In biomedical research and lifetime data analysis, the comparison of two hazard functions usually plays an important role in practice. In this thesis, we consider the standard independent two-sample framework under right censoring. We construct efficient and useful confidence intervals for the ratio and difference of two hazard functions using smoothed empirical likelihood methods. The empirical log-likelihood ratio is derived and its asymptotic distribution is a chi-squared distribution. Furthermore, the proposed method can be applied to medical diagnosis research. Simulation studies show that the proposed EL confidence intervals have better performance in terms of coverage accuracy and average length than the traditional normal approximation method. Finally, our methods are illustrated with real clinical trial data. It is concluded that the empirical likelihood methods provide better inferential outcomes.