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. The proportional hazards (PH) regression model has played a pivotal role in survival analysis since Cox first proposed it in 1972 (Cox, 1972, 1975). Its familiar form for the hazard function is given by: λ(t|x) = λ 0 (t) exp β T x , where λ 0 (t) is the unknown baseline hazard function corresponding to x = 0, the vector of regression coefficients, β, expresses the dependence of the distribution of a survival time on a known covariate vector x ∈ R An important but strong assumption of the Cox PH model is that the covariates have a linear effect on the log hazard function. However, this assumption is not always guaranteed, which can result in erroneous conclusions. To relax this assumption, nonparametric methods have been adopted to estimate the log hazard function. Research focused on this area has been carried out by O, among others. All of these authors based their approaches on the following common form for the conditional hazard function: λ(t|x) = λ 0 (t) · exp {ψ(x)} , where ψ(x) is an unspecified smooth function of x. However, unstructured nonparametric function estimation suffers from the so-called ''curse of dimensionality'' and, thus, is not practically useful when the dimension of the covariate vector x is high. The ''curse of dimensionality'' is 177 an important issue that concerns many statisticians, so substantial research has been done using structured instead of unstructured nonparametric models. For example, Sleeper and Harrington (1990) used …