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Lifetime Data Analysis

, Volume 1, Issue 2, pp 145–156 | Cite as

Score test of homogeneity for survival data

  • D. Commenges
  • P. K. Andersen
Article

Abstract

If follow-up is made for subjects which are grouped into units, such as familial or spatial units then it may be interesting to test whether the groups are homogeneous (or independent for given explanatory variables). The effect of the groups is modelled as random and we consider a frailty proportional hazards model which allows to adjust for explanatory variables. We derive the score test of homogeneity from the marginal partial likelihood and it turns out to be the sum of a pairwise correlation term of martingale residuals and an overdispersion term. In the particular case where the sizes of the groups are equal to one, this statistic can be used for testing overdispersion. The asymptotic variance of this statistic is derived using counting process arguments. An extension to the case of several strata is given. The resulting test is computationally simple; its use is illustrated using both simulated and real data. In addition a decomposition of the score statistic is proposed as a sum of a pairwise correlation term and an overdispersion term. The pairwise correlation term can be used for constructing a statistic more robust to departure from the proportional hazard model, and the overdispesion term for constructing a test of fit of the proportional hazard model.

Keywords

test of homogeneity overdispersion survival data partial likelihood counting processes 

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References

  1. O. O. Aalen, “Heterogeneity in survival analysis,”Statist. Med. vol. 7 pp. 1121–1137, 1988.Google Scholar
  2. P. K. Andersen, Ø. Borgan, R. D. Gill and N. Keiding,Statistical Models Based on Counting Processes, Springer-Verlag: New York, 1993.Google Scholar
  3. J. R. Batchelor and M. Hackett, “HLA matching in treatment of burned patients with skin allografts,”Lancet vol. 2 pp. 581–583, 1970.Google Scholar
  4. N. E. Breslow, “Covariance analysis of censored survival data,”Biometrics vol. 30 pp. 89–99, 1974.Google Scholar
  5. D. G. Clayton, “A model for association in bivariate life-tables and its application in epidemiological studies of family tendency in chronic disease incidence,”Biometrika vol. 65 pp. 141–151, 1978.Google Scholar
  6. D. G. Clayton and J. Cuzick, “Multivariate generalizations of the proportional hazards model (with discussion),”J. Roy. Statist. Soc. A vol. 148 pp. 82–117, 1985.Google Scholar
  7. D. Commenges and P. K. Andersen, “Score test of homogeneity for survival data,” Research Report 93/5, Statistical Research Unit, University of Copenhaguen, 1993.Google Scholar
  8. D. Commenges, L. Letenneur, H. Jacqmin, T. Moreau, and J.-F. Dartigues, “Test of homogeneity of binary data with explanatory variables,”Biometrics vol. 50 pp. 613–620, 1994.Google Scholar
  9. D. R. Cox, “Partial likelihood,”Biometrika vol. 66 pp. 269–276, 1975.Google Scholar
  10. D. R. Cox, “Some remarks on overdispersion.’Biometrika vol. 70 pp. 269–74, 1983.Google Scholar
  11. C. B. Dean, “Testing for Overdispersion in Poisson and Binomial Regression Models,”Journal of the American Statistical Association vol. 87 pp. 451–457, 1992.Google Scholar
  12. T. R. Fleming and D. P. Harrington,Counting Processes and Survival Analysis Wiley: New York, 1991.Google Scholar
  13. R. J. Gray, “Test for Variation Over Groups in Survival Data,”Journal of the American Statistical Association vol. 90 pp. 198–203, 1995.Google Scholar
  14. P. Hougaard, “A class of multivariate failure time distributions,”Biometrika vol. 73 pp. 671–678, 1986.Google Scholar
  15. H. Jacqmin and D. Commenges, “Tests of Homogeneity for Generalized Linear Models,”Journal of the American Statistical Association, in press.Google Scholar
  16. D. Y. Lin and L. J. Wei, “Goodness-of-Fit Tests for the General Cox Regression Model,”Statistica Sinica vol. 1 pp. 1–17, 1991.Google Scholar
  17. G. G. Nielsen, R. D. Gill, P. K. Andersen, and T. I. A. Sørensen, “A counting process approach to maximum likelihood estimation in frailty models,”Scand. J. Statist. vol. 19 pp. 25–43, 1992.Google Scholar
  18. V. S. Perkel, M. H. Gail, J. Lubin, D. Y. Pee, R. Weinstein, E. Shore-Freeman and A. B. Schneider, “Radiation-induced thyroid neoplasms: Evidence for familial susceptibility factors,’J. Clin. Endocrin. and Metabolism vol. 66 pp. 1316–1322, 1988.Google Scholar
  19. R. L. Prentice and S. G. Self, “Contribution to the discussion of the paper by D. G. Clayton and J. Cuzick,”J. Roy. Statist. Soc. A vol. 148 pp. 112–113. 1985.Google Scholar
  20. J. W. Vaupel, K. G. Manton and E. Stallard, “The impact of heterogeneity in individual frailty on the dynamics of mortality,”Demography vol. 16 pp. 439–454, 1979.Google Scholar
  21. J. W. Vaupel and A. I. Yashin, “Heterogeneity ruses: some surprising effects of selection on population dynamics,”Amer. Statist. vol. 39 pp. 176–185, 1985.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • D. Commenges
    • 1
  • P. K. Andersen
    • 2
  1. 1.INSERM U330BordeauxFrance
  2. 2.Statistical Research UnitUniversity of CopenhagenCopenhagen NDenmark

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