Empirical Likelihood Methods for Survival Analysis and Generalized Linear Models with External Auxiliary Information

In clinical studies, how to evaluate the treatment effect is a crucial topic. Nowadays, the cumulative hazard ratio and the difference of survival functions are often applied to accom- plish this task, especially when those hazards may be nonproportional. The stratified Cox proportional hazards model, as an important extension of the classical Cox model, has the ability to flexibly handle non-proportional hazards. In this dissertation, we propose a novel empirical likelihood (EL) method to construct the confidence intervals for the cumulative hazard ratio and difference of survival functions under the stratified Cox model. On the other hand, Generalized linear models (GLMs) are regression models widely used in medi- cal research, machine learning, and other fields. Recently, leveraging auxiliary information from external sources to enhance the estimation efficiency of model parameters has become a prominent research topic. In this dissertation, we construct estimating equations based on auxiliary information from external source and combine these with the quasi-likelihood estimating equations derived from individual-level data to form a unified estimating equa- tion. To address the population heterogeneity among different sources of information, we define bias parameters and include them in the unified estimating equation. We utilize a generalized method of moment (GMM) based on the unified estimating equation to estimate the coefficients in GLMs. Simulation studies are conducted under various settings to examine the finite sample properties of the proposed methods. We also demonstrate the large sample properties of the proposed method and provide proofs for them. The proposed methods are finally applied to perform statistical analyses on several real-world datasets.