Integrative analysis of gene-environment interactions under a multi-response partially linear varying coefficient model

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).

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.

Integrative deep learning with prior assisted feature selection

Integrative analysis has emerged as a prominent tool in biomedical research, offering a solution to the "smalln $$ n $$ and largep $$ p $$ " challenge. Leveraging the powerful capabilities of deep learning in extracting complex relationship between genes and diseases, our objective in this study is to incorporate deep learning into the framework of integrative analysis. Recognizing the redundancy within candidate features, we introduce a dedicated feature selection layer in the proposed integrative deep learning method. To further improve the performance of feature selection, the rich previous researches are utilized by an ensemble learning method to identify "prior information". This leads to the proposed prior assisted integrative deep learning (PANDA) method. We demonstrate the superiority of the PANDA method through a series of simulation studies, showing its clear advantages over competing approaches in both feature selection and outcome prediction. Finally, a skin cutaneous melanoma (SKCM) dataset is extensively analyzed by the PANDA method to show its practical application.

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.

Robust Bayesian variable selection for gene-environment interactions

Gene-environment (G× E) interactions have important implications to elucidate the etiology of complex diseases beyond the main genetic and environmental effects. Outliers and data contamination in disease phenotypes of G× E studies have been commonly encountered, leading to the development of a broad spectrum of robust regularization methods. Nevertheless, within the Bayesian framework, the issue has not been taken care of in existing studies. We develop a fully Bayesian robust variable selection method for G× E interaction studies. The proposed Bayesian method can effectively accommodate heavy-tailed errors and outliers in the response variable while conducting variable selection by accounting for structural sparsity. In particular, for the robust sparse group selection, the spike-and-slab priors have been imposed on both individual and group levels to identify important main and interaction effects robustly. An efficient Gibbs sampler has been developed to facilitate fast computation. Extensive simulation studies, analysis of diabetes data with single-nucleotide polymorphism measurements from the Nurses' Health Study, and The Cancer Genome Atlas melanoma data with gene expression measurements demonstrate the superior performance of the proposed method over multiple competing alternatives.

Sparse group variable selection for gene-environment interactions in the longitudinal study

Penalized variable selection for high-dimensional longitudinal data has received much attention as it can account for the correlation among repeated measurements while providing additional and essential information for improved identification and prediction performance. Despite the success, in longitudinal studies, the potential of penalization methods is far from fully understood for accommodating structured sparsity. In this article, we develop a sparse group penalization method to conduct the bi-level gene-environment (G  × $\times $  E) interaction study under the repeatedly measured phenotype. Within the quadratic inference function framework, the proposed method can achieve simultaneous identification of main and interaction effects on both the group and individual levels. Simulation studies have shown that the proposed method outperforms major competitors. In the case study of asthma data from the Childhood Asthma Management Program, we conduct G  × $\times $  E study by using high-dimensional single nucleotide polymorphism data as genetic factors and the longitudinal trait, forced expiratory volume in 1 s, as the phenotype. Our method leads to improved prediction and identification of main and interaction effects with important implications.

Multivariate partial linear varying coefficients model for gene-environment interactions with multiple longitudinal traits

Correlated phenotypes often share common genetic determinants. Thus, a multi‐trait analysis can potentially increase association power and help in understanding pleiotropic effect. When multiple traits are jointly measured over time, the correlation information between multivariate longitudinal responses can help to gain power in association analysis, and the longitudinal traits can provide insights on the dynamic gene effect over time. In this work, we propose a multivariate partially linear varying coefficients model to identify genetic variants with their effects potentially modified by environmental factors. We derive a testing framework to jointly test the association of genetic factors and illustrated with a bivariate phenotypic trait, while taking the time varying genetic effects into account. We extend the quadratic inference functions to deal with the longitudinal correlations and used penalized splines for the approximation of nonparametric coefficient functions. Theoretical results such as consistency and asymptotic normality of the estimates are established. The performance of the testing procedure is evaluated through Monte Carlo simulation studies. The utility of the method is demonstrated with a real data set from the Twin Study of Hormones and Behavior across the menstrual cycle project, in which single nucleotide polymorphisms associated with emotional eating behavior are identified.

Interep: An R Package for High-Dimensional Interaction Analysis of the Repeated Measurement Data

We introduce interep, an R package for interaction analysis of repeated measurement data with high-dimensional main and interaction effects. In G × E interaction studies, the forms of environmental factors play a critical role in determining how structured sparsity should be imposed in the high-dimensional scenario to identify important effects. Zhou et al. (2019) (PMID: 31816972) proposed a longitudinal penalization method to select main and interaction effects corresponding to the individual and group structure, respectively, which requires a mixture of individual and group level penalties. The R package interep implements generalized estimating equation (GEE)-based penalization methods with this sparsity assumption. Moreover, alternative methods have also been implemented in the package. These alternative methods merely select effects on an individual level and ignore the group-level interaction structure. In this software article, we first introduce the statistical methodology corresponding to the penalized GEE methods implemented in the package. Next, we present the usage of the core and supporting functions, which is followed by a simulation example with R codes and annotations. The R package interep is available at The Comprehensive R Archive Network (CRAN).

