Topics in Application of Nonparametric Smoothing Splines

There are two topics in this dissertation. The first topic is 'Smoothing Parameter Selection in Nonparametric Generalized Linear Models via Sixth-order Laplace Approximation' and the second topic is 'Smoothing Spline-based Score Tests for Proportional Hazards Models'.We present a new approach for the automatic selection of the smoothing parameter in nonparametric smoothing spline Generalized Linear Models (GLMs), using the Restricted Maximum Likelihood (REML) method and the sixth-order Laplace approximation of Raudenbush et al. (2000). The proposed approach is compared with Generalized Additive Mixed Model (GAMM, Lin and Zhang 1999) and Generalized Approximate Cross-Validation (GACV, Gu and Xiang 2001) through simulations and is shown to be effective. We propose 'score-type' tests for the proportional hazards assumption and for covariate effects in the Cox model, using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate. The tests are based on the penalized partial likelihood. By treating the inverse of the smoothing parameter as a variance component, we derive the score tests by testing an equivalent null hypothesis that the corresponding variance component is zero. The tests are shown to have size close to the nominal level and to provide good power against general alternatives in simulations. We apply the proposed tests to data from a cancer clinical trial.