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Lifetime Data Analysis

, 15:534 | Cite as

Model checks for Cox-type regression models based on optimally weighted martingale residuals

  • Axel Gandy
  • Uwe Jensen
Article

Abstract

We introduce directed goodness-of-fit tests for Cox-type regression models in survival analysis. “Directed” means that one may choose against which alternatives the tests are particularly powerful. The tests are based on sums of weighted martingale residuals and their asymptotic distributions. We derive optimal tests against certain competing models which include Cox-type regression models with different covariates and/or a different link function. We report results from several simulation studies and apply our test to a real dataset.

Keywords

Cox-regression Goodness-of-fit Martingale residuals Survival analysis 

References

  1. Aalen OO (1980) A model for nonparametric regression analysis of counting processes. In: Klonecki W, Kozek A, Rosinski J(eds) Mathematical statistics and probability theory. Lecture notes in statistics. Springer, New York, pp 1–25Google Scholar
  2. Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Statist 10(4): 1100–1120zbMATHCrossRefMathSciNetGoogle Scholar
  3. Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkzbMATHGoogle Scholar
  4. Bahadur RR (1960) Stochastic comparison of tests. Ann Math Statist 31: 276–295zbMATHCrossRefMathSciNetGoogle Scholar
  5. Barlow WE, Prentice RL (1988) Residuals for relative risk regression. Biometrika 75: 65–74zbMATHCrossRefMathSciNetGoogle Scholar
  6. Cox DR (1972) Regression models and life-tables. J Roy Statist Soc Ser B 34(2): 187–220zbMATHMathSciNetGoogle Scholar
  7. Davison AC, Hinkley DV (1997) Bootstrap methods and their application. Cambridge series in statistical and probabilistic mathematics, vol 1. Cambridge University Press, CambridgeGoogle Scholar
  8. Fine JP (2002) Comparing nonnested Cox models. Biometrika 89(3): 635–647zbMATHCrossRefMathSciNetGoogle Scholar
  9. Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, New YorkzbMATHGoogle Scholar
  10. Gandy A (2006) Directed model checks for regression models from survival analysis. Logos Verlag, Berlin, Dissertation, Universität UlmGoogle Scholar
  11. Gandy A (2009) Sequential implementation of Monte Carlo tests with uniformly bounded resampling risk. J Am Statist Assoc (to appear)Google Scholar
  12. Gandy A, Jensen U (2005a) Checking a semiparametric additive risk model. Lifetime Data Anal 11(4): 451–472zbMATHCrossRefMathSciNetGoogle Scholar
  13. Gandy A, Jensen U (2005b) On goodness of fit tests for Aalen’s additive risk model. Scand J Statist 32: 425–445zbMATHCrossRefMathSciNetGoogle Scholar
  14. Grambsch PM, Therneau TM (1994) Proportional hazards tests and diagnostics based on weighted residuals. Biometrika 81: 515–526 (correction: Volume 82 page 668)zbMATHCrossRefMathSciNetGoogle Scholar
  15. Grønnesby JK, Borgan Ø (1996) A method for checking regression models in survival analysis based on the risk score. Lifetime Data Anal 2(4): 315–328CrossRefGoogle Scholar
  16. Hall W, Wellner J (1980) Confidence bands for a survival curve from censored data. Biometrika 67: 133–144zbMATHCrossRefMathSciNetGoogle Scholar
  17. Khmaladze EV (1981) Martingale approach in the theory of goodness-of-fit tests. Theory Probab Appl 26: 240–257CrossRefMathSciNetGoogle Scholar
  18. Lin DY, Wei LJ (1989) The robust inference for the Cox proportional hazards model. J Am Statist Assoc 84(408): 1074–1078zbMATHCrossRefMathSciNetGoogle Scholar
  19. Lin DY, Wei LJ, Ying Z (1993) Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 80(3): 557–572zbMATHCrossRefMathSciNetGoogle Scholar
  20. Martinussen T, Aalen OO, Scheike TH (2008) The Mizon-Richard encompassing test for the Cox and Aalen additive hazards models. Biometrics 64(1): 164–171zbMATHCrossRefMathSciNetGoogle Scholar
  21. Marzec L, Marzec P (1997) Generalized martingale-residual processes for goodness-of-fit inference in Cox’s type regression models. Ann Statist 25(2): 683–714zbMATHCrossRefMathSciNetGoogle Scholar
  22. Nikitin Y (1995) Asymptotic efficiency of nonparametric tests. Cambridge University Press, CambridgezbMATHGoogle Scholar
  23. Prentice RL, Self SG (1983) Asymptotic distribution theory for Cox-type regression models with general relative risk form. Ann Statist 11(3): 804–813zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Institute of Applied Mathematics and StatisticsUniversity of HohenheimStuttgartGermany

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