Bayesian hierarchical structured variable selection methods with application to MIP studies in breast cancer

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.

Bayesian interaction selection model for multimodal neuroimaging data analysis

Multimodality or multiconstruct data arise increasingly in functional neuroimaging studies to characterize brain activity under different cognitive states. Relying on those high-resolution imaging collections, it is of great interest to identify predictive imaging markers and intermodality interactions with respect to behavior outcomes. Currently, most of the existing variable selection models do not consider predictive effects from interactions, and the desired higher-order terms can only be included in the predictive mechanism following a two-step procedure, suffering from potential misspecification. In this paper, we propose a unified Bayesian prior model to simultaneously identify main effect features and intermodality interactions within the same inference platform in the presence of high-dimensional data. To accommodate the brain topological information and correlation between modalities, our prior is designed by compiling the intermediate selection status of sequential partitions in light of the data structure and brain anatomical architecture, so that we can improve posterior inference and enhance biological plausibility. Through extensive simulations, we show the superiority of our approach in main and interaction effects selection, and prediction under multimodality data. Applying the method to the Adolescent Brain Cognitive Development (ABCD) study, we characterize the brain functional underpinnings with respect to general cognitive ability under different memory load conditions.

A hierarchical spike-and-slab model for pan-cancer survival using pan-omic data

Background

Pan-omics, pan-cancer analysis has advanced our understanding of the molecular heterogeneity of cancer. However, such analyses have been limited in their ability to use information from multiple sources of data (e.g., omics platforms) and multiple sample sets (e.g., cancer types) to predict clinical outcomes. We address the issue of prediction across multiple high-dimensional sources of data and sample sets by using molecular patterns identified by BIDIFAC+, a method for integrative dimension reduction of bidimensionally-linked matrices, in a Bayesian hierarchical model. Our model performs variable selection through spike-and-slab priors that borrow information across clustered data. We use this model to predict overall patient survival from the Cancer Genome Atlas with data from 29 cancer types and 4 omics sources and use simulations to characterize the performance of the hierarchical spike-and-slab prior.

Results

We found that molecular patterns shared across all or most cancers were largely not predictive of survival. However, our model selected patterns unique to subsets of cancers that differentiate clinical tumor subtypes with markedly different survival outcomes. Some of these subtypes were previously established, such as subtypes of uterine corpus endometrial carcinoma, while others may be novel, such as subtypes within a set of kidney carcinomas. Through simulations, we found that the hierarchical spike-and-slab prior performs best in terms of variable selection accuracy and predictive power when borrowing information is advantageous, but also offers competitive performance when it is not.

Conclusions

We address the issue of prediction across multiple sources of data by using results from BIDIFAC+ in a Bayesian hierarchical model for overall patient survival. By incorporating spike-and-slab priors that borrow information across cancers, we identified molecular patterns that distinguish clinical tumor subtypes within a single cancer and within a group of cancers. We also corroborate the flexibility and performance of using spike-and-slab priors as a Bayesian variable selection approach.

Bayesian hierarchical models for high-dimensional mediation analysis with coordinated selection of correlated mediators

We consider Bayesian high-dimensional mediation analysis to identify among a large set of correlated potential mediators the active ones that mediate the effect from an exposure variable to an outcome of interest. Correlations among mediators are commonly observed in modern data analysis; examples include the activated voxels within connected regions in brain image data, regulatory signals driven by gene networks in genome data, and correlated exposure data from the same source. When correlations are present among active mediators, mediation analysis that fails to account for such correlation can be suboptimal and may lead to a loss of power in identifying active mediators. Building upon a recent high-dimensional mediation analysis framework, we propose two Bayesian hierarchical models, one with a Gaussian mixture prior that enables correlated mediator selection and the other with a Potts mixture prior that accounts for the correlation among active mediators in mediation analysis. We develop efficient sampling algorithms for both methods. Various simulations demonstrate that our methods enable effective identification of correlated active mediators, which could be missed by using existing methods that assume prior independence among active mediators. The proposed methods are applied to the LIFECODES birth cohort and the Multi-Ethnic Study of Atherosclerosis (MESA) and identified new active mediators with important biological implications.

