. Biometrics 65(3):737–745MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Liu L, Ma JZ, O’Quigley J (2008) Joint analysis of multi-level repeated measures data and survival: an application to the end stage renal disease (esrd) data. Stat Med 27(27):5679–5691MathSciNetCrossRefGoogle Scholar
  23. 23.
    Prentice RL, Kalbfleisch JD, Peterson AV Jr, Flournoy N, Farewell V, Breslow N (1978) The analysis of failure times in the presence of competing risks. Biometrics 34:541–554Google Scholar
  24. 24.
    Ratcliffe SJ, Guo W, Ten Have TR (2004) Joint modeling of longitudinal and survival data via a common frailty. Biometrics 60(4):892–899MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Rizopoulos D (2010) Joint modelling of longitudinal and time-to-event data: challenges and future directions. In: 45th Scientific meeting of the Italian statistical society. Universitadi Padova, PadovaGoogle Scholar
  26. 26.
    Rosansky SJ, Glassock RJ (2014) Is a decline in estimated gfr an appropriate surrogate end point for renoprotection trials & quest. Kidney Int 85(4):723–727CrossRefGoogle Scholar
  27. 27.
    Schluchter MD (1992) Methods for the analysis of informatively censored longitudinal data. Stat Med 11(14–15):1861–1870CrossRefGoogle Scholar
  28. 28.
    Song X, Davidian M, Tsiatis AA (2002) A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data. Biometrics 58(4):742–753MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Stevens LA, Greene T, Levey AS (2006) Surrogate end points for clinical trials of kidney disease progression. Clin J Am Soc Nephrol 1(4):874–884CrossRefGoogle Scholar
  30. 30.
    Taylor JMG, Park Y, Ankerst DP, Proust-Lima C, Williams S, Kestin L, Bae K, Pickles T, Sandler H (2013) Real-time individual predictions of prostate cancer recurrence using joint modelsGoogle Scholar
  31. 31.
    Trautmann H, Steuer D, Mersmann O, Bornkamp B (2014) truncnorm: truncated normal distribution. R package version 1.0-7Google Scholar
  32. 32.
    Tseng Y-K, Hsieh F, Wang J-L (2005) Joint modelling of accelerated failure time and longitudinal data. Biometrika 92(3):587–603MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tsiatis A (1975) A nonidentifiability aspect of the problem of competing risks. Proc Nat Acad Sci 72(1):20–22MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tsiatis AA, Davidian M (2001) A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error. Biometrika 88(2):447–458MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Tsiatis AA, Davidian M (2004) Joint modeling of longitudinal and time-to-event data: an overview. Stat Sin 14(3):809–834MathSciNetMATHGoogle Scholar
  36. 36.
    Tsiatis AA, DeGruttola V, Wulfsohn MS (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):27CrossRefMATHGoogle Scholar
  37. 37.
    Vonesh EF, Greene T, Schluchter MD (2006) Shared parameter models for the joint analysis of longitudinal data and event times. Stat Med 25(1):143–163MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wang Y, Taylor JMG (2001) Jointly modeling longitudinal and event time data with application to acquired immunodeficiency syndrome. J Am Stat Assoc 96(455):895–905Google Scholar
  39. 39.
    Wilhelm S, Manjunath BG (2015) tmvtnorm: truncated multivariate normal and student t distribution. R package version 1.4-10Google Scholar
  40. 40.
    Williamson PR, Kolamunnage-Dona R, Philipson P, Marson AG (2008) Joint modelling of longitudinal and competing risks data. Stat Med 27(30):6426–6438MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wright JT Jr, Bakris G, Greene T, Agodoa LY, Appel LJ, Charleston J, Cheek D, Douglas-Baltimore JG, Gassman J, Glassock R et al (2002) Effect of blood pressure lowering and antihypertensive drug class on progression of hypertensive kidney disease: results from the aask trial. JAMA 288(19):2421–2431CrossRefGoogle Scholar
  42. 42.
    Wu L, Liu W, Yi GY, Huang Y (2012) Analysis of longitudinal and survival data: joint modeling, inference methods, and issues. J Probab Stat 1–17:2012MathSciNetMATHGoogle Scholar
  43. 43.
    Yu M, Law NJ, Taylor JM, Sandler HM (2004) Joint longitudinal-survival-cure models and their application to prostate cancer. Stat Sin 14(3):835–862MathSciNetMATHGoogle Scholar

Copyright information

© International Chinese Statistical Association 2016

Authors and Affiliations

  • Wei Yang
    • 1
  • Dawei Xie
    • 1
  • Qiang Pan
    • 1
  • Harold I. Feldman
    • 1
  • Wensheng Guo
    • 1
  1. 1.Department of Biostatistics and EpidemiologyUniversity of Pennsylvania School of MedicinePhiladelphiaUSA

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