Semiparametric Regression Models for Censored Data

Abstract

In this talk, I will introduce two classes of semiparametric regression models for censored or survival data analysis. The first class is the partially linear single-index proportional hazards (PSI-PH) models, the second class is the partially linear singleindex accelerated failure time (PSI-AFT) models. They are the extended Cox PH models and AFT models respectively. The single-index model provides a flexible way for modelling the association between a response and a set of predictor variables when the link function is unknown. It presents a technique for “dimension reduction” in multivariate models and generalizes the existing linear models for data analysis. The proposed partially linear single-index models generalize both the partially linear model and the single-index model in censored data analysis. The PSI-PH models are considered in two cases: the baseline hazard function is parameterized or totally unknown. In the former case, using techniques of local linear fit, we can estimate all the parameters (in the baseline hazard function, the linear part and the non-parametric part) simultaneously. An asymptotic distribution theory for the proposed estimators is established. The estimators are proved to be semiparametric efficient. In the latter case, we propose to use B-splines to approximate the nonparametric function, so that Cox's estimation procedure can still be used. The estimators are approximately semiparametric efficient. For the PSI-AFT models, we use a synthetic data method to transform the censored data into synthetic data unbiasedly, so that we can apply the semiparametric least squares principle to estimate all the parameters. The asymptotic distribution of the estimators is obtained as well. In all these cases, the unknown regression function is estimated with the same efficiency as if all the parameters were known. Monte Carlo simulations are conducted to illustrate the proposed methodologies. Real data analyses are shown for the importance of the proposed methods in applications.