Polynomial spline estimation of partially linear single-index proportional hazards regression models

The Cox proportional hazards (PH) model usually assumes linearity of the covariates on the log hazard function, which may be violated because linearity cannot always be guaranteed. We propose a partially linear single-index proportional hazards regression model, which can model both linear and nonlinear covariate effects on the log hazard in the proportional hazards model. We adopt a polynomial spline smoothing technique to model the structured nonparametric single-index component for the nonlinear covariate effects. This method can reduce the dimensionality of the covariates being modeled, while, at the same time, can provide efficient estimates of the covariate effects. A two-step iterative algorithm to estimate the nonparametric component and the covariate effects is used for facilitating implementation. Asymptotic properties of the estimators are derived. Monte Carlo simulation studies are presented to compare the new method with the standard Cox linear PH model and some other comparable models. A case study with clinical trial data is presented for illustration.