Robust functional Cox regression model

Survival analysis with functional covariates has emerged as an important extension of the classical Cox proportional hazards model, allowing one to assess how entire trajectories or curves influence time-to-event outcomes. However, existing functional Cox models are typically fitted using non-robust techniques and can be highly sensitive to outliers or aberrant observations in the data. In this paper, we propose a robust functional Cox regression model that addresses this limitation. The proposed methodology combines a projection-pursuit-based robust functional principal component analysis with robust Cox regression estimation in a finite-dimensional subspace. By adopting the robust functional principal component analysis approach for dimension reduction, we obtain principal components that resist the influence of outlying functional observations. Then, a robust partial likelihood approach which additionally downweights the effects of outliers is used to estimate the parameters of a Cox regression model constructed using the robust functional principal components and scalar covariates. We establish the asymptotic properties of the proposed estimator, including Fisher consistency, \(\sqrt{n}\)-consistency, and asymptotic normality, under a set of mild and practically verifiable regularity conditions. Furthermore, we derive and analyze the influence function to assess the robustness characteristics of the estimator. Through an extensive Monte Carlo simulation study, we provide compelling evidence that the proposed method outperforms classical functional linear Cox regression and penalized functional regression techniques, particularly in the presence of outliers. We further demonstrate the proposed method’s effectiveness using accelerometry-based survival data from the National Health and Nutrition Examination Survey. Our method has been implemented in the \(\texttt{rfcox}\) package.