Competing risk model with a nonparametric form of relative risks.

Competing risks, referring to multiple mutually exclusive cause-specific events in survival data, are commonly seen in medical and engineering studies. In the presence of covariates, the Cox proportional hazards model is the most common approach to modeling covariate effects as the relative risk component for each cause-specific hazard function. However, the linear form of covariate effects in the existing models can be too restrictive to satisfy in practice. In this work, we propose a Cox regression model for competing risk survival data where the relative risk of each cause-specific hazard takes a flexible nonparametric form. The effects of multiple covariates are modeled under a smoothing spline ANOVA framework allowing for nonparametric effect components among covariates. For each cause-specific hazard, we develop an estimation procedure based on the optimization of penalized partial likelihood. We show that our spline estimators of the log relative risk functions can achieved the optimal nonparametric rates of convergence. Through simulation studies, we demonstrate the numerical performance of the proposed method. We then apply our approach to a multiple myeloma dataset, where the effects of gene expressions on cancer and non-cancer related death are investigated. The nonlinear trend revealed in our analysis offers a strong support to the need of a nonparametric modeling approach like ours.