Skip to main content

Advertisement

SpringerLink
Log in

Accelerated failure time models for recurrent event data analysis and joint modeling

  • Original paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

There are two commonly encountered problems in survival analysis: (a) recurrent event data analysis, where an individual may experience an event multiple times over follow-up; and (b) joint modeling, where the event time distribution depends on a longitudinally measured internal covariate. The proportional hazards (PH) family offers an attractive modeling paradigm for recurrent event data analysis and joint modeling. Although there are well-known techniques to test the PH assumption for standard survival data analysis, checking this assumption for joint modeling has received less attention. An alternative framework involves considering an accelerated failure time (AFT) model, which is particularly useful when the PH assumption fails. Note that there are AFT models that can describe data with wide ranging characteristics but have received far less attention in modeling recurrent event data and joint analysis of time-to-event and longitudinal data. In this paper, we develop methodology to analyze these types of data using the AFT family of distributions. Fitting these models is computationally and numerically much more demanding compared to standard survival data analysis. In particular, fitting a joint model is a computationally intensive task as it requires to approximate multiple integrals that do not have an analytic solution except in very special cases. We propose computational algorithms for statistical inference, and develop a software package to fit these models. The proposed methodology is demonstrated using both simulated and real data.

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

Similar content being viewed by others

References

  • Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723

    Article  MathSciNet  MATH  Google Scholar 

  • Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10(4):1100–1120

  • Broström G (2018) eha: event history analysis. https://CRAN.R-project.org/package=eha, r package version 2.6.0

  • Cai Q, Wang MC, Chan KCG (2017) Joint modeling of longitudinal, recurrent events and failure time data for survivor’s population. Biometrics 73(4):1150–1160

  • Chen MH, Ibrahim J, Sinha D (2004) A new joint model for longitudinal and survival data with a cure fraction. J Multivar Anal 91(1):18–34

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng SH (2004) Estimating marginal effects in accelerated failure time models for serial sojourn times among repeated events. Lifetime Data Anal 10:175–190

    Article  MathSciNet  MATH  Google Scholar 

  • Chi YY, Ibrahim JG (2006) Joint models for multivariate longitudinal and multivariate survival data. Biometrics 62(2):432–445

    Article  MathSciNet  MATH  Google Scholar 

  • Cook RJ, Lawless J (2007) The statistical analysis of recurrent events, 1st edn. Springer, New York

    MATH  Google Scholar 

  • Cox DR (1972) Regression models and life-tables. J Roy Stat Soc Ser B (Methodol) 34(2):187–220

    MathSciNet  MATH  Google Scholar 

  • Cox C, Matheson M (2014) A comparison of the generalized gamma and exponentiated Weibull distributions. Stat Med 33(21):3772–3780

    Article  MathSciNet  Google Scholar 

  • Cox DR, Oakes D (1984) Analysis of survival data. Chapman and Hall, London

    Google Scholar 

  • Dong JJ, Wang S, Wang L, Gill J, Cao J (2019) Joint modelling for organ transplantation outcomes for patients with diabetes and the end-stage renal disease. Stat Methods Med Res 28(9):2724–2737

    Article  MathSciNet  Google Scholar 

  • Elashoff RM, Li G, Li N (2008) A joint model for longitudinal measurements and survival data in the presence of multiple failure types. Biometrics 64(3):762–771

    Article  MathSciNet  MATH  Google Scholar 

  • Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7:457–511

    Article  MATH  Google Scholar 

  • Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2013) Bayesian data analysis, 3rd edn. Chapman and Hall/CRC, New York

    Book  MATH  Google Scholar 

  • Gelman A, Hwang J, Vehtari A (2014) Understanding predictive information criteria for Bayesian models. Stat Comput 24:997–1016

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh D (2004) Accelerated rates regression models for recurrent events. Lifetime Data Anal 10:247–261

    Article  MathSciNet  MATH  Google Scholar 

  • González JR, Fernandez E, Moreno V, Ribes J, Peris M, Navarro M, Cambray M, Borràs JM (2005) Sex differences in hospital readmission among colorectal cancer patients. J Epidemiol Commun Health 59(6):506–511

    Article  Google Scholar 

  • Gould LA, Boye ME, Crowther MJ, Ibrahim JG, Quartey G, Micallef S, Bois FY (2015) Joint modeling of survival and longitudinal non-survival data: current methods and issues. Stat Med 34(14):2181–2195

    Article  MathSciNet  Google Scholar 

  • Guo X, Carlin BP (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 

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

    Article  MATH  Google Scholar 

  • Hsieh F, Tseng YK, Wang JL (2006) Joint modeling of survival and longitudinal data: likelihood approach revisited. Biometrics 62:1037–1043

    Article  MathSciNet  MATH  Google Scholar 

  • Huang Y, Peng L (2009) Accelerated recurrence time models. Scand J Stat 36:636–648

    Article  MathSciNet  MATH  Google Scholar 

  • Hwang BS, Pennell ML (2014) Semiparametric Bayesian joint modeling of a binary and continuous outcome with applications in toxicological risk assessment. Stat Med 33:1162–1175

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53(282):457–481

    Article  MathSciNet  MATH  Google Scholar 

  • Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Li L, XiaoYun M, LiuQuan S (2010) A class of additive-accelerated means regression models for recurrent event data. Sci China Math 53:3139–3151