Identifying Gene-Environment Interactions With Robust Marginal Bayesian Variable Selection

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 Interaction: A Variable Selection Perspective

Gene-environment interactions have important implications for elucidating the genetic basis of complex diseases beyond the joint function of multiple genetic factors and their interactions (or epistasis). In the past, G × E interactions have been mainly conducted within the framework of genetic association studies. The high dimensionality of G × E interactions, due to the complicated form of environmental effects and the presence of a large number of genetic factors including gene expressions and SNPs, has motivated the recent development of penalized variable selection methods for dissecting G × E interactions, which has been ignored in the majority of published reviews on genetic interaction studies. In this article, we first survey existing studies on both gene-environment and gene-gene interactions. Then, after a brief introduction to the variable selection methods, we review penalization and relevant variable selection methods in marginal and joint paradigms, respectively, under a variety of conceptual models. Discussions on strengths and limitations, as well as computational aspects of the variable selection methods tailored for G × E studies, have also been provided.

Integrative functional linear model for genome-wide association studies with multiple traits

In recent biomedical research, genome-wide association studies (GWAS) have demonstrated great success in investigating the genetic architecture of human diseases. For many complex diseases, multiple correlated traits have been collected. However, most of the existing GWAS are still limited because they analyze each trait separately without considering their correlations and suffer from a lack of sufficient information. Moreover, the high dimensionality of single nucleotide polymorphism (SNP) data still poses tremendous challenges to statistical methods, in both theoretical and practical aspects. In this article, we innovatively propose an integrative functional linear model for GWAS with multiple traits. This study is the first to approximate SNPs as functional objects in a joint model of multiple traits with penalization techniques. It effectively accommodates the high dimensionality of SNPs and correlations among multiple traits to facilitate information borrowing. Our extensive simulation studies demonstrate the satisfactory performance of the proposed method in the identification and estimation of disease-associated genetic variants, compared to four alternatives. The analysis of type 2 diabetes data leads to biologically meaningful findings with good prediction accuracy and selection stability.

Semiparametric Bayesian variable selection for gene-environment interactions

Many complex diseases are known to be affected by the interactions between genetic variants and environmental exposures beyond the main genetic and environmental effects. Study of gene-environment (G×E) interactions is important for elucidating the disease etiology. Existing Bayesian methods for G×E interaction studies are challenged by the high-dimensional nature of the study and the complexity of environmental influences. Many studies have shown the advantages of penalization methods in detecting G×E interactions in "large p, small n" settings. However, Bayesian variable selection, which can provide fresh insight into G×E study, has not been widely examined. We propose a novel and powerful semiparametric Bayesian variable selection model that can investigate linear and nonlinear G×E interactions simultaneously. Furthermore, the proposed method can conduct structural identification by distinguishing nonlinear interactions from main-effects-only case within the Bayesian framework. Spike-and-slab priors are incorporated on both individual and group levels to identify the sparse main and interaction effects. The proposed method conducts Bayesian variable selection more efficiently than existing methods. Simulation shows that the proposed model outperforms competing alternatives in terms of both identification and prediction. The proposed Bayesian method leads to the identification of main and interaction effects with important implications in a high-throughput profiling study with high-dimensional SNP data.

Penalized Variable Selection for Lipid-Environment Interactions in a Longitudinal Lipidomics Study

Lipid species are critical components of eukaryotic membranes. They play key roles in many biological processes such as signal transduction, cell homeostasis, and energy storage. Investigations of lipid-environment interactions, in addition to the lipid and environment main effects, have important implications in understanding the lipid metabolism and related changes in phenotype. In this study, we developed a novel penalized variable selection method to identify important lipid-environment interactions in a longitudinal lipidomics study. An efficient Newton-Raphson based algorithm was proposed within the generalized estimating equation (GEE) framework. We conducted extensive simulation studies to demonstrate the superior performance of our method over alternatives, in terms of both identification accuracy and prediction performance. As weight control via dietary calorie restriction and exercise has been demonstrated to prevent cancer in a variety of studies, analysis of the high-dimensional lipid datasets collected using 60 mice from the skin cancer prevention study identified meaningful markers that provide fresh insight into the underlying mechanism of cancer preventive effects.

