Title: Model diagnostics for censored regression via randomized survival probabilities Show Chinese title

Title: 审查回归的模型诊断通过随机生存概率

Abstract: Residuals in normal regression are used to assess a model's goodness-of-fit (GOF) and discover directions for improving the model. However, there is a lack of residuals with a characterized reference distribution for censored regression. In this paper, we propose to diagnose censored regression with normalized randomized survival probabilities (RSP). The key idea of RSP is to replace the survival probability of a censored failure time with a uniform random number between 0 and the survival probability of the censored time. We prove that RSPs always have the uniform distribution on $(0,1)$ under the true model with the true generating parameters. Therefore, we can transform RSPs into normally-distributed residuals with the normal quantile function. We call such residuals by normalized RSP (NRSP residuals). We conduct simulation studies to investigate the sizes and powers of statistical tests based on NRSP residuals in detecting the incorrect choice of distribution family and non-linear effect in covariates. Our simulation studies show that, although the GOF tests with NRSP residuals are not as powerful as a traditional GOF test method, a non-linear test based on NRSP residuals has significantly higher power in detecting non-linearity. We also compared these model diagnostics methods with a breast-cancer recurrent-free time dataset. The results show that the NRSP residual diagnostics successfully captures a subtle non-linear relationship in the dataset, which is not detected by the graphical diagnostics with CS residuals and existing GOF tests. Show Chinese abstract

Abstract: 在普通回归中，残差用于评估模型的拟合优度 (GOF) 并发现改进模型的方向。然而，在截尾回归中缺乏具有明确参考分布的残差。本文提出利用归一化随机生存概率（RSP）来诊断截尾回归。RSP 的关键思想是用一个介于 0 和截尾时间的生存概率之间的均匀随机数替换截尾失效时间的生存概率。我们证明了在真实模型和真实生成参数下，RSP 总是在$(0,1)$上具有均匀分布。因此，我们可以使用正态分位函数将 RSP 转换为正态分布的残差，称为归一化 RSP（NRSP 残差）。我们进行了模拟研究，以调查基于 NRSP 残差的统计检验的大小和功效，用于检测分布族选择错误和协变量中的非线性效应。我们的模拟研究表明，尽管基于 NRSP 残差的 GOF 检验不如传统的 GOF 检验方法强大，但基于 NRSP 残差的非线性检验在检测非线性方面具有显著更高的功效。我们还通过乳腺癌无复发时间数据集比较了这些模型诊断方法。结果显示，NRSP 残差诊断成功捕捉到数据集中微妙的非线性关系，而图形诊断与现有 GOF 检验未检测到该关系。