A Bayesian group lasso classification for ADNI volumetrics data

The primary objective of this paper is to develop a statistically valid classification procedure for analyzing brain image volumetrics data obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) in elderly subjects with cognitive impairments. The Bayesian group lasso method thereby proposed for logistic regression efficiently selects an optimal model with the use of a spike and slab type prior. This method selects groups of attributes of a brain subregion encouraged by the group lasso penalty. We conduct simulation studies for high- and low-dimensional scenarios where our method is always able to select the true parameters that are truly predictive among a large number of parameters. The method is then applied on dichotomous response ADNI data which selects predictive atrophied brain regions and classifies Alzheimer's disease patients from healthy controls. Our analysis is able to give an accuracy rate of 80% for classifying Alzheimer's disease. The suggested method selects 29 brain subregions. The medical literature indicates that all these regions are associated with Alzheimer's patients. The Bayesian method of model selection further helps selecting only the subregions that are statistically significant, thus obtaining an optimal model.

Variable selection and estimation in causal inference using Bayesian spike and slab priors

Unbiased estimation of causal effects with observational data requires adjustment for confounding variables that are related to both the outcome and treatment assignment. Standard variable selection techniques aim to maximize predictive ability of the outcome model, but they ignore covariate associations with treatment and may not adjust for important confounders weakly associated to outcome. We propose a novel method for estimating causal effects that simultaneously considers models for both outcome and treatment, which we call the bilevel spike and slab causal estimator (BSSCE). By using a Bayesian formulation, BSSCE estimates the posterior distribution of all model parameters and provides straightforward and reliable inference. Spike and slab priors are used on each covariate coefficient which aim to minimize the mean squared error of the treatment effect estimator. Theoretical properties of the treatment effect estimator are derived justifying the prior used in BSSCE. Simulations show that BSSCE can substantially reduce mean squared error over numerous methods and performs especially well with large numbers of covariates, including situations where the number of covariates is greater than the sample size. We illustrate BSSCE by estimating the causal effect of vasoactive therapy vs. fluid resuscitation on hypotensive episode length for patients in the Multiparameter Intelligent Monitoring in Intensive Care III critical care database.

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.

A Bayesian hierarchical variable selection prior for pathway-based GWAS using summary statistics

While genome-wide association studies (GWASs) have been widely used to uncover associations between diseases and genetic variants, standard SNP-level GWASs often lack the power to identify SNPs that individually have a moderate effect size but jointly contribute to the disease. To overcome this problem, pathway-based GWASs methods have been developed as an alternative strategy that complements SNP-level approaches. We propose a Bayesian method that uses the generalized fused hierarchical structured variable selection prior to identify pathways associated with the disease using SNP-level summary statistics. Our prior has the flexibility to take in pathway structural information so that it can model the gene-level correlation based on prior biological knowledge, an important feature that makes it appealing compared to existing pathway-based methods. Using simulations, we show that our method outperforms competing methods in various scenarios, particularly when we have pathway structural information that involves complex gene-gene interactions. We apply our method to the Wellcome Trust Case Control Consortium Crohn's disease GWAS data, demonstrating its practical application to real data.

Knowledge-Guided Statistical Learning Methods for Analysis of High-Dimensional -Omics Data in Precision Oncology

High-dimensional -omics data such as genomic, transcriptomic, and metabolomic data offer great promise in advancing precision medicine. In particular, such data have enabled the investigation of complex diseases such as cancer at an unprecedented scale and in multiple dimensions. However, a number of analytical challenges complicate analysis of high-dimensional -omics data. One is the growing recognition that complex diseases such as cancer are multifactorial and may be attributed to harmful changes on multiple -omics levels and on the pathway level. When individual genes in an important pathway have relatively weak signals, it can be challenging to detect them on their own, but the aggregated signal in the pathway can be considerably stronger and hence easier to detect with the same sample size. To address these challenges, there is a growing body of literature on knowledge-guided statistical learning methods for analysis of high-dimensional -omics data that can incorporate biological knowledge such as functional genomics and functional proteomics. These methods have been shown to improve predication and classification accuracy and yield biologically more interpretable results compared with statistical learning methods that do not use biological knowledge. In this review, we survey current knowledge-guided statistical learning methods, including both supervised learning and unsupervised learning, and their applications to precision oncology, and we discuss future research directions.

Structured Genome-Wide Association Studies with Bayesian Hierarchical Variable Selection

It becomes increasingly important in using genome-wide association studies (GWAS) to select important genetic information associated with qualitative or quantitative traits. Currently, the discovery of biological association among SNPs motivates various strategies to construct SNP-sets along the genome and to incorporate such set information into selection procedure for a higher selection power, while facilitating more biologically meaningful results. The aim of this paper is to propose a novel Bayesian framework for hierarchical variable selection at both SNP-set (group) level and SNP (within group) level. We overcome a key limitation of existing posterior updating scheme in most Bayesian variable selection methods by proposing a novel sampling scheme to explicitly accommodate the ultrahigh-dimensionality of genetic data. Specifically, by constructing an auxiliary variable selection model under SNP-set level, the new procedure utilizes the posterior samples of the auxiliary model to subsequently guide the posterior inference for the targeted hierarchical selection model. We apply the proposed method to a variety of simulation studies and show that our method is computationally efficient and achieves substantially better performance than competing approaches in both SNP-set and SNP selection. Applying the method to the Alzheimers Disease Neuroimaging Initiative (ADNI) data, we identify biologically meaningful genetic factors under several neuroimaging volumetric phenotypes. Our method is general and readily to be applied to a wide range of biomedical studies.

