A brief survey on methods to handle non‐proportional hazards in survival analysis is given with emphasis on short‐term and long‐term hazard ratio modelling. A drawback of the existing model of this nature is that except at time zero or infinity, the hazard ratio for a unit increase in the value of a covariate depends on the starting value. With two or more covariates, the hazard ratio for a unit increase in one covariate with other covariates held fixed depends in an unintended way on the values of the other covariates. We propose an alternative way to model short‐term and long‐term hazard ratios without the above drawbacks through a judicious choice of covariate‐time interactions. Under the new model, it is easier to describe the time‐varying effect of each covariate on the hazard. Nonparametric maximum likelihood estimation for the new model can be carried out in the same way as for the existing model. We also propose a product version of the existing model, which overcomes its second drawback but not the first. The advocated covariate–time interaction model provides a better fit to the Veterans Administration lung cancer data set than the original and product versions of the existing model.

中文翻译：

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具有无协变量值的危害比函数的非比例危害模型

简要介绍了生存分析中处理非比例风险的方法，重点是短期和长期风险比建模。这种性质的现有模型的一个缺点是，除了在时间为零或无穷大时，协变量值增加一个单位的风险比取决于起始值。对于两个或多个协变量，一个协变量在其他协变量保持不变的情况下单位增加的风险比以一种非预期的方式取决于其他协变量的值。我们提出了一种替代方法，通过对协变量时间相互作用的明智选择，对短期和长期风险比进行建模而没有上述缺点。在新模型下，更容易描述每个协变量对风险的时变效应。新模型的非参数最大似然估计可以按照与现有模型相同的方式进行。我们还提出了现有模型的产品版本，它克服了它的第二个缺点，但不是第一个。与现有模型的原始版本和产品版本相比，提倡的协变量时间交互模型更适合退伍军人管理局肺癌数据集。