Comparison of Some Improved Estimators for Linear Regression Model under Different Conditions

Multiple linear regression model plays a key role in statistical inference and it has extensive applications in business, environmental, physical and social sciences. Multicollinearity has been a considerable problem in multiple regression analysis. When the regressor variables are multicollinear, it becomes difficult to make precise statistical inferences about the regression coefficients. There are some statistical methods that can be used, which are discussed in this thesis are ridge regression, Liu, two parameter biased and LASSO estimators. Firstly, an analytical comparison on the basis of risk was made among ridge, Liu and LASSO estimators under orthonormal regression model. I found that LASSO dominates least squares, ridge and Liu estimators over a significant portion of the parameter space for large dimension. Secondly, a simulation study was conducted to compare performance of ridge, Liu and two parameter biased estimator by their mean squared error criterion. I found that two parameter biased estimator performs better than its corresponding ridge regression estimator. Overall, Liu estimator performs better than both ridge and two parameter biased estimator.

Reverse Engineering of Modified Genes by Bayesian Network Analysis Defines Molecular Determinants Critical to the Development of Glioblastoma

In this study we have identified key genes that are critical in development of astrocytic tumors. Meta-analysis of microarray studies which compared normal tissue to astrocytoma revealed a set of 646 differentially expressed genes in the majority of astrocytoma. Reverse engineering of these 646 genes using Bayesian network analysis produced a gene network for each grade of astrocytoma (Grade I–IV), and ‘key genes’ within each grade were identified. Genes found to be most influential to development of the highest grade of astrocytoma, Glioblastoma multiforme were: COL4A1, EGFR, BTF3, MPP2, RAB31, CDK4, CD99, ANXA2, TOP2A, and SERBP1. All of these genes were up-regulated, except MPP2 (down regulated). These 10 genes were able to predict tumor status with 96–100% confidence when using logistic regression, cross validation, and the support vector machine analysis. Markov genes interact with NFkβ, ERK, MAPK, VEGF, growth hormone and collagen to produce a network whose top biological functions are cancer, neurological disease, and cellular movement. Three of the 10 genes - EGFR, COL4A1, and CDK4, in particular, seemed to be potential ‘hubs of activity’. Modified expression of these 10 Markov Blanket genes increases lifetime risk of developing glioblastoma compared to the normal population. The glioblastoma risk estimates were dramatically increased with joint effects of 4 or more than 4 Markov Blanket genes. Joint interaction effects of 4, 5, 6, 7, 8, 9 or 10 Markov Blanket genes produced 9, 13, 20.9, 26.7, 52.8, 53.2, 78.1 or 85.9%, respectively, increase in lifetime risk of developing glioblastoma compared to normal population. In summary, it appears that modified expression of several ‘key genes’ may be required for the development of glioblastoma. Further studies are needed to validate these ‘key genes’ as useful tools for early detection and novel therapeutic options for these tumors.