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

<p><b>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).</b></p> <p>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.</p>

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

<p><b>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).</b></p> <p>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.</p>

Bias Corrected H-likelihood Approach for Joint Models of Longitudinal and Survival Data, With Application to Community Acquired Pneumonia

Time-to-event coupled with longitudinal trajectories are often of interest in biomedicine, and one popular approach to analysing such data is with a Joint Model (JM). JMs often have intractable marginal likelihoods, and one way to tackle this issue is by using the hierarchical likelihood (HL) estimation approach by Lee and Nelder [12]. The HL approximation sometimes results in biased estimates, and we propose a biascorrection approach (C-HL) that has been used for other models (eg, frailty models). We have applied, for the first time, the C-HL in the context of joint modelling of time-to-event and repeated measures data. Our C-HL method shows efficiency improvement, which comes at a cost of a more expensive computation than the existing HL approach. Additionally, we illustrate our method with a new MIMIC-IV CAP dataset