    Article  MathSciNet  MATH  Google Scholar 

  • Lin DY, Wei LJ, Ying Z (1998) Accelerated failure time models for counting processes. Biometrika 85:605–618

    Article  MathSciNet  MATH  Google Scholar 

  • Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure rate data. IEEE Trans Reliab 42(2):299–302

    Article  MATH  Google Scholar 

  • Mudholkar GS, Srivastava DK (1995) The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37(4):436–445

    Article  MATH  Google Scholar 

  • Prentice RL, Williams BJ, Peterson AV (1981) On the regression analysis of multivariate failure time data. Biometrika 68(2):373–379

    Article  MathSciNet  MATH  Google Scholar 

  • Press W, Teukolsky S, Vetterling W, Flannery B (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, New York

    MATH  Google Scholar 

  • R Core Team (2020) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/

  • Rizopoulos D (2010) JM: an R package for the joint modelling of longitudinal and time-to-event data. J Stat Softw 35(9):1–33

    Article  Google Scholar 

  • Rizopoulos D (2012) Joint Models for longitudinal and time-to-event data with applications in R, 1st edn. Chapman and Hall/CRC, Florida

    Book  MATH  Google Scholar 

  • Rizopoulos D, Hatfield LA, Carlin BP, Takkenberg JJM (2014) Combining dynamic predictions from joint models for longitudinal and time-to-event data using Bayesian model averaging. J Am Stat Assoc 109(508):1385–1397

    Article  MathSciNet  Google Scholar 

  • Rondeau V, Gonzalez J, Mazroui Y, Mauguen A, Diakite A, Laurent A, Lopez M, Król A, Sofeu C (2019) frailtypack: general frailty models: shared, joint and nested frailty models with prediction; evaluation of failure-time surrogate endpoints. https://CRAN.R-project.org/package=frailtypack, r package version 3.0.3

  • Schoenfeld DA (1980) Chi-squared goodness-of-fit tests for the proportional hazards regression model. Biometrika 67:145–153

    Article  MathSciNet  MATH  Google Scholar 

  • Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit (with discussion). J R Stat Soc Ser B (Methodol) 64(4):583–639

    Article  MathSciNet  MATH  Google Scholar 

  • Spiegelhalter D, Thomas A, Best N, Lunn D (2003) WinBUGS user manual. University of Cambridge, MRC Biostatistics Unit

    Google Scholar 

  • Stacy EW (1962) A generalization of gamma distribution. Ann Math Stat 33(3):1187–1192

    Article  MathSciNet  MATH  Google Scholar 

  • Stan Development Team (2020a) RStan: the R interface to Stan, R package version 2.21.2. https://mc-stan.org

  • Stan Development Team (2020b) Stan modeling language users guide and reference manual, Version 2.27. https://mc-stan.org

  • Sun L, Su B (2008) A class of accelerated means regression models for recurrent event data. Lifetime Data Anal 14:357–375

    Article  MathSciNet  MATH  Google Scholar 

  • Therneau TM, Grambsch PM (2000) Modelling survival Ddta: extending the Cox model, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  • Tseng YK, Hsieh F, Wang JL (2005) Joint modelling of accelerated failure time and longitudinal data. Biometrika 92:587–603

    Article  MathSciNet  MATH  Google Scholar 

  • Tsiatis AA, Davidian M (2004) Joint modeling of longitudinal and time-to-event data: an overview. Stat Sin 14(3):809–834

    MathSciNet  MATH  Google Scholar 

  • Watanabe S (2010) Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J Mach Learn Res 11:3571–3594

    MathSciNet  MATH  Google Scholar 

  • Wei LJ, Lin DY, Weissfeld L (1989) Regression analysis of multivariate incomplete failure time data by modelling marginal distributions. J Am Stat Assoc 84(408):1065–1073

    Article  Google Scholar 

  • Wu L, Liu W, Hu XJ (2010) Joint inference on hiv viral dynamics and immune suppression in presence of measurement errors. Biometrics 66(2):327–335

    Article  MathSciNet  MATH  Google Scholar 

  • Wulfsohn MS, Tsiatis AA (1997) A joint model for survival and longitudinal data measured with error. Biometrics 57(1):330–339

    Article  MathSciNet  MATH  Google Scholar 

  • Xu G, Chiou SH, Huang CY, Wang MC, Yan J (2017) Joint scale-change models for recurrent events and failure time. J Am Stat Assoc 112(518):794–805

    Article  MathSciNet  Google Scholar 

  • Zhang H, Wu L (2019) Joint model of accelerated failure time and mechanistic nonlinear model for censored covariates, with application in HIV/AIDS. Ann Appl Stat 13(4):2140–2157

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao M, Wang Y, Zhou Y (2016) Accelerated failure time model with quantile information. Ann Inst Stat Math 68:1001–1024

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant to SA Khan.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, S7N 5E6, Canada

    Shahedul A. Khan & Nyla Basharat

Authors
  1. Shahedul A. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nyla Basharat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shahedul A. Khan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant to SA Khan.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 254 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khan, S.A., Basharat, N. Accelerated failure time models for recurrent event data analysis and joint modeling. Comput Stat 37, 1569–1597 (2022). https://doi.org/10.1007/s00180-021-01171-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-021-01171-7

Keywords

Search

Navigation