Doctor of Philosophy
Statistics and Actuarial Sciences
Survival regression models usually assume that covariate effects have a linear form. In many circumstances, however, the assumption of linearity may be violated. The present work addresses this limitation by adding nonlinear covariate effects to survival models. Nonlinear covariates are handled using a single index structure, which allows high-dimensional nonlinear effects to be reduced to a scalar term. The nonlinear single index approach is applied to modeling of survival data with multivariate responses, in three popular models: the proportional hazards (PH) model, the proportional odds (PO) model, and the generalized transformation model. Another extension of the PH and PO model is the handling of the baseline function. Instead of modeling it in a parametric way, which is fairly restrictive, or leaving it unspecified, which makes it impossible to calculate the survival and hazard functions, a weakly parametric approach is used here. As a result, the full likelihood can be applied for inference.
The new developments are realized by adding a number of weakly parametric elements to the standard parametric regression models. The marginal baseline hazard functions are modeled using piecewise constants. Marginal survival functions are combined in using copula models, such as the Clayton model, to incorporate association among the multivariate responses. The nonlinear covariate effect is brought into the model through a smooth function with the single-index structure as the input. The smooth function is modeled using a spline.
The performance of the PH, PO, and transformation models with the proposed extensions is evaluated through extensive simulation studies. The PH and PO models are also applied to a real-world data set. The results suggest that the proposed methods can capture the nonlinear covariate effects well, and that there is benefit to modeling the association between the correlated responses. Individual-level survival or hazard function estimates also provide information of interest to researchers. The proposed transformation model in particular is very promising. Some discussion of how this model may be further developed is provided.
Lei, Na, "Flexible Partially Linear Single Index Regression Models for Multivariate Survival Data" (2013). Electronic Thesis and Dissertation Repository. 1802.