Model Checking Techniques for Parametric Regression with Censored Data

We develop a broad class of graphical and numerical methods for checking individual components of parametric survival models as well as omnibus tests for assessing the overall goodness of fit. The general goodness-of-fit process for testing the distributional assumption is the difference between the parametric maximum likelihood estimator and the Aalen-Breslow type estimator for the baseline survival function. We approximate the null distribution of this process with a zero-mean Gaussian process whose distribution can be easily generated through simulation, which enables us to compute the P-value for the supremum test. In order to examine the deterministic model components and to construct global omnibus tests, we study cumulative sums of martingale-based residuals over the failure time or/and covariates. Under the assumed model, the distributions of these partial-sum processes can again be approximated through simulating certain zero-mean Gaussian processes. One may then plot each observed process along with a few realizations from the corresponding Gaussian process to assess visually how unusual the observed residual pattern is. Supremum tests may also be performed. Extensive Monte Carlo studies demonstrate that the proposed test have proper sizes and are highly sensitive to model mis-specification. Applications to two well-known real data sets yield some new insights.