Chung Yunro, Ivanova Anastasia, Hudgens Michael G, Fine Jason P
Public Health Sciences Division, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue North, Seattle, Washington 98109, U.S.A.
Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-7420, U.S.A.
Biometrika. 2018 Mar 1;105(1):133-148. doi: 10.1093/biomet/asx064. Epub 2017 Dec 5.
We consider the estimation of the semiparametric proportional hazards model with an unspecified baseline hazard function where the effect of a continuous covariate is assumed to be monotone. Previous work on nonparametric maximum likelihood estimation for isotonic proportional hazard regression with right-censored data is computationally intensive, lacks theoretical justification, and may be prohibitive in large samples. In this paper, partial likelihood estimation is studied. An iterative quadratic programming method is considered, which has performed well with likelihoods for isotonic parametric regression models. However, the iterative quadratic programming method for the partial likelihood cannot be implemented using standard pool-adjacent-violators techniques, increasing the computational burden and numerical instability. The iterative convex minorant algorithm which uses pool-adjacent-violators techniques has also been shown to perform well in related parametric likelihood set-ups, but evidences computational difficulties under the proportional hazards model. An alternative pseudo-iterative convex minorant algorithm is proposed which exploits the pool-adjacent-violators techniques, is theoretically justified, and exhibits computational stability. A separate estimator of the baseline hazard function is provided. The algorithms are extended to models with time-dependent covariates. Simulation studies demonstrate that the pseudo-iterative convex minorant algorithm may yield orders-of-magnitude reduction in computing time relative to the iterative quadratic programming method and the iterative convex minorant algorithm, with moderate reductions in the bias and variance of the estimators. Analysis of data from a recent HIV prevention study illustrates the practical utility of the isotonic methodology in estimating nonlinear, monotonic covariate effects.
我们考虑对具有未指定基线风险函数的半参数比例风险模型进行估计，其中假设连续协变量的效应是单调的。先前关于右删失数据的等距比例风险回归的非参数最大似然估计的工作计算量很大，缺乏理论依据，并且在大样本中可能不可行。在本文中，我们研究了偏似然估计。我们考虑了一种迭代二次规划方法，该方法在等距参数回归模型的似然估计中表现良好。然而，用于偏似然的迭代二次规划方法不能使用标准的相邻违反者合并技术来实现，这增加了计算负担和数值不稳定性。使用相邻违反者合并技术的迭代凸下逼近算法在相关的参数似然设置中也已被证明表现良好，但在比例风险模型下存在计算困难。我们提出了一种替代的伪迭代凸下逼近算法，该算法利用了相邻违反者合并技术，具有理论依据，并且具有计算稳定性。我们提供了基线风险函数的单独估计器。这些算法被扩展到具有随时间变化协变量的模型。模拟研究表明，相对于迭代二次规划方法和迭代凸下逼近算法，伪迭代凸下逼近算法可能会使计算时间减少几个数量级，同时估计器的偏差和方差会有适度降低。对最近一项艾滋病毒预防研究的数据进行分析，说明了等距方法在估计非线性、单调协变量效应方面的实际效用。
