Barker Peter, Henderson Robin
Department of Mathematics and Statistics, Lancaster University.
Lifetime Data Anal. 2005 Jun;11(2):265-84. 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.
伽马脆弱模型是生存分析中Cox比例风险模型的自然扩展。由于脆弱性是不可观测的，因此通常采用期望最大化（E-M）方法进行估计。结果表明，这种方法会导致脆弱性方差在有限样本中被低估，相应的回归参数也会因此被低估。对于单变量情况，我们通过模拟研究和完全枚举来探究偏差的来源。我们注意到，基于秩的E-M方法仅通过失效发生的顺序来识别脆弱性；生存时间中明显存在的额外脆弱性被忽略了，结果导致脆弱性方差被低估。我们建议对标准E-M方法进行一种改进，即将非参数Breslow估计替换为用于基线风险的局部似然公式，这使得生存时间本身能够进入模型。模拟表明，即使在小样本量的情况下，这种方法也能大幅减少偏差。所开发的方法被应用于来自西北区域白血病登记处的生存数据。
