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Nonparametric change point estimation for survival distributions with a partially constant hazard rate

  • Alessandra R. Brazzale
  • Helmut Küchenhoff
  • Stefanie Krügel
  • Tobias S. Schiergens
  • Heiko Trentzsch
  • Wolfgang Hartl
Article

Abstract

We present a new method for estimating a change point in the hazard function of a survival distribution assuming a constant hazard rate after the change point and a decreasing hazard rate before the change point. Our method is based on fitting a stump regression to p values for testing hazard rates in small time intervals. We present three real data examples describing survival patterns of severely ill patients, whose excess mortality rates are known to persist far beyond hospital discharge. For designing survival studies in these patients and for the definition of hospital performance metrics (e.g. mortality), it is essential to define adequate and objective end points. The reliable estimation of a change point will help researchers to identify such end points. By precisely knowing this change point, clinicians can distinguish between the acute phase with high hazard (time elapsed after admission and before the change point was reached), and the chronic phase (time elapsed after the change point) in which hazard is fairly constant. We show in an extensive simulation study that maximum likelihood estimation is not robust in this setting, and we evaluate our new estimation strategy including bootstrap confidence intervals and finite sample bias correction.

Keywords

Change point Survival Hazard rate ICU Acute phase 

Notes

Acknowledgements

We would like to thank the Associate Editor and the two anonymous Referees for their careful reading of the paper and the most useful comments which greatly helped us improving it.

References

  1. Altun M, Comert SV (2016) A change-point based reliability prediction model using field return data. Reliab Eng Syst Saf 156:175–184CrossRefGoogle Scholar
  2. Antoniadis A, Gijbels I, MacGibbon B (2000) Non-parametric estimation for the location of a change-point in an otherwise smooth hazard function under random censoring. Scand J Stat 27:501–519MathSciNetCrossRefzbMATHGoogle Scholar
  3. Callcut RA, Wakam G, Conroy AS, Kornblith L, Howard BM, Campion EM, Nelson MF, Mell MW, Cohen MJ (2016) Discovering the truth about life after discharge: long-term trauma-related mortality. J Trauma Acute Care Surg 80:210–215CrossRefGoogle Scholar
  4. Chang IS, Chen CH, Hsiung CA (1994) Estimation in change-point hazard rate models with random censorship. In: Carlstein E, Müuller HG, Siegmund D (eds) Change point problems, Institute of mathematical statistics lecture notes, vol 23, pp 78–92.Google Scholar
  5. Davison AC, Hinkley DV (1997) Bootstrap methods and their application. Cambridge University Press, New YorkGoogle Scholar
  6. Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman & Hall, New YorkGoogle Scholar
  7. Eriksson M, Brattström O, Larsson E, Oldner A (2016) Causes of excessive late death after trauma compared with a matched control cohort. Br J Surg 103:1282–1289CrossRefGoogle Scholar
  8. Gijbels I, Gürler U (2003) Estimation of a change point in a hazard function based on censored data. Life Time Data Anal 9:395–411MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gürler U, Yenigün CD (2011) Full and conditional likelihood approaches for hazard change point estimation with truncated and censored data. Comput Stat Data Anal 55:2856–2870MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kleyner A, Sandborn P (2005) A warranty forecasting model based on piecewise statistical distributions and stochastic simulation. Reliab Eng Syst Saf 88:207–214CrossRefGoogle Scholar
  11. Li Y, Qian L, Zhang W (2013) Estimation in a change-point hazard regression model with long-term survivors. Stat Probab Lett 83:1683–1691MathSciNetCrossRefzbMATHGoogle Scholar
  12. Loader CR (1991) Inference for a hazard rate change point. Biometrika 78:749–757MathSciNetCrossRefzbMATHGoogle Scholar
  13. Mallik A, Sen B, Banerjee M, Michailidis G (2011) Threshold estimation based on a p-value framework in dose-response and regression settings. Biometrika 98:887–900MathSciNetCrossRefzbMATHGoogle Scholar
  14. Matthews DE, Farewell VT (1982) On testing for a constant hazard rate against a change point alternative. Biometrics 38:463–468CrossRefGoogle Scholar
  15. Müller HG, Wang JL (1990) Nonparametric analysis of changes in hazard rates for censored survival data. An alternative to change point models. Biometrika 77:305–314MathSciNetCrossRefzbMATHGoogle Scholar
  16. Noura AA, Read KLQ (1990) Proportional hazards change point models in survival analysis. J Roy Stat Soc Ser C (Appl Stat) 39:241–253zbMATHGoogle Scholar
  17. Pouw ME, Peelen LM, Moons KG, Kalkman CJ, Lingsma HF (2013) Including post-discharge mortality in calculation of hospital standardised mortality ratios: retrospective analysis of hospital episode statistics. Br Med J 347:f5913CrossRefGoogle Scholar
  18. Schiergens TS, Dörsch M, Mittermeier L, Brand K, Küchenhoff H, Lee SML, Feng H, Jauch KW, Werner J, Thasler WE (2015) Thirty-day mortality leads to underestimation of postoperative death after liver resection: a novel method to define the acute postoperative period. Surgery 158:1530–1537CrossRefGoogle Scholar
  19. Schneider CP, Fertmann J, Geiger S, Wolf H, Biermaier H, Hofner B, Küchenhoff H, Jauch KW, Hartl WH (2010) Long-term survival after surgical critical illness: the impact of prolonged preceding organ support therapy. Ann Surg 251:1145–1153CrossRefGoogle Scholar
  20. Wang J, Zheng M, Yu W (2014) Wavelet analysis of change points in nonparametric hazard rate models under random censorship. Commun Stat Theory Methods 43:1956–1978MathSciNetCrossRefzbMATHGoogle Scholar
  21. Yang CH, Yuan T, Kuo W, Kuo Y (2012) Non-parametric Bayesian modeling of hazard rate with a change point for nanoelectronic devices. IIE Trans 44:496–506CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Scienze StatisticheUniversità degli Studi di PadovaPadovaItaly
  2. 2.Statistical Consulting Unit, Department of StatisticsLudwig-Maximilians-Universität MünchenMunichGermany
  3. 3.Department of StatisticsLudwig-Maximilians-Universität MünchenMunichGermany
  4. 4.Department of General, Visceral, Transplantation and Vascular Surgery, University School of Medicine, Grosshadern CampusLudwig-Maximilians-Universität MünchenMunichGermany
  5. 5.Institut für Notfallmedizin und Medizinmanagement INM, Klinikum der Universität MünchenLudwig-Maximilians-Universität MünchenMunichGermany

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