Proportional hazards model with the biomarker–treatment interaction plays an important role in the survival analysis of the subset treatment effect. A threshold parameter for a continuous biomarker variable defines the subset of patients who can benefit or lose from a certain new treatment. In this article, we focus on a continuous threshold effect using the rectified linear unit and propose a gradient descent method to obtain the maximum likelihood estimation of the regression coefficients and the threshold parameter simultaneously. Under certain regularity conditions, we prove the consistency, asymptotic normality and provide a robust estimate of the covariance matrix when the model is misspecified. To illustrate the finite sample properties of the proposed methods, we simulate data to evaluate the empirical biases, the standard errors and the coverage probabilities for both the correctly specified models and misspecified models. The proposed continuous threshold model is applied to a prostate cancer data with serum prostatic acid phosphatase as a biomarker.

中文翻译：

---

生存分析中具有双向交互作用的连续阈值模型

具有生物标志物与治疗相互作用的比例风险模型在亚治疗效果的生存分析中起着重要作用。连续生物标志物变量的阈值参数定义了可以从某种新疗法中受益或失去的患者子集。在本文中，我们集中在使用整流线性单元的连续阈值效应上，并提出了一种梯度下降方法来同时获得回归系数和阈值参数的最大似然估计。在某些规则性条件下，我们证明了一致性，渐近正态性，并在模型指定不正确时提供了协方差矩阵的可靠估计。为了说明所提出方法的有限样本属性，我们通过模拟数据来评估经验偏差，正确指定的模型和错误指定的模型的标准误差和覆盖率。所提出的连续阈值模型以血清前列腺酸磷酸酶为生物标记物，应用于前列腺癌数据。