Hazard models are the most commonly used tool to analyse time-to-event data. If more than one time scale is relevant for the event under study, models are required that can incorporate the dependence of a hazard along two (or more) time scales. Such models should be flexible to capture the joint influence of several times scales and nonparametric smoothing techniques are obvious candidates. P-splines offer a flexible way to specify such hazard surfaces, and estimation is achieved by maximizing a penalized Poisson likelihood. Standard observations schemes, such as right-censoring and left-truncation, can be accommodated in a straightforward manner. The model can be extended to proportional hazards regression with a baseline hazard varying over two scales. Generalized linear array model (GLAM) algorithms allow efficient computations, which are implemented in a companion R-package.

翻译：风险模型是分析时间-事件数据最常用的工具。当所研究的事件涉及多个相关时间尺度时，需要采用能够同时考虑两个（或更多）时间尺度上风险依赖性的模型。此类模型需具备灵活性以捕捉多个时间尺度的联合影响，而非参数平滑技术显然是候选方法。P样条提供了一种灵活定义此类风险曲面的方式，并通过最大化惩罚泊松似然实现参数估计。标准观测模式（如右删失和左截断）可被直接纳入该框架。该模型还可扩展为具有双时间尺度基线风险的比例风险回归模型。广义线性阵列模型算法能够实现高效计算，并已在配套的R语言软件包中完成实现。