This research was dedicated to analyze two types of biomedical data: curved data lying on a manifold and censored survival data from clinical trials. The main part of the research aims at developing a general regression framework for the analysis of a manifold-valued response in a Riemannian symmetric space (RSS) and its association with Euclidean covariates of interest, such as age. Such data arises frequently in medical imaging, computational biology, and computer vision, among many others. We developed an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution on RSS. We proposed various link functions from the Euclidean space of covariates to the RSS of responses. We constructed parameter estimates and test statistics, and determined their asymptotic distributions and geometric invariant properties. Simulation studies were used to evaluate the finite sample properties of our method. We applied our model to investigate the association between covariates, including gender, age, and diagnosis, and the shape of the Corpus Callosum contours from the Alzheimer's Disease Neuroimaging Initiative dataset, in both cross-sectional and longitudinal cases. In oncology clinical trials, progression-free survival (PFS) has been a key endpoint to support licensing approval, and it is recommended to have the investigator's tumor assessments verified by an independent review committee blinded to study treatments, especially in open-label studies. Agreement between these evaluations may vary for subjects with short or long PFS, while there exist no such statistical quantities that can completely account for this temporal pattern of agreements. We proposed a new method to assess temporal agreement between two time-to-event endpoints, assuming they have a positive probability of being identical. Overall scores of agreement over a period of time are also proposed. We used maximum likelihood estimation to infer the proposed agreement measures using empirical data, accounting for different censoring mechanisms including reader's censoring (event from one reader dependently censored by event from the other reader). The proposed method is demonstrated to perform well in small-sample via extensive simulation studies and is illustrated through a head and neck cancer trial.