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DNA analysis is now a data intensive discipline. New technology has transformed biomedical research by making a plethora of molecular data available at reduced costs and great speeds. Large consortiums and many individual laboratories have already generated vast datasets: as an example, one such database, the Gene Expression Omnibus (GEO contains more than 1.8 million samples. This data is readily, publicly available but analyzing it requires computational and statistical resources. A popular concern in biological research is to identify those genomic pathways that are related to the organism’s reaction to treatment or disease. There are numerous techniques that try to reduce the false positive errors and rank the pathways according to the degree of the phenotype relationship strength. This goal is accompanied by several challenges: finding parsimonious models with a good balance between simplicity and complexity and designing methods for pathway selection using appropriate significance thresholds. Often, it is difficult to escape the temptation of "ad-hoc" procedures that may work for particular examples but cannot be properly expanded to general cases. Over the years, many methods have been proposed but over-representation analysis (ORA) remains the most popular. The underlying assumption of ORA is that pathways with an irregular number of differentially expressed genes are responsible for the phenotype to the detriment of lesser differentially expressed pathways. Under the umbrella of logistic regression, we propose a method that aims to improve ORA. We show that traditional hypergeometric ORA methods are fully described by and can be considered a special case of the logistic regression methods. Logistic regression presents the advantage that while it produces simple models, they are richer, and they describe the biological process in a more accurate fashion. While logistic regression has been proposed before as a solution for ORA, we prove the over-encompassing nature of the method and we also propose a flavor of regression that can be aimed at different scenarios. Furthermore, logistic regression has a solid mathematical basis and produces results that have biological justification.

Identifying genomic pathways that are related to various clinical phenotypes is an important concern in today's biological research. I will discuss a few statistical frameworks that are very well suited for microarray gene expression analysis. I will address common issues with this type of analysis by offering new mathematical perspectives and by developing new statistical models that help explain the biological phenomenons that are being studied. For the purpose of this thesis, I define a pathway as a collection or subset of genes with a common biological process, molecular function, or cellular location. Typically, over-representation(ORA) type methods are used to predict gene expression as a function of gene membership and to provide rankings of pathways based on estimated expression levels and/or p-values. I start by showing that traditional hypergeometric ORA methods are fully described by and can be considered a special case of the logistic regression methods. Logistic regression presents the advantage that while it produces simple models, they are more rich and they describe the biological process in a more accurate fashion. While logistic regression has been proposed before as an improvement over ORA, I prove the over-encompassing nature of the method and I also propose flavors of regression that can be aimed at different scenarios. Furthermore, logistic regression has a solid mathematical basis and produces results that have biological justification. I continue by developing a Bayesian hierarchical regression model that solves three important problems of ORA analysis: it reduces type I error rates, it disentangles effects in cases of overlapping pathways and it shrinks probability estimates depending on the length of the pathways, providing sensible estimates and rankings. Our method is able to emphasize pathways that are biologically relevant to the phenotype in a real study. Simulations further show when this method is preferable to over-representation analysis. I present the cases when our method is more accurate in ranking relevant pathways and when it is better at demoting pathways that are irrelevant to the phenotype.