Given the prominence of targeted therapy and immunotherapy in cancer treatment, it becomes imperative to consider heterogeneity in patients' responses to treatments, which contributes greatly to the widely used proportional hazard assumption invalidated as in several clinical trials. To address the challenge, we develop a Dual Cox model theory including a Dual Cox model and a fitting algorithm. As one of the finite mixture models, the proposed Dual Cox model consists of two independent Cox models based on patients' responses to one designated treatment (usually the experimental one) in the clinical trial. Responses of patients in the designated treatment arm can be observed and hence those patients are known responders or non-responders. From the perspective of subgroup classification, such a phenomenon renders the proposed model as a semi-supervised problem, compared to the typical finite mixture model where the subgroup classification is usually unsupervised. A specialized expectation-maximization algorithm is utilized for model fitting, where the initial parameter values are estimated from the patients in the designated treatment arm and then the iteratively reweighted least squares (IRLS) is applied. Under mild assumptions, the consistency and asymptotic normality of its estimators of effect parameters in each Cox model are established. In addition to strong theoretical properties, simulations demonstrate that our theory can provide a good approximation to a wide variety of survival models, is relatively robust to the change of censoring rate and response rate, and has a high prediction accuracy and stability in subgroup classification while it has a fast convergence rate. Finally, we apply our theory to two clinical trials with cross-overed KM plots and identify the subgroups where the subjects benefit from the treatment or not.

翻译：鉴于靶向治疗和免疫治疗在癌症治疗中的突出地位，考虑患者对治疗反应的异质性变得至关重要，这种异质性在很大程度上导致广泛使用的比例风险假设在多项临床试验中失效。为应对这一挑战，我们提出了一种双Cox模型理论，包括一个双Cox模型及相应的拟合算法。作为有限混合模型的一种，所提出的双Cox模型基于患者对临床试验中指定治疗（通常为实验性治疗）的反应，由两个独立的Cox模型构成。指定治疗组患者的反应是可观察的，因此这些患者被已知为反应者或非反应者。从亚组分类的角度看，与通常无监督的典型有限混合模型不同，这种现象使得所提模型成为一个半监督问题。模型拟合采用专门的期望最大化算法，其中初始参数值从指定治疗组的患者中估计得出，随后应用迭代加权最小二乘法（IRLS）。在温和假设下，我们建立了每个Cox模型中效应参数估计量的一致性和渐近正态性。除强理论性质外，模拟实验表明，我们的理论能为多种生存模型提供良好的近似，对删失率和反应率的变化相对稳健，在亚组分类中具有高预测精度和稳定性，同时收敛速度较快。最后，我们将该理论应用于两个具有交叉KM曲线的临床试验，识别出患者是否从治疗中获益的亚组。