Robust inference and model checking techniques for censored linear regression models

Regression models are among the most widespread statistical tools applied in the analysis of experimental and observational data. When the model is misspecified this can seriously affect the validity and efficiency of inference procedures. Unfortunately, investigators do not routinely check the adequacy of the specified model for their particular data analysis. Several authors have developed model checking techniques for regression models. In particular, for the linear regression model (Stute (1997); Stute et al. (1998)), the Cox model (Lin et al. (1993)) and generalized linear models (Su and Wei (1991); Lin et al. (2002); Stute and Zhu (2002)). The methodology of Lin et al. (1993) is based on martingale residuals and presently, robust estimation and model checking techniques for the censored linear regression model are not available.; In the first chapter we propose a new type of residual and an easily computed functional form test for the Cox proportional hazards model. The proposed test is a modification of the omnibus test for testing the overall fit of a parametric regression model, developed by Stute et al. (1998), and is based on what we call censoring consistent residuals. In addition, we develop residual plots that can be used to identify the correct functional forms of covariates. We compare our test with the functional form test of Lin et al. (1993) in a simulation study. The practical application of the proposed residuals and functional form test is illustrated using both a simulated data set and a real data set.; In the second chapter, we present robust inferences for certain covariate effects on the failure time in the presence of "nuisance" confounders under a semiparametric, partial linear regression setting. Specifically, the estimation procedures for the regression coefficients of interest are derived from a working linear model and are valid even when the function of the confounders in the model is not correctly specified. The new proposals are illustrated with two examples and their validity for cases with practical sample sizes is demonstrated via a simulation study.; The last chapter develops model checking techniques for assessing functional form specifications of covariates in censored linear regression models. These procedures are based on an censored data analog to taking cumulative sums of "robust" residuals over the space of the covariate under investigation. These cumulative sums are formed by integrating certain Kaplan-Meier estimators and may be viewed as "robust" censored data analogs to the processes considered by Lin et al. (2002). The null distributions of these stochastic processes can be approximated by the distributions of certain zero-mean Gaussian processes whose realizations can be generated by computer simulation. Each observed process can then be graphically compared with a few realizations from the Gaussian process. We also develop formal test statistics for numerical comparison. Such comparisons enable one to assess objectively whether an apparent trend seen in a residual plot reflects model misspecification or natural variation. We illustrate the methods with a well known dataset. In addition, we examine the finite sample performance of the proposed test statistics in simulation experiments. In our simulation experiments, the proposed test statistics have good power of detecting misspecification while at the same time controlling the size of the test.