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Introduction: Since the introduction of the LASSO, computational approaches to variable selection have been rigorously developed in the statistical literature. The need for such methods has become increasingly important with the advent of highthroughput technologies in genomics and brain imaging studies where it is believed that the number of truly important variables is small relative to the total number of variables. While the focus of these methods has been on additive models, there are several applications where interaction models can reflect biological phenomena and improve statistical power. For example, genome wide association studies (GWAS) have been unable to explain a large proportion of heritability (the variance in phenotype attributable to genetic variants) and it has been suggested that this missing heritability may in part be due to geneenvironment interactions. Furthermore, diseases are now thought to be the result of entire biological networks whose states are affected by environmental factors. These systemic changes can induce or eliminate strong correlations between elements in a network without necessarily affecting their mean levels.
Methods: Therefore, we propose a multivariate penalization procedure for detecting interactions between high dimensional data ($p >> n$) and an environmental factor, where the effect of this environmental factor on the high dimensional data is widespread and plays a role in predicting the response. Our approach improves on existing procedures for detecting such interactions in several ways; 1) it simultaneously performs model selection and estimation 2) it automatically enforces the strong heredity property, i.e., an interaction term can only be included in the model if the corresponding main effects are in the model 3) it reduces the dimensionality of the problem and leverages the high correlations by transforming the input feature space using network connectivity measures and 4) it leads to interpretable models which are biologically meaningful.
Results: An extensive simulation study shows that our method outperforms LASSO, Elastic Net and Group LASSO in terms of both prediction accuracy and feature selection. We apply our methods to the NIH pediatric brain development study to refine estimates of which regions of the frontal cortex are associated with intelligence scores, and a sample of motherchild pairs from a prospective birth cohort to identify epigenetic marks observed at birth that help predict childhood obesity. Our method is implemented in an \texttt{R} package http://sahirbhatnagar.com/eclust/
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