Longitudinal study is an experimental design which takes repeated measurements of some variables from a study cohort over a specified time period. Collected data is most often modelled using a mixed effects model, which permits heterogeneity analysis of the variables over time. In this thesis, we apply the linear mixed effects models to applications that cover different domains of research. First, we consider the problem of estimating a multivariate probit model in a longitudinal data setting with emphasis on sampling a high-dimensional correlation matrix, and improving the overall efficiency of the posterior sampling approach via a dynamic variance reduction technique. The proposed method is used to analyse stated preference of female contraceptive products by Australian general practitioners, and hence provide insights to their behaviour in decision-making. Additionally, we introduced a multiclass classification model for growth trajectory that flexibly extends a piecewise linear model popular in the literature by allowing the number of classes to be data driven. Individual-specific random change points are introduced to model heterogeneity in growth phases realistically. The model is then applied on a birth cohort from the Healthy Birth, Growth and Development knowledge integration (HBGDki) project funded by the Bill and Melinda Gates Foundation. Finally, we investigate the evolution of unobserved executive functions of male soccer players representing a professional German Bundesliga club using a latent variable model, where cognitive outcomes from a test battery of neuropsychological assessments undergone by the players are manifestation of some underlying curves representing executive functions. This is the first study of its kind in soccer research that permits a longitudinal analysis of domain-generic and domain-specific executive functions.