Advertisement

Lifetime Data Analysis

, Volume 13, Issue 1, pp 1–16 | Cite as

Data-driven smooth tests of the proportional hazards assumption

  • David Kraus
Article

Abstract

A new test of the proportional hazards assumption in the Cox model is proposed. The idea is based on Neyman’s smooth tests. The Cox model with proportional hazards (i.e. time-constant covariate effects) is embedded in a model with a smoothly time-varying covariate effect that is expressed as a combination of some basis functions (e.g., Legendre polynomials, cosines). Then the smooth test is the score test for significance of these artificial covariates. Furthermore, we apply a modification of Schwarz’s selection rule to choosing the dimension of the smooth model (the number of the basis functions). The score test is then used in the selected model. In a simulation study, we compare the proposed tests with standard tests based on the score process.

Keywords

Cox model Neyman’s smooth test Proportional hazards assumption Schwarz’s selection rule 

Notes

Acknowledgements

I would like to thank two anonymous reviewers for their detailed insightful comments that lead to a significant improvement of the paper and inspired me for further research. I gratefully acknowledge that the work has been supported by the GAČR Grant No. 201/05/H007 and the GAAV Grand No. IAA101120604. Computations have been carried out in METACentrum (Czech academic supercomputer network).

References

  1. Abrahamowicz M, MacKenzie T, Esdaile JM (1996) Time-dependent hazard ratio: modeling and hypothesis testing with application in lupus nephritis. J Am Stat Assoc 91:1432–1439zbMATHCrossRefGoogle Scholar
  2. Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10:1100–1120zbMATHMathSciNetGoogle Scholar
  3. Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkzbMATHGoogle Scholar
  4. Claeskens G, Hjort NL (2004) Goodness of fit via non-parametric likelihood ratios. Scand J Stat 31:487–513zbMATHCrossRefMathSciNetGoogle Scholar
  5. Cox DR (1972) Regression models and life-tables. J Roy Stat Soc Ser B 34:187–220zbMATHGoogle Scholar
  6. Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, New YorkzbMATHGoogle Scholar
  7. Inglot T, Ledwina T (2004) Refined version of data driven Neyman’s test. Preprint 001, Institute of Mathematics, Wrocław University of TechnologyGoogle Scholar
  8. Inglot T, Kallenberg WCM, Ledwina T (1997) Data driven smooth tests for composite hypotheses. Ann Stat 25:1222–1250zbMATHCrossRefMathSciNetGoogle Scholar
  9. Janssen A (2003) Which power of goodness of fit tests can really be expected: intermediate versus contiguous alternatives. Stat Decisions 21:301–325zbMATHCrossRefMathSciNetGoogle Scholar
  10. Kallenberg WCM, Ledwina T (1995) On data driven Neyman’s tests. Prob Math Stat 15:409–426zbMATHMathSciNetGoogle Scholar
  11. Kallenberg WCM, Ledwina T (1997) Data-driven smooth tests when the hypothesis is composite. J Am Stat Assoc 92:1094–1104zbMATHCrossRefMathSciNetGoogle Scholar
  12. Kraus D (2006) Identifying nonproportional covariates in the Cox model. Research Report 2170, Institute of Information Theory and Automation, PragueGoogle Scholar
  13. Kvaløy JT, Neef LR (2004) Tests for the proportional intensity assumption based on the score process. Lifetime Data Anal 10:139–157zbMATHCrossRefMathSciNetGoogle Scholar
  14. Ledwina T (1994) Data-driven version of Neyman’s smooth test of fit. J Am Stat Assoc 89:1000–1005zbMATHCrossRefMathSciNetGoogle Scholar
  15. Lin DY, Wei LJ, Ying Z (1993) Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 80:557–572zbMATHCrossRefMathSciNetGoogle Scholar
  16. Martinussen T, Scheike TH, Skovgaard IM (2002) Efficient estimation of fixed and time-varying covariate effects in multiplicative intensity models. Scand J Stat 29:57–74zbMATHCrossRefMathSciNetGoogle Scholar
  17. Peña EA (1998a) Smooth goodness-of-fit tests for composite hypothesis in hazard based models. Ann Stat 26:1935–1971zbMATHCrossRefGoogle Scholar
  18. Peña EA (1998b) Smooth goodness-of-fit tests for the baseline hazard in Cox’s proportional hazards model. J Am Stat Assoc 93:673–692zbMATHCrossRefGoogle Scholar
  19. Peña EA (2003) Classes of fixed-order and adaptive smooth goodness-of-fit tests with discrete right-censored data. In: Mathematical and statistical methods in reliability (Trondheim, 2002). World Sci Publishing, River EdgeGoogle Scholar
  20. Scheike TH, Martinussen T (2004) On estimation and tests of time-varying effects in the proportional hazards model. Scand J Stat 31:51–62zbMATHCrossRefMathSciNetGoogle Scholar
  21. Woodroofe M (1978) Large deviations of likelihood ratio statistics with applications to sequential testing. Ann Stat 6:72–84zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Institute of Information Theory and AutomationPrague 8Czech Republic
  2. 2.Department of StatisticsCharles University in PraguePragueCzech Republic

Personalised recommendations