Censored quantile regression with auxiliary information

In Survival analysis, it is vital to understand the effect of the covariates on the survival time. Commonly studied models are the Cox [1972] proportional hazards model and the accelerated failure time model. These methods mainly focus on one characteristic of the survival time. In reality, the association between the response and risk factors is not homogeneous always. This leads to the use of quantile regression [Koenker and Basset, 1978] models, which provide a global description of the association. In quantile regression modeling of the survival data, the problem of estimating the regression coefficients for extreme quantiles can be affected by severe censoring [Portnoy, 2003], especially when the sample size is small. In epidemiological studies, however, there are often times when only a subset of the whole study cohort is accurately observed. The rest of the cohort has only some auxiliary covariate available. The naive use of the auxiliary covariate in the model without the accurately measured covariate could lead to biased estimates. To deal with this problem in censored quantile regression, we propose a regression calibration based method when there is a linear relationship between the auxiliary covariate and the accurately measured covariate. When the relationship is non-linear, we propose a non-parametric kernel smoothing technique. We also propose an empirical likelihood [Owen, 1998, 2001] based weighted censored quantile regression to improve the efficiency of the censored quantile regression estimation by utilizing the auxiliary information about the target population parameters available through scientific facts/previous studies. The proposed estimators are consistent and have asymptotically Gaussian distributions. The efficiency gain compared to the existing methods is remarkable. These methods provide the possibilities of looking into extreme quantiles of the survival distribution. We also applied our proposed methods in real case examples