Full likelihood inferences in the Cox model: an empirical likelihood approach

Abstract

For the regression parameter β 0 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 (β 0, F 0), where F 0 is the baseline distribution in the Cox model. Using the empirical likelihood parameterization, we explicitly profile out nuisance parameter F 0 to obtain the full-profile likelihood function for β 0 and the maximum likelihood estimator (MLE) for (β 0, F 0). The relation between the MLE and Cox’s partial likelihood estimator for β 0 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 full-likelihood 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.