Variable selection and inference with a new robust Bayesian elastic net

In high-dimensional genomics studies, elastic net has gained wide popularity for its ability to accommodate structured sparsity, including multicollinearity among omics features. Recently, efforts to robustify elastic net have garnered considerable attention, as data heterogeneity in terms of outliers and heavy-tailed errors in disease phenotypes are frequently encountered. However, statistical inference procedures for robust elastic net remain underdeveloped. To fill the gap, we propose a new robust Bayesian elastic net that leads to superior performance in model fitting and especially statistical inference in the presence of heavy-tailed errors. Specifically, we have developed a fully Bayesian method that builds on a robust likelihood function to safeguard against heterogeneity of complex diseases while accounting for strong correlations. Incorporation of spike-and-slab priors in the Bayesian hierarchical model has significantly improved accuracy in shrinkage estimation, variable selection and statistical inference by inducing exact sparsity through posterior estimates generated from the Metropolis-within-Gibbs sampling. Our numeric study suggests that the new robust elastic net yields valid Bayesian credible intervals with nominal coverage probabilities even on finite samples contaminated with outliers. Furthermore, the analysis of SNP data from the Nurses' Health Study (NHS) has demonstrated the superiority of the proposed method over alternative approaches.