Variable selection methods have been extensively developed for and applied to cancer genomics data to identify important omics features associated with complex disease traits, including cancer outcomes. However, the reliability and reproducibility of the findings are in question if valid inferential procedures are not available to quantify the uncertainty of the findings. In this article, we provide a gentle but systematic review of high-dimensional frequentist and Bayesian inferential tools under sparse models which can yield uncertainty quantification measures, including confidence (or Bayesian credible) intervals, p values and false discovery rates (FDR). Connections in high-dimensional inferences between the two realms have been fully exploited under the “unpenalized loss function + penalty term” formulation for regularization methods and the “likelihood function × shrinkage prior” framework for regularized Bayesian analysis. In particular, we advocate for robust Bayesian variable selection in cancer genomics studies due to its ability to accommodate disease heterogeneity in the form of heavy-tailed errors and structured sparsity while providing valid statistical inference. The numerical results show that robust Bayesian analysis incorporating exact sparsity has yielded not only superior estimation and identification results but also valid Bayesian credible intervals under nominal coverage probabilities compared with alternative methods, especially in the presence of heavy-tailed model errors and outliers.

In high-throughput genetics studies, an important aim is to identify gene–environment interactions associated with the clinical outcomes. Recently, multiple marginal penalization methods have been developed and shown to be effective in G×E studies. However, within the Bayesian framework, marginal variable selection has not received much attention. In this study, we propose a novel marginal Bayesian variable selection method for G×E studies. In particular, our marginal Bayesian method is robust to data contamination and outliers in the outcome variables. With the incorporation of spike-and-slab priors, we have implemented the Gibbs sampler based on Markov Chain Monte Carlo (MCMC). The proposed method outperforms a number of alternatives in extensive simulation studies. The utility of the marginal robust Bayesian variable selection method has been further demonstrated in the case studies using data from the Nurse Health Study (NHS). Some of the identified main and interaction effects from the real data analysis have important biological implications.

Gene-environment (G×E) interaction is critical for understanding the genetic basis of complex disease beyond genetic and environment main effects. In addition to existing tools for interaction studies, penalized variable selection emerges as a promising alternative for dissecting G×E interactions. Despite the success, variable selection is limited in terms of accounting for multidimensional measurements. Published variable selection methods cannot accommodate structured sparsity in the framework of integrating multiomics data for disease outcomes. In this paper, we have developed a novel variable selection method in order to integrate multi-omics measurements in G×E interaction studies. Extensive studies have already revealed that analyzing omics data across multi-platforms is not only sensible biologically, but also resulting in improved identification and prediction performance. Our integrative model can efficiently pinpoint important regulators of gene expressions through sparse dimensionality reduction, and link the disease outcomes to multiple effects in the integrative G×E studies through accommodating a sparse bi-level structure. The simulation studies show the integrative model leads to better identification of G×E interactions and regulators than alternative methods. In two G×E lung cancer studies with high dimensional multi-omics data, the integrative model leads to an improved prediction and findings with important biological implications.

There is a growing interest in applying statistical machine learning methods, such as LASSO regression and its extensions, to analyze healthcare datasets. One representative recent study conducted by Huang et al. has examined LASSO and group LASSO regression with categorical predictors that are widely used in healthcare studies to represent variables with nominal or ordinal categories. Despite the success of these studies, statistical inference procedures and quantifying uncertainty for regression with categorical predictors have largely been overlooked, partly due to the theoretical challenges practitioners face when applying these methods in behavioral research. In this article, we aim to fill this gap by investigating from a Bayesian perspective. Specifically, we conduct Bayesian LASSO analysis with categorical predictors under different coding strategies, and thoroughly investigate the impact of four representative coding strategies on variable selection and prediction. In particular, we have conducted uncertainty quantification in terms of marginal Bayesian credible intervals by leveraging the advantage that fully Bayesian analysis can enable exact statistical inference even on finite samples. In this study, we demonstrate that the variable selection, estimation and prediction of Bayesian LASSO are influenced by the coding strategies with the real-world Medical Expenditure Panel Survey (MEPS) data. The performance of Bayesian LASSO has also been compared with LASSO and linear regression.