In this paper, we measure the extent to which a biological marker is a surrogate endpoint for a clinical event by the proportional reduction in the regression coefficient for the treatment indicator due to the inclusion of the marker in the Cox regression model. We estimate this proportion by applying the partial likelihood function to two Cox models postulated on the same failure time variable. We show that the resultant estimator is asymptotically normal with a simple variance estimator. One can construct confidence intervals for the proportion by using the direct normal approximation to the point estimator or by using Fieller's theorem. Extensive simulation studies demonstrate that the proposed methods are appropriate for practical use. We provide applications to HIV/AIDS clinical trials.