Lifetime Data Analysis

, Volume 11, Issue 2, pp 265–284

Small Sample Bias in the Gamma Frailty Model for Univariate Survival


DOI: 10.1007/s10985-004-0387-7

Cite this article as:
Barker, P. & Henderson, R. Lifetime Data Anal (2005) 11: 265. doi:10.1007/s10985-004-0387-7


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.


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

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

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