Incorporating Grouping Information in Bayesian Variable Selection with Applications in Genomics

Abstract

In many applications it is of interest to determine a limited number of important explanatory factors (representing groups of potentially overlapping predictors) rather than original predictor variables. The often imposed requirement that the clustered predictors should enter the model simultaneously may be limiting as not all the variables within a group need to be associated with the outcome. Within-group sparsity is often desirable as well. Here we propose a Bayesian variable selection method, which uses the grouping information as a means of introducing more equal competition to enter the model within the groups rather than as a source of strict regularization constraints. This is achieved within the context of Bayesian LASSO (least absolute shrinkage and selection operator) by allowing each regression coefficient to be penalized differentially and by considering an additional regression layer to relate individual penalty parameters to a group identification matrix. The proposed hierarchical model therefore enables inference simultaneously on two levels: (1) the regression layer for the continuous outcome in relation to the predictors and (2) the regression layer for the penalty parameters in relation to the grouping information. Both situations with overlapping and non-overlapping groups are applicable. The method does not assume within-group homogeneity across the regression coefficients, which is implicit in many structured penalized likelihood approaches. The smoothness here is enforced at the penalty level rather than within the regression coefficients. To enhance the potential of the proposed method we develop two rapid computational procedures based on the expectation maximization (EM) algorithm, which offer substantial time savings in applications where the high-dimensionality renders Markov chain Monte Carlo (MCMC) approaches less practical. We demonstrate the usefulness of our method in predicting time to death in glioblastoma patients using pathways of genes.