Robust Bayesian variable selection for gene-environment interactions

BHCox: Bayesian heredity-constrained Cox proportional hazards models for detecting gene-environment interactions

BACKGROUND

Gene-environment (G × E) interactions play a critical role in understanding the etiology of diseases and exploring the factors that affect disease prognosis. There are several challenges in detecting G × E interactions for censored survival outcomes, such as the high dimensionality, complexity of environmental effects, and specificity of survival analysis. The effect heredity, which incorporates the dependence of the main effects and interactions in the analysis, has been widely applied in the study of interaction detection. However, it has not yet been applied to Bayesian Cox proportional hazards models for detecting interactions for censored survival outcomes.

RESULTS

In this study, we propose Bayesian heredity-constrained Cox proportional hazards (BHCox) models with novel spike-and-slab and regularized horseshoe priors that incorporate effect heredity to identify and estimate the main and interaction effects. The no-U-turn sampler (NUTS) algorithm, which has been implemented in the R package brms, was used to fit the proposed model. Extensive simulations were performed to evaluate and compare our proposed approaches with other alternative models. The simulation studies illustrated that BHCox models outperform other alternative models. We applied the proposed method to real data of non-small-cell lung cancer (NSCLC) and identified biologically plausible G × smoking interactions associated with the prognosis of patients with NSCLC.

CONCLUSIONS

In summary, BHCox can be used to detect the main effects and interactions and thus have significant implications for the discovery of high-dimensional interactions in censored survival outcome data.

The spike-and-slab quantile LASSO for robust variable selection in cancer genomics studies

Data irregularity in cancer genomics studies has been widely observed in the form of outliers and heavy-tailed distributions in the complex traits. In the past decade, robust variable selection methods have emerged as powerful alternatives to the nonrobust ones to identify important genes associated with heterogeneous disease traits and build superior predictive models. In this study, to keep the remarkable features of the quantile LASSO and fully Bayesian regularized quantile regression while overcoming their disadvantage in the analysis of high-dimensional genomics data, we propose the spike-and-slab quantile LASSO through a fully Bayesian spike-and-slab formulation under the robust likelihood by adopting the asymmetric Laplace distribution (ALD). The proposed robust method has inherited the prominent properties of selective shrinkage and self-adaptivity to the sparsity pattern from the spike-and-slab LASSO (Roc̆ková and George, J Am Stat Associat, 2018, 113(521): 431-444). Furthermore, the spike-and-slab quantile LASSO has a computational advantage to locate the posterior modes via soft-thresholding rule guided Expectation-Maximization (EM) steps in the coordinate descent framework, a phenomenon rarely observed for robust regularization with nondifferentiable loss functions. We have conducted comprehensive simulation studies with a variety of heavy-tailed errors in both homogeneous and heterogeneous model settings to demonstrate the superiority of the spike-and-slab quantile LASSO over its competing methods. The advantage of the proposed method has been further demonstrated in case studies of the lung adenocarcinomas (LUAD) and skin cutaneous melanoma (SKCM) data from The Cancer Genome Atlas (TCGA).

BHAFT: Bayesian heredity-constrained accelerated failure time models for detecting gene-environment interactions in survival analysis

In addition to considering the main effects, understanding gene-environment (G × E) interactions is imperative for determining the etiology of diseases and the factors that affect their prognosis. In the existing statistical framework for censored survival outcomes, there are several challenges in detecting G × E interactions, such as handling high-dimensional omics data, diverse environmental factors, and algorithmic complications in survival analysis. The effect heredity principle has widely been used in studies involving interaction identification because it incorporates the dependence of the main and interaction effects. However, Bayesian survival models that incorporate the assumption of this principle have not been developed. Therefore, we propose Bayesian heredity-constrained accelerated failure time (BHAFT) models for identifying main and interaction (M-I) effects with novel spike-and-slab or regularized horseshoe priors to incorporate the assumption of effect heredity principle. The R package rstan was used to fit the proposed models. Extensive simulations demonstrated that BHAFT models had outperformed other existing models in terms of signal identification, coefficient estimation, and prognosis prediction. Biologically plausible G × E interactions associated with the prognosis of lung adenocarcinoma were identified using our proposed model. Notably, BHAFT models incorporating the effect heredity principle could identify both main and interaction effects, which are highly useful in exploring G × E interactions in high-dimensional survival analysis. The code and data used in our paper are available at https://github.com/SunNa-bayesian/BHAFT.

Is Seeing Believing? A Practitioner's Perspective on High-Dimensional Statistical Inference in Cancer Genomics Studies

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.