Bayesian Joint Spike-and-Slab Graphical Lasso

In this article, we propose a new class of priors for Bayesian inference with multiple Gaussian graphical models. We introduce Bayesian treatments of two popular procedures, the group graphical lasso and the fused graphical lasso, and extend them to a continuous spike-and-slab framework to allow self-adaptive shrinkage and model selection simultaneously. We develop an EM algorithm that performs fast and dynamic explorations of posterior modes. Our approach selects sparse models efficiently and automatically with substantially smaller bias than would be induced by alternative regularization procedures. The performance of the proposed methods are demonstrated through simulation and two real data examples.

Bayesian variable selection based on clinical relevance weights in small sample studies-Application to colon cancer

Using clinical data to model the medical decisions behind sequential treatment actions raises methodological challenges. Physicians often have access to many covariates that may be used when making sequential treatment decisions for individual patients. Statistical variable selection methods may help finding which of these variables are used for this decision in everyday practice. When the sample size is not large, Bayesian variable selection methods can address this setting and allow for expert information to be incorporated into prior distributions. Motivated by clinical practice data involving repeated dose adaptation for Irinotecan in colorectal metastatic cancer, we propose a modification of the stochastic search variable selection (SSVS) method, which we call weight-based SSVS (WBS). We use clinical relevance weights elicited from physician experts to construct prior distributions, with the goal to identify the most influential toxicities and other covariates used for dose adjustment. We evaluate and compare the WBS model performance to the Lasso and SSVS through an extensive simulation study. The simulations show that WBS has better performance and lower rates of false positives and false negatives than the other methods but depends strongly on the covariate weights.

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.

Using Regularization to Infer Cell Line Specificity in Logical Network Models of Signaling Pathways

Understanding the functional properties of cells of different origins is a long-standing challenge of personalized medicine. Especially in cancer, the high heterogeneity observed in patients slows down the development of effective cures. The molecular differences between cell types or between healthy and diseased cellular states are usually determined by the wiring of regulatory networks. Understanding these molecular and cellular differences at the systems level would improve patient stratification and facilitate the design of rational intervention strategies. Models of cellular regulatory networks frequently make weak assumptions about the distribution of model parameters across cell types or patients. These assumptions are usually expressed in the form of regularization of the objective function of the optimization problem. We propose a new method of regularization for network models of signaling pathways based on the local density of the inferred parameter values within the parameter space. Our method reduces the complexity of models by creating groups of cell line-specific parameters which can then be optimized together. We demonstrate the use of our method by recovering the correct topology and inferring accurate values of the parameters of a small synthetic model. To show the value of our method in a realistic setting, we re-analyze a recently published phosphoproteomic dataset from a panel of 14 colon cancer cell lines. We conclude that our method efficiently reduces model complexity and helps recovering context-specific regulatory information.

A Bayesian Gene-Based Genome-Wide Association Study Analysis of Osteosarcoma Trio Data Using a Hierarchically Structured Prior

Osteosarcoma is considered to be the most common primary malignant bone cancer among children and young adults. Previous studies suggest growth spurts and height to be risk factors for osteosarcoma. However, studies on the genetic cause are still limited given the rare occurrence of the disease. In this study, we investigated in a family trio data set that is composed of 209 patients and their unaffected parents and conducted a genome-wide association study (GWAS) to identify genetic risk factors for osteosarcoma. We performed a Bayesian gene-based GWAS based on the single-nucleotide polymorphism (SNP)-level summary statistics obtained from a likelihood ratio test of the trio data, which uses a hierarchically structured prior that incorporates the SNP-gene hierarchical structure. The Bayesian approach has higher power than SNP-level GWAS analysis due to the reduced number of tests and is robust by accounting for the correlations between SNPs so that it borrows information across SNPs within a gene. We identified 217 genes that achieved genome-wide significance. Ingenuity pathway analysis of the gene set indicated that osteosarcoma is potentially related to TP53, estrogen receptor signaling, xenobiotic metabolism signaling, and RANK signaling in osteoclasts.

