We are living in an era in which many mysteries related to science, technologies and design can be answered by "learning" the huge amount of data accumulated over the past few decades. In the processes of those endeavors, highly-correlated high-dimensional data are frequently observed in many areas including predicting shelf life, controlling manufacturing processes, and identifying important pathways related with diseases. We define a "set" as a group of highly-correlated high-dimensional (HCHD) variables that possess a certain practical meaning or control a certain process, and define an "element" as one of the HCHD variables within a certain set. Such an elements-within-a-set structure is very complicated because: (i) the dimensions of elements in different sets can vary dramatically, ranging from two to hundreds or even thousands; (ii) the true relationships, include element-wise associations, set-wise interactions, and element-set interactions, are unknown; (iii) and the sample size (n) is usually much smaller than the dimension of the elements (p). The goal of this dissertation is to provide a systematic way to identify both the set effects and the element effects associated with survival outcomes from heterogeneous populations using Bayesian survival kernel models. By connecting kernel machines with semiparametric Bayesian hierarchical models, the proposed unified model frameworks can identify significant elements as well as sets regardless of mis-specifications of distributions or kernels. The proposed methods can potentially be applied to a vast range of fields to solve real-world problems. / PHD / We are living in an era in which many mysteries related to science, technologies and design can be answered by “learning” the huge amount of data accumulated over the past few decades. In the processes of those endeavors, highly-correlated high-dimensional data are frequently observed in many areas including predicting shelf life, controlling manufacturing processes, and identifying important pathways related with diseases. For example, for a group of 30 patients in a medical study, values for an immense number of variables like gender, age, height, weight, and blood pressure of each patient are recorded. High-dimensional means the number of variables (i.e. p) could be very large (e.g. p > 500), while the number of subjects or the sample size (i.e. n) is small (n = 30). We define a “set” as a group of highly-correlated high-dimensional (HCHD) variables that possess a certain practical meaning or control a certain process, and define an “element” as one of the HCHD variables within a certain set. Such an elements-within-a-set structure is very complicated because: (i) the dimensions of elements in different sets can vary dramatically, ranging from two to hundreds or even thousands; (ii) the true relationships, include element-wise associations, set-wise interactions, and element-set interactions, are unknown; (iii) and the sample size (n) is usually much smaller than the dimension of the elements (p). The goal of this dissertation is to provide a systematic way to identify both the set effects and the element effects associated with survival outcomes from heterogeneous populations using different proposed statistical models. The proposed models can incorporate prior knowledge to boost the model performance. The proposed methods can potentially be applied to a vast range of fields to solve real-world problems.

Pathway is a set of genes which are predefined and serve a particular celluar or physiological function. Ranking pathways relevant to a particular phenotype can help researchers focus on a few sets of genes in pathways. In this thesis, a Bayesian hierarchical latent model was proposed using generalized linear random effects model. The advantage of the approach was that it can easily incorporate prior knowledges when the sample size was small and the number of genes was large. For the covariance matrix of a set of random variables, two Gaussian random processes were considered to construct the dependencies among genes in a pathway. One was based on the polynomial kernel and the other was based on the Gaussian kernel. Then these two kernels were compared with constant covariance matrix of the random effect by using the ratio, which was based on the joint posterior distribution with respect to each model. For mixture models, log-likelihood values were computed at different values of the mixture proportion, compared among mixtures of selected kernels and point-mass density (or constant covariance matrix). The approach was applied to a data set (Mootha et al., 2003) containing the expression profiles of type II diabetes where the motivation was to identify pathways that can discriminate between normal patients and patients with type II diabetes. / Master of Science

The wide application of the genomic microarray technology triggers a tremendous need in the development of the high dimensional genetic data analysis. Many statistical methods for the microarray data analysis consider one gene at a time, but they may miss subtle changes at the single gene level. This limitation may be overcome by considering a set of genes simultaneously where the gene sets are derived from the prior biological knowledge and are called "pathways". We have made contributions on two specific research topics related to the high dimensional genetic pathway data. One is to propose a semi- parametric model for identifying pathways related to the zero inflated clinical outcomes; the other is to propose a multilevel Gaussian graphical model for exploring both pathway and gene level network structures. For the first problem, we develop a semiparametric model via a Bayesian hierarchical framework. We model the pathway effect nonparametrically into a zero inflated Poisson hierarchical regression model with unknown link function. The nonparametric pathway effect is estimated via the kernel machine and the unknown link function is estimated by transforming a mixture of beta cumulative density functions. Our approach provides flexible semiparametric settings to describe the complicated association between gene microarray expressions and the clinical outcomes. The Metropolis-within-Gibbs sampling algorithm and Bayes factor are used to make the statistical inferences. Our simulation results support that the semiparametric approach is more accurate and flexible than the zero inflated Poisson regression with the canonical link function, this is especially true when the number of genes is large. The usefulness of our approaches is demonstrated through its applications to a canine gene expression data set (Enerson et al., 2006). Our approaches can also be applied to other settings where a large number of highly correlated predictors are present. Unlike the first problem, the second one is to take into account that pathways are not independent of each other because of shared genes and interactions among pathways. Multi-pathway analysis has been a challenging problem because of the complex dependence structure among pathways. By considering the dependency among pathways as well as genes within each pathway, we propose a multi-level Gaussian graphical model (MGGM): one level is for pathway network and the second one is for gene network. We develop a multilevel L1 penalized likelihood approach to achieve the sparseness on both levels. We also provide an iterative weighted graphical LASSO algorithm (Guo et al., 2011) for MGGM. Some asymptotic properties of the estimator are also illustrated. Our simulation results support the advantages of our approach; our method estimates the network more accurate on the pathway level, and sparser on the gene level. We also demonstrate usefulness of our approach using the canine genes-pathways data set. / Ph. D.