“…To investigate the dynamic relationship between repeated measurements in an outcome process and covariates, various inference procedures have been developed in the form of regression analysis over the last several decades. This includes parametric methods for linear and nonlinear regressions (Laird, Donnelly, and Ware 1992; Diggle, Liang, and Zeger 1994; Davidian and Giltinan 1995), nonparametric regressions with kernel, spline and other smoothing techniques (Rice and Silverman 1991; Altman and Casella 1995; Silverman 1996; Lin and Carroll 2000, 2001), smoothing methods for non-parametric regressions with smooth time-varying coefficients (Hoover, Rice, Wu, and Yang 1998; Wu, Chiang, and Hoover 1998; Huang, Wu and Zhou 2002, 2004; Xue and Zhu 2007), functional linear models (Fan and Zhang 2000; James, Wang and Zhu 2009), and inference methods on semiparametric regressions (Moyeed and Diggle 1994; Zeger and Diggle 1994; Lin and Ying 2001; Lin and Carroll 2006; Li 2011). In particular, semiparametric regression models follow the idea of the Cox (1972) proportional hazards model and keep the regression form of covariate effects while leave the baseline (or intercept) unspecified, hence remove limitations of parametric and nonparametric regressions in practical analysis of longitudinal observations (Wu, Chiang and Hoover 1998).…”