Nonparametric Estimation of Relative Risk Using Splines and Cross-Validation

An alternative to the local scoring method of Hastie and Tibshirani [J. Statist. Sci., 1 (1986), pp. 297–318] is provided for nonparametric estimation of the relative risk in the Cox model. The method involves penalized partial likelihood. Computations are carried out using a damped Newton–Raphson iteration. Each iterate is evaluated using an appropriately preconditioned conjugate gradient algorithm. The algorithm is globally convergent under mild conditions. One-step diagnostics are developed and cross-validation criteria are provided to guide the evaluation of the degree of smoothness of the estimator. These cross-validation scores have potential application to model selection in standard Cox regression contexts also. Bayesian confidence intervals akin to those of Wahba [J. Roy. Statist. Soc., 45 (1983), pp. 133–150] are defined. The performance of the methodology is illustrated on real and simulated data.