There is a growing interest in precision medicine, where a potentially censored survival time is often the most important outcome of interest. To discover optimal treatment regimens for such an outcome, we propose a semiparametric proportional hazards model by incorporating the interaction between treatment and a single index of covariates through an unknown monotone link function. This model is flexible enough to allow non-linear treatment-covariate interactions and yet provides a clinically interpretable linear rule for treatment decision. We propose a sieve maximum likelihood estimation approach, under which the baseline hazard function is estimated nonparametrically and the unknown link function is estimated via monotone quadratic B-splines. We show that the resulting estimators are consistent and asymptotically normal with a covariance matrix that attains the semiparametric efficiency bound. The optimal treatment rule follows naturally as a linear combination of the maximum likelihood estimators of the model parameters. Through extensive simulation studies and an application to an AIDS clinical trial, we demonstrate that the treatment rule derived from the single-index model outperforms the treatment rule under the standard Cox proportional hazards model.

人们对精准医学越来越感兴趣，其中潜在的审查生存时间往往是最重要的关注结果。为了发现这种结果的最佳治疗方案，我们提出了一种半参数比例风险模型，通过未知的单调链接函数合并治疗和单个协变量指数之间的相互作用。该模型足够灵活，允许非线性治疗-协变量相互作用，并为治疗决策提供临床可解释的线性规则。我们提出了一种筛最大似然估计方法，在该方法下以非参数方式估计基线危险函数，并通过单调二次 B 样条估计未知链接函数。我们证明，所得到的估计量与达到半参数效率界限的协方差矩阵是一致且渐近正态的。最佳处理规则自然遵循模型参数的最大似然估计量的线性组合。通过广泛的模拟研究和艾滋病临床试验的应用，我们证明单指标模型得出的治疗规则优于标准 Cox 比例风险模型下的治疗规则。