Lifetime Data Analysis

, Volume 11, Issue 2, pp 265–284

Small Sample Bias in the Gamma Frailty Model for Univariate Survival

Article

Abstract

The gamma frailty model is a natural extension of the Cox proportional hazards model in survival analysis. Because the frailties are unobserved, an E-M approach is often used for estimation. Such an approach is shown to lead to finite sample underestimation of the frailty variance, with the corresponding regression parameters also being underestimated as a result. For the univariate case, we investigate the source of the bias with simulation studies and a complete enumeration. The rank-based E-M approach, we note, only identifies frailty through the order in which failures occur; additional frailty which is evident in the survival times is ignored, and as a result the frailty variance is underestimated. An adaption of the standard E-M approach is suggested, whereby the non-parametric Breslow estimate is replaced by a local likelihood formulation for the baseline hazard which allows the survival times themselves to enter the model. Simulations demonstrate that this approach substantially reduces the bias, even at small sample sizes. The method developed is applied to survival data from the North West Regional Leukaemia Register.

Keywords

bias censoring E-M algorithm gamma frailty local likelihood life history data proportional hazards model smoothing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen, P., Klein, J., Knudsen, K., Palacious, R. 1997“Estimation of variance in Cox’s regression model with shared gamma frailties”Biometrics5314751484Google Scholar
  2. Betensky, R., Lindsey, L.M., Ryan, J., Wand, M. 2002“A local likelihood proportional hazards model for interval censored data”Statistics in Medicine21263275Google Scholar
  3. Betensky, R., Lindsey, J., Ryan, L., Wand, M. 1999“Local EM estimation of the hazard function for interval-censored data”Biometrics55238245Google Scholar
  4. Cleveland, W. 1979“Robust locally weighted regression and smoothing scatterplots”Journal of the American Statistical Association74829836Google Scholar
  5. Dempster, A.P., Laird, N.M., Rubin, D.B. 1977“Maximum likelihood from incomplete data via the EM algorithm”Journal of the Royal Statistical Society SeriesB39138Google Scholar
  6. Elbers, C., Ridder, G. 1982“True and spurious duration dependence The identifiability of the proportional hazard model”Review of Economic Studies49403409Google Scholar
  7. Heckman, J., Singer, B. 1984identifiability of the proportional hazard model”Review of Economic Studies51231241Google Scholar
  8. Henderson, R., Oman, P. 1999“Effect of frailty on marginal regression estimates in survival analysis”.Journal of the Royal Statistical Society Series B61367379Google Scholar
  9. Hougaard, P. 2000Analysis of Multivariate Survival DataSpringer-VerlagNew YorkGoogle Scholar
  10. Klein, J. 1992“Semiparametric estimation of random effects using the Cox model based on the E-M algorithm”Biometrics48795806Google Scholar
  11. Klein, J.P., Moeschberger, M.L. 1997Survival Analysis: Techniques for Censored and Truncated DataSpringer-VerlagNew YorkGoogle Scholar
  12. Nielsen, G.G., Gill, R.G., Andersen, P.K. 1992“A counting process approach to maximum likelihood estimation in frailty models”Scandinavian Journal of Statistics192543Google Scholar
  13. Parner, E. 1998“Asymptotic theory for the correlated gamma-frailty model”The Annals of Statistics26183214Google Scholar
  14. Rondeau, V., Commenges, D., Joly, P. 2003“Maximum penalized lieklihood estimation in a gamma frailty model”Lifetime data analysis9139153Google Scholar
  15. Therneau, T., Grambsch, P. 2000Modelling Survival DataSpringer-VerlagNew Yorkchap.9.Google Scholar
  16. Tibshirani, R., Hastie, T. 1987“Local likelihood estimation”Journal of the American Statistical Society82559567Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.TE1 J/3, ParklandsMacclesfieldU.K
  2. 2.Department of Mathematics and StatisticsLancaster UniversityU.K

Personalised recommendations