An important target of many biomedical and clinical research paradigms is to identify biomarkers, including risk scores, with strong prognostic capabilities. Biomarker evaluations are usually utilized to predict the progression of the disease under study. In such clinical studies, one major research objective is to identify immune response biomarkers measured longitudinally that may be associated with the risk of death, infection, or any other interesting event. Joint modeling of longitudinal and survival data has become incredibly beneficial for analyzing such clinical trial data. In this thesis, we propose and explore innovative joint models and methods for analyzing longitudinal response and time-to-event data, and also address the computational challenges associated with the likelihood inference. Specifically, we consider two approaches to obtain the estimates of the model parameters, the maximum likelihood (ML) method and approximate likelihood method based on the so-called h-likelihood. In both cases, we consider a joint model that assumes a latent process based on random effects to describe the association among longitudinal and survival data. We use the linear mixed-effects model for longitudinal measurements and Weibull frailty model for survival outcomes. The two submodels are linked through shared random effects, where the longitudinal and survival processes are conditionally independent given the random effects. We also investigate the impact of the association parameter on the regression estimators in a separate analysis of each outcome. In the case of strong associations, the separate analysis provides biased parameter estimates with poor coverage probabilities of the confidence interval. In this work, we also investigate the effect of a misspecified random effects distribution on the parameter estimates in joint models. We assess the robustness properties of the maximum likelihood estimators obtained under misspecified random effects distributions, i.e., when the random effect follows a non-normal distribution, such as the Chisquare or Gamma distribution. Our empirical study shows that the use of the t-distribution for random effects gives more robust estimates of model parameters, as compared to the commonly used normal distribution. As an application, we analyze a large clinical dataset of spesis patients obtained from a longitudinal study.