Residual Analysis for Regression with Censored Data via Randomized Survival Probabilities
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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.
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