Estimating the number of components in a mixture and analysis of recurrent events with time dependent covariates in the presence of dependent censoring

Part I of this dissertation deals with estimating the number of components (k) in a normal univariate mixture. The problem of estimating the number of components is motivated by the challenge of detecting non-responders to alendronate treatment of osteoporosis based on the repeat measurements of bone mineral density (BMD). Motivated by the data from Fracture Intervention Trial (FIT), clinical trial of alendronate, we describe a model of treatment response that is based on normal mixture models. As a part of model checking technique in our non-response model, we want to select the optimal number of mixture components. We discuss available methods for choosing the number of components including bootstrapping the likelihood ratio test statistic and optimizing a variety of validity functionals such as AIC, BIC, MDL and ICOMP. We investigate the minimization of distance between fitted mixture model and the true density as a method for estimating k. The distances we consider are Kullback-Leibler and L ₂. We estimate these distances using cross validation. We show great performance of this method in comparison with other available methods and apply all the discussed methods to show that there is no evidence of non-response to alendronate therapy among osteoporosis patients.; Part II of this dissertation is concerned with modeling of recurrent events occurrence. We discuss current modeling approaches to recurrent events analysis as well as their aims and limitations. We are interested in the problem of recurrent events because of the recurrent lung infections that we refer to as exacerbations in cystic fibrosis patients. Provided the data from the Epidemiologic Study of Cystic Fibrosis (ESCF), we use different models to relate occurrence of lung exacerbations to the observed covariates indicating stage of the disease or health status of the patient. We evaluate different approaches based on their applicability in the presence of time dependent covariates and a large number of recurrent events. We present the general estimating function approach to model parameter estimation in the presence of informative censoring. While this methodology is not novel, we discuss several different approaches to applying it in the context of recurrent events. We also propose specific estimating functions with desirable properties in the case of independent censoring. Finally, we discuss the effect of lung function, presence of various microorganisms in respiratory cultures, and general developmental and health status of a cystic fibrosis patient on the occurrence of lung exacerbations.