The hazard function represents one of the main quantities of interest in the analysis of survival data. We propose a general approach for parametrically modelling the dynamics of the hazard function using systems of autonomous ordinary differential equations (ODEs). This modelling approach can be used to provide qualitative and quantitative analyses of the evolution of the hazard function over time. Our proposal capitalises on the extensive literature of ODEs which, in particular, allow for establishing basic rules or laws on the dynamics of the hazard function via the use of autonomous ODEs. We show how to implement the proposed modelling framework in cases where there is an analytic solution to the system of ODEs or where an ODE solver is required to obtain a numerical solution. We focus on the use of a Bayesian modelling approach, but the proposed methodology can also be coupled with maximum likelihood estimation. A simulation study is presented to illustrate the performance of these models and the interplay of sample size and censoring. Two case studies using real data are presented to illustrate the use of the proposed approach and to highlight the interpretability of the corresponding models. We conclude with a discussion on potential extensions of our work and strategies to include covariates into our framework.

翻译：风险函数是生存数据分析中的核心关注对象之一。我们提出了一种通用方法，利用自治常微分方程组对风险函数的动态变化进行参数化建模。该建模方法可对风险函数随时间的演变提供定性与定量分析。我们的方案借鉴了常微分方程领域的丰富文献，特别是通过使用自治常微分方程建立风险函数动态变化的基本规则或定律。我们展示了在常微分方程组存在解析解或需借助数值求解器获取数值解时，如何实施所提出的建模框架。虽然本研究重点采用贝叶斯建模方法，但该技术框架亦可与极大似然估计相结合。通过模拟研究展示了这些模型的性能表现及样本量与删失数据的相互作用。基于真实数据的两项案例研究，既验证了所提方法的实用性，也凸显了相应模型的可解释性。最后，我们讨论了工作的潜在扩展方向及将协变量纳入该框架的策略。