Title: Residual Analysis for Regression with Censored Data via Randomized Survival Probabilities Show Chinese title

Title: 右删失数据回归的残差分析与随机生存概率

Abstract: Residual analysis is extremely important in regression modelling. Residuals are used to graphically and numerically check the overall goodness-of-fit of a model, to discover the direction for improving the model, and to identify outlier observations. Cox-Snell residuals, which are transformed from survival probabilities (SPs), are typically used for checking survival regression models for failure times. Survival probabilities are uniformly distributed under the true model when there is no censored failure time. However, the SPs for censored failure times are no longer uniformly distributed. This is attributed to non-zero probability masses on the censored observations. Several non-random methods have been proposed to modify CS residuals or SPs in the literature. However, their sampling distributions under the true model are not characterized, resulting in a lack of reference distributions for analysis with these modified residuals. In this paper, we propose to use randomized survival probabilities (RSP) to define residuals for censored data. We will show that RSPs always have the uniform distribution under the true model even with censored times. Therefore, they can be transformed into residuals with the normal quantile function. We call such residuals by normally-transformed RSP (NRSP) residuals. We conduct extensive simulation studies to demonstrate that NRSP residuals are normally distributed when the fitted model is correctly specified; consequently, the Shapiro-Wilk normality test applied to NRSP residuals is well-calibrated. Our simulation studies also show the versatility of NRSP residuals in detecting many kinds of model mis-specifications. We also demonstrate the effectiveness of NRSP residuals in assessing three AFT models for a breast-cancer recurrent-free failure times dataset. Show Chinese abstract

Abstract: 残差分析在回归建模中极为重要。残差被用来从图形和数值上检查模型的整体拟合优度，发现改进模型的方向，并识别异常观测值。Cox-Snell 残差是由生存概率（SP）转换而来的，通常用于检查失效时间的生存回归模型。当没有截尾失效时间时，在真实模型下生存概率是均匀分布的。然而，对于截尾失效时间，这些生存概率不再均匀分布。这是由于截尾观测值具有非零的概率质量。文献中已经提出了几种非随机方法来修改 Cox-Snell 残差或生存概率。然而，它们在真实模型下的抽样分布未被刻画，导致缺乏用于分析这些修改残差的参考分布。本文提出使用随机化生存概率（RSP）来定义截尾数据的残差。我们将展示即使有截尾时间，RSP 在真实模型下始终具有均匀分布。因此，它们可以使用正态分位函数转化为残差。我们称这种残差为正态变换的 RSP 残差（NRSP）。我们进行了广泛的模拟研究，以证明当拟合模型正确指定时，NRSP 残差呈正态分布；因此，应用于 NRSP 残差的 Shapiro-Wilk 正态性检验是校准良好的。我们的模拟研究还展示了 NRSP 残差在检测多种模型误指定方面的多功能性。我们还演示了 NRSP 残差在评估乳腺癌无复发生存时间数据集中的三种加速失效时间（AFT）模型的有效性。