Joint models for time-to-event and multivariate longitudinal data

The joint models for longitudinal and time-to-event data are a recent family models that jointly analyse the longitudinal and the survival data. The models are composed by two sub-models, the longitudinal and the survival sub-model. A proportional hazard model can be used for survival sub-model and it is expressed in function of the true and unobserved value of the longitudinal outcome, while concerning the longitudinal sub-model a linear mixed model is often proposed. After analysing some univariate cases, it is interesting to study the situation in which one of the two sub-models or both are multivariate. Thus different scenarios are possible. Firstly it is possible to consider a situation in which only the longitudinal sub-model is multivariate, in this situation a multivariate linear mixed model or another type of multivariate longitudinal model can be considered. Choosing a multivariate linear mixed-effects model, a different longitudinal outcome must be considered for each linear predictor. Accordingly the survival sub-model is composed by several parameters that express the relation of each true and unobserved value of the longitudinal outcome with the hazard function. Secondly a situation in which only the survival sub-model is multivariate is possible, thus the survival sub-model may consider two situations, competing risks or recurrent events. Lastly a situation in which both the longitudinal and the survival model are multivariate must be considered. The sub-models are composed by the multivariate longitudinal and survival sub-models which are jointly analysed. The biggest problem related to the multivariate situation concerns the computational aspect of the estimation. In fact considering that the univariate case is computational demanding, increasing the number of the parameters or the dimension of the sub-models will lead to higher computational demanding situations. This problem could be solved with the implementation of some algorithms in the R software that could reduce the time and the memory requested. In this thesis the focus is on the situation in which only the longitudinal sub-model is multivariate. The aim is to find new methods of estimation and some algorithms that could help to solve the problem of the computational aspect. At first a two stages approach is implemented as it permits to obtain very fast and significant estimations. The most of the applications of joint models focus on the biostatistical area, thus the event analysed is death or the manifestation of a disease and the influence of some biomarkers on it. In this thesis the focus is on the undergraduates' career, analysing the careers of the undergraduate students in an Italian university, using jointly the time to graduation and the student's path, focusing on the marks and on the number of exams that the student has already passed before a fixed time. The algorithms are implemented also on a well-known biostatistical data set available in the package JM of the software R the test the reliability and the efficiency.