Due to rapid developments in machine learning, and in particular neural networks, a number of new methods for time-to-event predictions have been developed in the last few years. As neural networks are parametric models, it is more straightforward to integrate parametric survival models in the neural network framework than the popular semi-parametric Cox model. In particular, discrete-time survival models, which are fully parametric, are interesting candidates to extend with neural networks. The likelihood for discrete-time survival data may be parameterized by the probability mass function (PMF) or by the discrete hazard rate, and both of these formulations have been used to develop neural network-based methods for time-to-event predictions. In this paper, we review and compare these approaches. More importantly, we show how the discrete-time methods may be adopted as approximations for continuous-time data. To this end, we introduce two discretization schemes, corresponding to equidistant times or equidistant marginal survival probabilities, and two ways of interpolating the discrete-time predictions, corresponding to piecewise constant density functions or piecewise constant hazard rates. Through simulations and study of real-world data, the methods based on the hazard rate parametrization are found to perform slightly better than the methods that use the PMF parametrization. Inspired by these investigations, we also propose a continuous-time method by assuming that the continuous-time hazard rate is piecewise constant. The method, named PC-Hazard, is found to be highly competitive with the aforementioned methods in addition to other methods for survival prediction found in the literature.

由于机器学习，特别是神经网络的快速发展，在过去几年中开发了许多用于事件时间预测的新方法。由于神经网络是参数模型，因此在神经网络框架中集成参数生存模型比流行的半参数 Cox 模型更直接。特别是，完全参数化的离散时间生存模型是用神经网络扩展的有趣候选者。离散时间生存数据的可能性可以通过概率质量函数 (PMF) 或离散风险率进行参数化，并且这两种公式都已用于开发基于神经网络的事件时间预测方法。在本文中，我们回顾并比较了这些方法。更重要的是，我们展示了如何将离散时间方法用作连续时间数据的近似值。为此，我们引入了两种离散化方案，对应于等距时间或等距边际生存概率，以及两种插值离散时间预测的方法，对应于分段常数密度函数或分段常数危险率。通过对真实世界数据的模拟和研究，发现基于危险率参数化的方法比使用 PMF 参数化的方法性能稍好。受这些调查的启发，我们还提出了一种连续时间方法，假设连续时间危险率是分段常数。该方法名为 PC-Hazard，