Profile Likelihood Estimation Applied To The Semi-parametric Models With Survival Data

In this thesis, we have investigated the efficiency of profile likelihood in the estimation of parameters from the Cox Proportional Hazards (PH) cure model and joint model of longitudinal and survival data. For the profile likelihood approach in the joint model of longitudinal and survival data, Hsieh et al. (2006) stated “No distributional or asymptotic theory is available to date, and even the standard errors (SEs), defined as the standard deviations of the parametric estimators, are difficult to obtain”. The reason behind this difficulty is the estimator of baseline hazard which involves implicit function in the profile likelihood estimation (Hirose and Liu, 2020). Hence finding the estimated SE of the parametric estimators from the Cox PH cure model and joint model using profile likelihood approach is a great challenge. Therefore, bootstrap method has been suggested to get the estimated standard errors while using the profile likelihood approach (Hsieh et al., 2006).

To solve the difficulty, we have expanded the profile likelihood function directly without assuming the derivative of the profile likelihood score function and obtain the explicit form of the SE estimator using the profile likelihood score function. Our proposed alternative approach gives us not only analytical understanding of the profile likelihood estimation, but also provides closed form formula to compute the standard error of the profile likelihood maximum likelihood estimator in terms of profile likelihood score function. To show the advantage of our proposed approach in medical and clinical studies, we have analysed the simulated and real-life data, and compared our results with the output obtained from the smcure, JM(method: ’Cox-PH-GH’) and joineRML R-packages. The outputs suggest that the bootstrap method and our proposed approach have provided similar and comparable results. In addition, the average computation times of our approach are much less compared to the above mentioned R-packages.