Identification of gene-environment interactions with marginal penalization

Gene-environment (G-E) interaction analysis has been extensively conducted for complex diseases. In marginal analysis, the common practice is to conduct likelihood-based (and other "standard") estimation with each marginal model, and then select significant G-E interactions and main effects based on p values and multiple comparisons adjustment. One limitation of this approach is that the identification results often do not respect the "main effects, interactions" hierarchy, which has been stressed in recent G-E interaction analyses. There is some recent effort tackling this problem, however, with very complex formulations. Another limitation of the common practice is that it may not perform well when regularization is needed, for example, because of "non-normal" distributions. In this article, we propose a marginal penalization approach which adopts a novel penalty to directly tackle the aforementioned problems. The proposed approach has a framework more coherent with that of the recently developed joint analysis methods and an intuitive formulation, and can be effectively realized. In simulation, it outperforms the popular significance-based analysis and simple penalization-based alternatives. Promising findings are made in the analysis of a single-nucleotide polymorphism and a gene expression data.

Structured gene-environment interaction analysis

For the etiology, progression, and treatment of complex diseases, gene-environment (G-E) interactions have important implications beyond the main G and E effects. G-E interaction analysis can be more challenging with higher dimensionality and need for accommodating the "main effects, interactions" hierarchy. In recent literature, an array of novel methods, many of which are based on the penalization technique, have been developed. In most of these studies, however, the structures of G measurements, for example, the adjacency structure of single nucleotide polymorphisms (SNPs; attributable to their physical adjacency on the chromosomes) and the network structure of gene expressions (attributable to their coordinated biological functions and correlated measurements) have not been well accommodated. In this study, we develop structured G-E interaction analysis, where such structures are accommodated using penalization for both the main G effects and interactions. Penalization is also applied for regularized estimation and selection. The proposed structured interaction analysis can be effectively realized. It is shown to have consistency properties under high-dimensional settings. Simulations and analysis of GENEVA diabetes data with SNP measurements and TCGA melanoma data with gene expression measurements demonstrate its competitive practical performance.

Penalized integrative semiparametric interaction analysis for multiple genetic datasets

In this article, we consider a semiparametric additive partially linear interaction model for the integrative analysis of multiple genetic datasets. The goals are to identify important genetic predictors and gene-gene interactions and to estimate the nonparametric functions that describe the environmental effects at the same time. To find the similarities and differences of the genetic effects across different datasets, we impose a group structure on the regression coefficients matrix under the homogeneity assumption, ie, models for different datasets share the same sparsity structure, but the coefficients may differ across datasets. We develop an iterative approach to estimate the parameters of main effects, interactions and nonparametric functions, where a reparametrization of interaction parameters is implemented to meet the strong hierarchy assumption. We demonstrate the advantages of the proposed method in identification, estimation, and prediction in a series of numerical studies. We also apply the proposed method to the Skin Cutaneous Melanoma data and the lung cancer data from the Cancer Genome Atlas.

A Selective Review of Multi-Level Omics Data Integration Using Variable Selection

High-throughput technologies have been used to generate a large amount of omics data. In the past, single-level analysis has been extensively conducted where the omics measurements at different levels, including mRNA, microRNA, CNV and DNA methylation, are analyzed separately. As the molecular complexity of disease etiology exists at all different levels, integrative analysis offers an effective way to borrow strength across multi-level omics data and can be more powerful than single level analysis. In this article, we focus on reviewing existing multi-omics integration studies by paying special attention to variable selection methods. We first summarize published reviews on integrating multi-level omics data. Next, after a brief overview on variable selection methods, we review existing supervised, semi-supervised and unsupervised integrative analyses within parallel and hierarchical integration studies, respectively. The strength and limitations of the methods are discussed in detail. No existing integration method can dominate the rest. The computation aspects are also investigated. The review concludes with possible limitations and future directions for multi-level omics data integration.

Network-based regularization for high dimensional SNP data in the case-control study of Type 2 diabetes

Background

Over the past decades, the prevalence of type 2 diabetes mellitus (T2D) has been steadily increasing around the world. Despite large efforts devoted to better understand the genetic basis of the disease, the identified susceptibility loci can only account for a small portion of the T2D heritability. Some of the existing approaches proposed for the high dimensional genetic data from the T2D case–control study are limited by analyzing a few number of SNPs at a time from a large pool of SNPs, by ignoring the correlations among SNPs and by adopting inefficient selection techniques.

Methods

We propose a network constrained regularization method to select important SNPs by taking the linkage disequilibrium into account. To accomodate the case control study, an iteratively reweighted least square algorithm has been developed within the coordinate descent framework where optimization of the regularized logistic loss function is performed with respect to one parameter at a time and iteratively cycle through all the parameters until convergence.

Results

In this article, a novel approach is developed to identify important SNPs more effectively through incorporating the interconnections among them in the regularized selection. A coordinate descent based iteratively reweighed least squares (IRLS) algorithm has been proposed.

Conclusions

Both the simulation study and the analysis of the Nurses’s Health Study, a case–control study of type 2 diabetes data with high dimensional SNP measurements, demonstrate the advantage of the network based approach over the competing alternatives.