Improved Approximation Methods for Tied Data in Continuous-Time Cox Regressions

Abstract

The Breslow (1974) and Efron (1977) approximations are two of the most widely-used methods for calculating the computationally-intensive partial likelihood function of continuous-time Cox (1972)-type proportional hazards regressions in the presence of tied data. This article develops analytical improvements to these approximations based on the Taylor expansion of the exact continuous-time partial likelihood function. This article also proposes a partial scaling of the Breslow approximation, the Breslow-Midpoint approximation, with the same computational complexity but that exhibits higher approximation accuracy in numerical simulations.