The approximation of partial likelihood with emphasis on case-control studies

For many case-control applications, previously suggested approximations to Cox's (1972, 1975) partial likelihood in prospective applications are shown to be seriously inadequate. Alternative approximations based on unconditional maximum likelihood estimation and Taylor series expansion are developed and shown to yield an improvement over the previous methods. In particular, an approximation to the maximum partial likelihood estimator is proposed using the log partial likelihood and its derivatives at the origin. Theoretical calculations for data from matched pairs and triplets and some limited simulation illustrate the potential value of the new approximations.