Small Sample Bias in the Gamma Frailty Model for Univariate Survival

Authors: Barker, Peter¹; Henderson, Robin²

Source: Lifetime Data Analysis, Volume 11, Number 2, June 2005 , pp. 265-284(20)

Publisher: Springer

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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

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10985-004-0387-7

Affiliations: 1: TE1 J/3, Parklands, Alderley Park, Macclesfield, Cheshire, SK10 4TF, U.K, Email: Peter.N.Barker@astrazeneca.com 2: Department of Mathematics and Statistics, Lancaster University, U.K,

Publication date: 2005-06-01