Longitudinal and repeated measurement data commonly arise in many scientific researchareas. Traditional methods have focused on estimating single mean response as a function ofa time related variable and other covariates in a homogeneous population. However, in manysituations the homogeneity assumption may not be appropriate. Latent mixture modelscombine latent class modeling and conventional mixture modeling. They accommodate thepopulation heterogeneity by modeling each subpopulation with a mixing component. Inthis paper, we developed a hybrid Markov Chain Monte Carlo algorithm to estimate theparameters of the latent mixture model. We show through simulation studies that MCMCalgorithm is superior than the EM algorithm when missing value percentage is large.As an extension of latent mixture models, we also propose the use of cubic splines asa curve fitting technique instead of classic polynomial fitting. We show that this methodgives better fits to the data, and our MCMC algorithm estimates the model efficiently. Weapply the cubic spline technique to a data set which was collected in a study of alcoholism.Our MCMC algorithm shows several different P300 amplitude trajectory patterns amongchildren and adolescents.Other topics that are covered in this thesis include the identifiability of the latent mixturemodel and the use of such model to predict a binary outcome. We propose a bivariate versionof the latent mixture model, where two courses of longitudinal responses can be modeled atthe same time. Computational aspects of such models remain to be completed in the future.
