Group Variable Selection Methods and Their Applications in Analysis of Genomic Data

1.1 Introduction Regression is a simple but the most useful statistical method in data analysis. The goal of regression analysis is to discover the relationship between a response y and a set of predictors x 1 , x 2 ,. .. , x p. When fitting a regression model, besides prediction accuracy, parsimony is another important criterion of goodness. Simpler models are preferred by researchers for easier interpretation of the relationship between x and y. Moreover, discarding irrelevant predictors often improves prediction accuracy [13]. Variable selection methods have long been used in regression analysis, for example forward selection, backward elimination, best subset regression. The number of variables p in the traditional setting is typically 10 or at most a few dozens. Modern scientific technology, led by the microarray, has produced data dramatically above the conventional scale. We have p = 1, 000 to 10,000 in gene expression microarray data, and p up to 500,000 in single nucleotide polymorphism (SNP) data. To make things more complicated, the large number of variables in the biological data are dependent. For example, it is well known that for genes that share a common biological function or participate in the same metabolic pathway, the pairwise correlations among them can be very high [14]. Traditional variable selection methods that select variables one by one may miss important group effects on pathways. Consequently, when traditional variable selection methods are applied in multiple data sets from a common biological system, the selected variables from the multiple studies may show little overlap. To overcome the challenges, we have developed a series of group variable selection methods, which construct highly correlated genes into a group and select the whole group once one gene among them is in the model. In this chapter, we introduce the idea of group variable selection and illustrate its utility by applying the methods to genomic data analysis.

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