Deep learning for the partially linear Cox model
Deep learning for the partially linear Cox model
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DOI：
10.1214/21-aos2153
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
Qixian Zhong;Jonas Mueller;Jane-Ling Wang
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文献类型：
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作者：
Qixian Zhong;Jonas Mueller;Jane-Ling Wang
虽然针对生存数据的深度学习方法在应用中取得了经验成功，但其中大多数方法很难解释，并且缺乏对它们的数学理解。本文研究了部分线性考克斯模型，其中模型的非线性部分使用深度神经网络实现。所提出的方法是灵活的，能够规避维数灾难，但它有利于解释治疗协变量对生存的影响。我们建立了最大偏似然估计的渐近理论，并证明了我们的非参数深度神经网络估计器达到了最小最大最优收敛速度（直到多对数因子）。此外，我们证明了相应的治疗协变量效应的有限维估计是协方差一致的，渐近正态的，并达到半参数效率。对两个真实的生存数据集的大量模拟研究和分析表明，所提出的估计量产生了具有上级覆盖率的置信区间以及与实际生存时间具有上级一致性的生存时间预测。
While deep learning approaches to survival data have demonstrated empirical success in applications, most of these methods are difficult to interpret and mathematical understanding of them is lacking. This paper studies the partially linear Cox model, where the nonlinear component of the model is implemented using a deep neural network. The proposed approach is flexi-ble and able to circumvent the curse of dimensionality, yet it facilitates interpretability of the effects of treatment covariates on survival. We establish asymptotic theories of maximum partial likelihood estimators and show that our nonparametric deep neural network estimator achieves the minimax optimal rate of convergence (up to a poly-logarithmic factor). Moreover, we prove that the corresponding finite-dimensional estimator for treatment co-variate effects is √ n -consistent, asymptotically normal, and attains semi-parametric efficiency. Extensive simulation studies and analyses of two real survival datasets show the proposed estimator produces confidence intervals with superior coverage as well as survival time predictions with superior concordance to actual survival times.