Growth Curves Curvesmixture Model Modelfor Longitudinal Data

Longitudinal data are collected by tracking individuals of interest either through time or over space,which means that individuals are measured repeatedly at different time points or under different experimental conditions.As a result,longitudinal data combine elements of time series and cross-sectional data.One of the fundamental tools for longitudinal studies is the growth curve model(GCM)which is an extension of multivariate regression model in the sense that an extra known grouping design matrix is added into the right hand side of the regression coefficient matrix to select an appropriate growth rate for each individual.Once been proposed,the GCM has been widely used in areas such as economics,biology,medical research,and epidemiology due to two obvious advantages.First,as a member of parametric statistics,GCM provides more accurate statistical inference.Second,with an explicit mathematical form of the parameter estimator,GCM is easier to execute in practice.Nowadays,the existing literatures mainly consider improvements in parameters estimation while little work has been done on rationalizing the strict,and sometimes inappropriate assumptions.Therefore,this paper focuses on discussing some of the most important model assumptions of GCM.More specifically,two major extensions of GCM assumptions are made in this dissertation.Firstly,the known grouping matrix is modified to be unknown.Secondly,the unstructured covariance matrix is modified to be structured.On the one hand,it is impractical to assume the grouping matrix in GCM to be known because usually neither is the treatment received clear nor is the feature keeping significant to subjects,so that the GCM which ask for a certain grouping design matrix is largely restricted in this kind of issues.On the other hand,it is inefficient to assume an unstructured covariance in GCM because many of the correlations among different observations within an individual can be well depicted by some of the existing covariance structures,the GCM with an unstructured covariance would miss some essential information.To make the model modification appropriate and to test the use of statistical information,these two assumptions would be extended combinedly.First,modeling the GCM with unknown grouping matrix is discussed respectively under uniform covariance structure(UCS)and serial covariance structure(SCS)which are widely used,and two new models obtained as a result.Then,parameters estimation is comprehensively studied in these two new models.The study techniques in this dissertation could be concluded as five.First,the model conversion,according to the character of the grouping matrix,the columns of it are assumed to follow independently categorical distribution.This prior assumption converts the GCM into a seemingly-like mixture model,which is referred to as growth curves mixture model.Second,the model simplification,under the specific covariance structures,namely the UCS and the SCS,the new growth curves mixture model can be simplified into two more parsimonious models.Third,the EM algorithm,to avoid the problems of missing information,the maximum likelihood estimation for these two parsimonious models is considered through the use of EM algorithm,along with clustering individuals.Fourth,the degenerate demonstration,the degenerate problem is discussed by substituting a known grouping matrix into these two parsimonious models to find the relationship between the GCM with specific covariance structures and the parsimonious models proposed.Fifth,the computational analysis,the proposed method is also applied to simulation studies and real data analysis to verify the efficiency of their estimates.The major achievement of this dissertation could be briefly summarized in four aspects.First,the growth curves mixture models separately under UCS and SCS are proposed.Second,the maximum likelihood estimates of two new parsimonious models are successfully obtained.Third,it is proved that the GCM with UCS and SCS are the special cases of the two corresponding parsimonious models illustrated in this dissertation.In other words,these two parsimonious models are generalized growth curve models.Fourth,the computational analysis has shown that the proposed procedure works well in simultaneous estimation and clustering.The main contribution of this dissertation are as follows.Above all,the GCM is generalized by extending the grouping matrix to be unknown so that it provides a solution to the real case where the grouping information is not available although it is with some obvious trend.Then,the role of covariance structure is proved to be crucial for the growth curves mixture models.The proposed growth curves mixture models under specific covariance structures provide a new approach to deal with the case when a certain correlation can be detected in different observations within individual.Furthermore,the generalized GCM proposed can simultaneously provide a new method to cluster longitudinal data,as the parameter estimation for the random grouping matrix is,in fact,the curve classification.