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Tests for Hazard Transformation


The semiparametric Cox proportional hazards model is routinely adopted to model time-to-event data. Proportionality is a strong assumption, especially when follow-up time, or study duration, is long. Zeng and Lin (J. R. Stat. Soc., Ser. B, 69:1–30, 2007) proposed a useful generalisation through a family of transformation models which allow hazard ratios to vary over time. In this paper we explore a variety of tests for the need for transformation, arguing that the Cox model is so ubiquitous that it should be considered as the default model, to be discarded only if there is good evidence against the model assumptions. Since fitting an alternative transformation model is more complicated than fitting the Cox model, especially as procedures are not yet incorporated in standard software, we focus mainly on tests which require a Cox fit only. A score test is derived, and we also consider performance of omnibus goodness-of-fit tests based on Schoenfeld residuals. These tests can be extended to compare different transformation models. In addition we explore the consequences of fitting a misspecified Cox model to data generated under a true transformation model. Data on survival of 1043 leukaemia patients are used for illustration.

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  1. Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York

    MATH  Google Scholar 

  2. Collett D (1994) Modelling survival data in medical research. Chapman & Hall, London, pp. 149–197

    Google Scholar 

  3. Commenges D, Andersen PK (1995) Score test of homogeneity for survival data. Lifetime Data Anal. 1:145–156

    Article  MATH  MathSciNet  Google Scholar 

  4. Commenges D, Jacqmin-Gadda H (1997) Generalized score test of homogeneity based on correlated random effects models. J. R. Stat. Soc. Ser. B 59:157–171

    Article  MATH  MathSciNet  Google Scholar 

  5. Cox DR (1972) Regression models and life tables. J. R. Stat. Soc. Ser. B 34:187–220

    MATH  Google Scholar 

  6. Cox DR, Hinkley DV (1974) Theoretical statistics. Chapman and Hall, London

    MATH  Google Scholar 

  7. Csörgő C, Faraway JJ (1996) The exact and asymptotic distribution of Cramér–Von Mises statistics. J. R. Stat. Soc. Ser. B 58:221–234

    Google Scholar 

  8. Drylewicz J, Commenges D, Thiébaut R (2010) Score tests for exploring complex models: application to HIV dynamics models. Biom. J. 52:10–21

    Article  Google Scholar 

  9. Gradshteyn IS Ryzhik IM (1994) Table of integrals, series, and products, 5th edn. Academic Press, London

    MATH  Google Scholar 

  10. Henderson R, Oman P (1999) Effect of frailty on marginal regression estimates in survival analysis. J. R. Stat. Soc. Ser. B 61:367–379

    Article  MATH  MathSciNet  Google Scholar 

  11. Henderson R, Shimakura S, Gorst D (2002) Modeling spatial variation in leukemia survival data. J. Am. Stat. Assoc. 97:965–972

    Article  MATH  MathSciNet  Google Scholar 

  12. Hjort NJ (1992) On inference for parametric survival data models. Int. Stat. Rev. 60:355–387

    Article  MATH  Google Scholar 

  13. Ho WK (2009) Transformation and dropout models for censored data. Unpublished PhD Thesis, Newcastle University, UK

  14. Hougaard P (2000) Analysis of multivariate survival data. Springer, New York

    MATH  Google Scholar 

  15. Klein JP, Moeschberger ML (2003) Survival analysis, 353-391. Springer, New York, pp. 353–391

    Google Scholar 

  16. O’Quigley J, Stare J (2003) Cumulative empirical process for survival models. in: Proceedings of the 25th international conference on information technology interfaces, pp. 205–210

  17. Schoenfeld D (1982) Partial residuals for the proportional hazards regression model. Biometrika 69:239–241

    Article  Google Scholar 

  18. Serfozo R (2009) Basics of applied stochastic processes. Springer, New York

    Book  MATH  Google Scholar 

  19. Solomon PJ (1984) Effects of misspecification of regression models in the analysis of survival data. Biometrika 71:291–298

    Article  MATH  MathSciNet  Google Scholar 

  20. Stare J, Pohar M, Henderson R (2005) Goodness of fit of relative survival models. Stat. Med. 24:1–15

    Article  MathSciNet  Google Scholar 

  21. Struthers CA, Kalbfleisch JD (1986) Misspecified proportional hazard models. Biometrika 73:363–369

    Article  MATH  MathSciNet  Google Scholar 

  22. Therneau TH, Grambsch PM (2000) Modeling survival data: extending the Cox model. Springer, New York

    MATH  Google Scholar 

  23. Zeng D, Lin DY (2006) Efficient estimation of semiparametric transformation models for counting processes. Biometrika 93:627–640

    Article  MATH  MathSciNet  Google Scholar 

  24. Zeng D, Lin DY (2007) Maximum likelihood estimation in semiparametric regression models with censored data. J. R. Stat. Soc. Ser. B 69:1–30

    Article  MathSciNet  Google Scholar 

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Correspondence to Robin Henderson.

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Ho, W.K., Henderson, R. & Philipson, P.M. Tests for Hazard Transformation. Stat Biosci 2, 41–64 (2010).

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  • Box–Cox transformation
  • Diagnostics
  • Score test
  • Schoenfeld residuals
  • Misspecification
  • Survival analysis