High-dimensional data are often encountered in biomedical, environmental, and other studies. For example, in biomedical studies that involve high-throughput omic data, an important problem is to search for genetic variables that are predictive of a particular phenotype. A conventional solution is to characterize such relationships through regression models in which a phenotype is treated as the response variable and the variables are treated as covariates; this approach becomes particularly challenging when the number of variables exceeds the number of samples. We propose a general framework for expressing the transformed mean of high-dimensional variables in an exponential distribution family via ANOVA models in which a low-rank interaction space captures the association between the phenotype and the variables. This alternative method transforms the variable selection problem into a well-posed problem with the number of observations larger than the number of variables. In addition, we propose a model selection criterion for the new model framework with a diverging number of parameters, and establish the consistency of the selection criterion. We demonstrate the appealing performance of the proposed method in terms of prediction and detection accuracy through simulations and real data analyses.
The restricted consistency property of leave-nv-out cross-validation for high-dimensional variable selection
