Full Likelihood Inferences in the Cox Model: an Empirical Likelihood Approach

For the regression parameter β0 in the Cox model, there have been several estimators constructed based on various types of approximated likelihood, but none of them has demonstrated small-sample advantage over Cox’s partial likelihood estimator. In this article, we derive the full likelihood function for (β0, F0), where F0 is the baseline distribution in the Cox model. Using the empirical likelihood parameterization, we explicitly profile out nuisance parameter F0 to obtain the full-profile likelihood function for β0 and the maximum likelihood estimator (MLE) for (β0, F0). The relation between the MLE and Cox’s partial likelihood estimator for β0 is made clear by showing that Taylor’s expansion gives Cox’s partial likelihood estimating function as the leading term of the full-profile likelihood estimating function. We show that the log fulllikelihood ratio has an asymptotic chi-squared distribution, while the simulation studies indicate that for small or moderate sample sizes, the MLE performs favorably over Cox’s partial likelihood estimator. In a real dataset example, our full likelihood ratio test and Cox’s partial likelihood ratio test lead to statistically different conclusions.