Robust estimation in the additive hazards model

In the additive hazards model the hazard function of a survival variable T is modeled additively as λ(t)=λ0(t)+β'z$\lambda (t)=\lambda _0(t)+{\bm \beta }^{\prime } {\bm z}$, where λ0(t) is a common non parametric baseline hazard function and z${\bm z}$ is a vector of independent variables. For this model, the pioneering work of Lin and Ying (1994) develops a closed-form estimator for the regression parameter β${\bm \beta }$ from a new estimating equation. That equation has a similar structure to the corresponding partial likelihood score function for the multiplicative model (Cox 1972) in that it exploits a martingale structure and it allows estimation of β${\bm \beta }$ separate from the baseline hazard function. Their estimator is asymptotically normal and highly efficient. However, a potential drawback is that it is very sensitive to outliers. In this paper we propose a family of robust alternatives for estimation of the parameter β${\bm \beta }$ in the additive hazards model which is robust to outliers and still highly efficient and asymptotically normal. We prove Fisher-consistency, obtain the influence function, and illustrate the estimation with simulated and real data. The latter corresponds to the time-honored Welsh Nickels Refiners dataset first introduced by Doll et al. (1970) and subsequently analyzed by Breslow and Day (1987) and Lin and Ying (1994), among others.