Hierarchical False Discovery Rate Control for High-dimensional Survival Analysis with Interactions

With the development of data collection techniques, analysis with a survival response and high-dimensional covariates has become routine. Here we consider an interaction model, which includes a set of low-dimensional covariates, a set of high-dimensional covariates, and their interactions. This model has been motivated by gene-environment (G-E) interaction analysis, where the E variables have a low dimension, and the G variables have a high dimension. For such a model, there has been extensive research on estimation and variable selection. Comparatively, inference studies with a valid false discovery rate (FDR) control have been very limited. The existing high-dimensional inference tools cannot be directly applied to interaction models, as interactions and main effects are not "equal". In this article, for high-dimensional survival analysis with interactions, we model survival using the Accelerated Failure Time (AFT) model and adopt a "weighted least squares + debiased Lasso" approach for estimation and selection. A hierarchical FDR control approach is developed for inference and respect of the "main effects, interactions" hierarchy. The asymptotic distribution properties of the debiased Lasso estimators are rigorously established. Simulation demonstrates the satisfactory performance of the proposed approach, and the analysis of a breast cancer dataset further establishes its practical utility.

The Bayesian Regularized Quantile Varying Coefficient Model

The quantile varying coefficient (VC) model can flexibly capture dynamical patterns of regression coefficients. In addition, due to the quantile check loss function, it is robust against outliers and heavy-tailed distributions of the response variable, and can provide a more comprehensive picture of modeling via exploring the conditional quantiles of the response variable. Although extensive studies have been conducted to examine variable selection for the high-dimensional quantile varying coefficient models, the Bayesian analysis has been rarely developed. The Bayesian regularized quantile varying coefficient model has been proposed to incorporate robustness against data heterogeneity while accommodating the non-linear interactions between the effect modifier and predictors. Selecting important varying coefficients can be achieved through Bayesian variable selection. Incorporating the multivariate spike-and-slab priors further improves performance by inducing exact sparsity. The Gibbs sampler has been derived to conduct efficient posterior inference of the sparse Bayesian quantile VC model through Markov chain Monte Carlo (MCMC). The merit of the proposed model in selection and estimation accuracy over the alternatives has been systematically investigated in simulation under specific quantile levels and multiple heavy-tailed model errors. In the case study, the proposed model leads to identification of biologically sensible markers in a non-linear gene-environment interaction study using the NHS data.

Springer: An R package for bi-level variable selection of high-dimensional longitudinal data

In high-dimensional data analysis, the bi-level (or the sparse group) variable selection can simultaneously conduct penalization on the group level and within groups, which has been developed for continuous, binary, and survival responses in the literature. Zhou et al. (2022) (PMID: 35766061) has further extended it under the longitudinal response by proposing a quadratic inference function-based penalization method in gene-environment interaction studies. This study introduces "springer," an R package implementing the bi-level variable selection within the QIF framework developed in Zhou et al. (2022). In addition, R package "springer" has also implemented the generalized estimating equation-based sparse group penalization method. Alternative methods focusing only on the group level or individual level have also been provided by the package. In this study, we have systematically introduced the longitudinal penalization methods implemented in the "springer" package. We demonstrate the usage of the core and supporting functions, which is followed by the numerical examples and discussions. R package "springer" is available at https://cran.r-project.org/package=springer.

Overlapping group screening for detection of gene-environment interactions with application to TCGA high-dimensional survival genomic data

BACKGROUND

In the context of biomedical and epidemiological research, gene-environment (G-E) interaction is of great significance to the etiology and progression of many complex diseases. In high-dimensional genetic data, two general models, marginal and joint models, are proposed to identify important interaction factors. Most existing approaches for identifying G-E interactions are limited owing to the lack of robustness to outliers/contamination in response and predictor data. In particular, right-censored survival outcomes make the associated feature screening even challenging. In this article, we utilize the overlapping group screening (OGS) approach to select important G-E interactions related to clinical survival outcomes by incorporating the gene pathway information under a joint modeling framework.

RESULTS

Simulation studies under various scenarios are carried out to compare the performances of our proposed method with some commonly used methods. In the real data applications, we use our proposed method to identify G-E interactions related to the clinical survival outcomes of patients with head and neck squamous cell carcinoma, and esophageal carcinoma in The Cancer Genome Atlas clinical survival genetic data, and further establish corresponding survival prediction models. Both simulation and real data studies show that our method performs well and outperforms existing methods in the G-E interaction selection, effect estimation, and survival prediction accuracy.

CONCLUSIONS

The OGS approach is useful for selecting important environmental factors, genes and G-E interactions in the ultra-high dimensional feature space. The prediction ability of OGS with the Lasso penalty is better than existing methods. The same idea of the OGS approach can apply to other outcome models, such as the proportional odds survival time model, the logistic regression model for binary outcomes, and the multinomial logistic regression model for multi-class outcomes.