Decision trees are popular in survival analysis for their interpretability and ability to model complex relationships. Survival trees, which predict the timing of singular events using censored historical data, are typically built through heuristic approaches. Recently, there has been growing interest in globally optimized trees, where the overall tree is trained by minimizing the error function over all its parameters. We propose a new soft survival tree model (SST), with a soft splitting rule at each branch node, trained via a nonlinear optimization formulation amenable to decomposition. Since SSTs provide for every input vector a specific survival function associated to a single leaf node, they satisfy the conditional computation property and inherit the related benefits. SST and the training formulation combine flexibility with interpretability: any smooth survival function (parametric, semiparametric, or nonparametric) estimated through maximum likelihood can be used, and each leaf node of an SST yields a cluster of distinct survival functions which are associated to the data points routed to it. Numerical experiments on 15 well-known datasets show that SSTs, with parametric and spline-based semiparametric survival functions, trained using an adaptation of the node-based decomposition algorithm proposed by Consolo et al. (2024) for soft regression trees, outperform three benchmark survival trees in terms of four widely-used discrimination and calibration measures. SSTs can also be extended to consider group fairness.

翻译：决策树因其可解释性和建模复杂关系的能力在生存分析中广受欢迎。生存树利用删失历史数据预测单一事件的发生时间，通常通过启发式方法构建。近年来，全局优化树的研究日益受到关注，此类树通过最小化所有参数的误差函数进行整体训练。本文提出一种新的软生存树模型（SST），其在每个分支节点采用软分裂规则，并通过可分解的非线性优化公式进行训练。由于SST为每个输入向量提供与单一叶节点关联的特定生存函数，因此满足条件计算特性并继承相关优势。SST及其训练框架兼具灵活性与可解释性：可通过最大似然估计使用任何平滑的生存函数（参数化、半参数化或非参数化），且SST的每个叶节点生成一组不同的生存函数簇，这些函数与路由至该节点的数据点相关联。在15个知名数据集上的数值实验表明，采用参数化及基于样条半参数化生存函数的SST，通过适配Consolo等人（2024）为软回归树提出的基于节点的分解算法进行训练，在四项广泛使用的区分度与校准度量上优于三种基准生存树。SST还可扩展至考虑群体公平性。