Relative risk estimation and inference using a generalized logrank statistic

When comparing two survival distributions with proportional hazard functions, the logrank test is optimal for testing the null hypothesis that the constant hazard ratio (relative risk) is one. In this paper, we focus on (i) testing for departures from a relative risk other than one, and (ii) estimation of the relative risk. The standard tool to address both (i) and (ii) is the Cox proportional hazards model. However, the performance of the Cox model can be less than optimal with small samples. We show why this is the case, and propose a simple alternative method of estimation and inference based on a generalized logrank (GLR) statistic. While the GLR and Cox model approaches are asymptotically similar, empirical results reveal that the GLR approach is notably more efficient than the Cox model when the number of subjects is small (< 100 subjects per treatment group). An example based on survival times of cervical cancer patients is used to illustrate the proposed methodology.