In survival analysis, the hazard function often depends on a set of covariates. Martingale and deviance residual are most widely used for examining the validity of the function form of covariates by checking whether there is a discernible trend in their scatterplot against continuous covariates. However, visual inspection of martingale and deviance residuals is often subjective. In addition, these residuals lack a reference distribution due to censoring. It is therefore challenging to derive numerical statistical tests based on martingale or deviance residuals. In this paper, we extend the idea of randomized survival probability (Li et al. 2021) and develop a residual diagnostic tool that can provide both graphical and numerical tests for checking the covariate functional form in semi-parametric shared frailty models. We develop a general function that calculates Z-residuals for semi-parametric shared frailty models based on the output from the \texttt{coxph} function in the \texttt{survival} package in R. Our extensive simulation studies indicate that the derived numerical test based on Z-residuals has great power for checking the functional form of covariates. In a real data application on modelling the survival time of acute myeloid leukemia patients, the Z-residual diagnosis results show that a model with log-transformation is inappropriate for modelling the survival time, which could not be detected by other diagnostic methods.

翻译：在生存分析中，风险函数通常依赖于一组协变量。鞅残差和偏差残差被广泛用于检验协变量函数形式的合理性，其方法是通过检查这些残差与连续协变量的散点图中是否存在可辨识的趋势。然而，对鞅残差和偏差残差的视觉检查往往具有主观性。此外，由于删失的存在，这些残差缺乏参照分布。因此，基于鞅残差或偏差残差推导数值统计检验具有挑战性。本文扩展了随机生存概率思想（Li等，2021），开发了一种残差诊断工具，可同时提供图形化检验和数值检验，用于半参数共享脆弱模型中协变量函数形式的检查。基于R语言survival包中coxph函数的输出，我们构建了一个通用函数来计算半参数共享脆弱模型的Z-残差。广泛模拟研究表明，基于Z-残差的数值检验在检查协变量函数形式方面具有强大功效。在针对急性髓系白血病患者生存时间建模的实际数据应用中，Z-残差诊断结果显示，采用对数变换的模型不适合拟合生存时间，而其他诊断方法未能检测出这一问题。