This dissertation focuses on two topics in semiparametric statistical methods and their applications in medical science: (1) prediction of patients' lifetimes based on their risk profiles; (2) estimation of dynamic exposure effects on survival outcomes. In Chapter II, we develops multiple imputation methods based on restricted mean models. The imputation method replaces each censored patient's event time with estimates for the true event time based on patient risk factors and observed survival information. Once multiple imputation is completed, the analyst has augmented uncensored datasets for standard statistical analyses. Simulation results show that our method outperforms its closest competitor in terms of bias and efficiency in both independent and dependent censoring scenarios. The proposed method is also much less subject to dependent censoring bias captured by covariates. This particular feature is observed in a full statistical analysis conducted in the context of the IBCSG Trial.

Dynamic exposure, Z(t), usually exerts complex effects on a chronic disease. Outcomes of interest are governed by a latent tumor initiation and progression process, modeled by frailty, U. The majority of literature treats U as a random variable independent of time. We believe that, with observed Z( t) being dynamic, U could represent a latent stochastic process in time that models development of a latent disease such as tumor growth in a cancer patient, characterized by observed Z( t). In Chapter III, we develop an imputation scheme for U(t) within the nonparametric maximum likelihood estimate (NPMLE) framework and establish general inference procedures for such models. In Chapter IV, we develop a mechanistic model to explain a dynamic effect of radiation when the disease has a latent development period before diagnosis. This work is motivated by Hormesis effect as discussed in Tsodikov and Muller (1998), where natural process of tumor growth is modulated by radiation. The non-linear interplay of Hormesis and carcinogenic effects may result in improved survival of the subjects under radiation. Our model reproduces the diversity of such complex effects. The advantage of this modeling approach is that we can interpret the time changing exposure as a stochastic process while retaining the power of rigorous statistical inference.