Du Pang, Ma Shuangge, Liang Hua
Department of Statistics, Virginia Tech, Blacksburg, VA 24061, USA,
Ann Stat. 2010 Aug 1;38(4):2092-2117. doi: 10.1214/09-AOS780.
We study the Cox models with semiparametric relative risk, which can be partially linear with one nonparametric component, or multiple additive or nonadditive nonparametric components. A penalized partial likelihood procedure is proposed to simultaneously estimate the parameters and select variables for both the parametric and the nonparametric parts. Two penalties are applied sequentially. The first penalty, governing the smoothness of the multivariate nonlinear covariate effect function, provides a smoothing spline ANOVA framework that is exploited to derive an empirical model selection tool for the nonparametric part. The second penalty, either the smoothly-clipped-absolute-deviation (SCAD) penalty or the adaptive LASSO penalty, achieves variable selection in the parametric part. We show that the resulting estimator of the parametric part possesses the oracle property, and that the estimator of the nonparametric part achieves the optimal rate of convergence. The proposed procedures are shown to work well in simulation experiments, and then applied to a real data example on sexually transmitted diseases.
我们研究具有半参数相对风险的Cox模型，该模型可以是带有一个非参数分量的部分线性模型，或者是具有多个可加或不可加非参数分量的模型。我们提出了一种惩罚偏似然方法，用于同时估计参数并为参数部分和非参数部分选择变量。我们依次应用两种惩罚。第一种惩罚用于控制多元非线性协变量效应函数的平滑度，它提供了一个平滑样条方差分析框架，利用该框架可以推导出非参数部分的经验模型选择工具。第二种惩罚，即平滑截断绝对偏差（SCAD）惩罚或自适应LASSO惩罚，用于在参数部分实现变量选择。我们证明，参数部分的所得估计量具有神谕性质，并且非参数部分的估计量达到了最优收敛速度。在模拟实验中，所提出的方法显示出良好的效果，然后将其应用于一个关于性传播疾病的实际数据示例。
