A partly linear model for censored regression quantiles

Partly linear models are useful as an extension to linear regression when the response cannot be easily parameterized in terms of all covariates. Their flexibility in keeping some linear terms, while at the same time introducing a nonparametric relationship makes them attractive for a number of applications. Their advantage over purely nonparametric models is that they avoid the curse of dimensionality. In quantile regression, examples of partly linear models appear for instance in He and Liang (2000) and He and Shi (1996), where normalized B-splines were utilized for the nonparametric estimation. Such quantile regression models can also be useful in exploring the distribution of survival data, typically characterized by right censoring. A method for quantile regression on censored data was developed by Portnoy (2003). The recursively reweighted Censored Regression Quantile (CRQ) estimator of Portnoy (2003), which is essentially a generalization of the Kaplan-Meier estimator, is a useful complement to the Cox proportional hazards model for survival data. This thesis proposes a partly linear model for censored regression quantiles as a method for modeling nonlinearities in duration data. As quantile regression does not impose a proportionality constraint on the hazard, it is less restrictive than the Cox method and it often illuminates aspects of the data that could otherwise be overlooked. This flexibility makes it an attractive alternative to existing methods like the accelerated failure time model and the proportional hazards model. B-splines are used in estimating the nonparametric part of the partly linear CRQ model and the resulting partly linear estimator is shown to be consistent. A simulation study shows that the partly linear model can be useful in improving the estimation of quantile treatment effects and of models with a nonlinear dependence on some of the covariates. Finally, three examples of biometrics and econometrics data are considered. Applications to Australian AIDS survival data, drugs relapse data and labor supply data demonstrate the usefulness of quantile regression for censored data.