Bayesian variable selection for post-analytic interrogation of susceptibility loci

Understanding the complex interplay among protein coding genes and regulatory elements requires rigorous interrogation with analytic tools designed for discerning the relative contributions of overlapping genomic regions. To this aim, we offer a novel application of Bayesian variable selection (BVS) for classifying genomic class level associations using existing large meta-analysis summary level resources. This approach is applied using the expectation maximization variable selection (EMVS) algorithm to typed and imputed SNPs across 502 protein coding genes (PCGs) and 220 long intergenic non-coding RNAs (lncRNAs) that overlap 45 known loci for coronary artery disease (CAD) using publicly available Global Lipids Gentics Consortium (GLGC) (Teslovich et al., 2010; Willer et al., 2013) meta-analysis summary statistics for low-density lipoprotein cholesterol (LDL-C). The analysis reveals 33 PCGs and three lncRNAs across 11 loci with >50% posterior probabilities for inclusion in an additive model of association. The findings are consistent with previous reports, while providing some new insight into the architecture of LDL-cholesterol to be investigated further. As genomic taxonomies continue to evolve, additional classes such as enhancer elements and splicing regions, can easily be layered into the proposed analysis framework. Moreover, application of this approach to alternative publicly available meta-analysis resources, or more generally as a post-analytic strategy to further interrogate regions that are identified through single point analysis, is straightforward. All coding examples are implemented in R version 3.2.1 and provided as supplemental material.

Elucidation of Genetic Interactions in the Yeast GATA-Factor Network Using Bayesian Model Selection

Understanding the structure and function of complex gene regulatory networks using classical genetic assays is an error-prone procedure that frequently generates ambiguous outcomes. Even some of the best-characterized gene networks contain interactions whose validity is not conclusively proven. Founded on dynamic experimental data, mechanistic mathematical models are able to offer detailed insights that would otherwise require prohibitively large numbers of genetic experiments. Here we attempt mechanistic modeling of the transcriptional network formed by the four GATA-factor proteins, a well-studied system of central importance for nitrogen-source regulation of transcription in the yeast Saccharomyces cerevisiae. To resolve ambiguities in the network organization, we encoded a set of five interactions hypothesized in the literature into a set of 32 mathematical models, and employed Bayesian model selection to identify the most plausible set of interactions based on dynamic gene expression data. The top-ranking model was validated on newly generated GFP reporter dynamic data and was subsequently used to gain a better understanding of how yeast cells organize their transcriptional response to dynamic changes of nitrogen sources. Our work constitutes a necessary and important step towards obtaining a holistic view of the yeast nitrogen regulation mechanisms; on the computational side, it provides a demonstration of how powerful Monte Carlo techniques can be creatively combined and used to address the great challenges of large-scale dynamical system inference.

Gene regulatory networks underlie all key processes that enable a cell to maintain long-term homeostasis in a changing environment. Understanding the structure and function of complex gene networks is an experimentally difficult and error-prone procedure. Mechanistic mathematical modeling promises to alleviate these problems, as we demonstrate here for the yeast GATA-factor network, the central controller of the cellular response to nitrogen source quality. Despite years of targeted studies, the interaction pattern of this network is still not known precisely. To resolve several still-remaining ambiguities, we generated a set of alternative mathematical models, and compared them against each other using Bayesian model selection based on dynamic gene expression data. The top-ranking model was then validated on a separate, newly generated dataset. Our work thus provides new insights to the mechanism of nitrogen regulation in yeast, while at the same time overcoming some key computational inference problems for large models in systems biology.

Bayesian hierarchical modelling for inferring genetic interactions in yeast

Quantitative fitness analysis (QFA) is a high throughput experimental and computational methodology for measuring the growth of microbial populations. QFA screens can be used to compare the health of cell populations with and without a mutation in a query gene to infer genetic interaction strengths genomewide, examining thousands of separate genotypes. We introduce Bayesian hierarchical models of population growth rates and genetic interactions that better reflect QFA experimental design than current approaches. Our new approach models population dynamics and genetic interaction simultaneously, thereby avoiding passing information between models via a univariate fitness summary. Matching experimental structure more closely, Bayesian hierarchical approaches use data more efficiently and find new evidence for genes which interact with yeast telomeres within a published data set.

Bayesian joint selection of genes and pathways: applications in multiple myeloma genomics

It is well-established that the development of a disease, especially cancer, is a complex process that results from the joint effects of multiple genes involved in various molecular signaling pathways. In this article, we propose methods to discover genes and molecular pathways significantly associated with clinical outcomes in cancer samples. We exploit the natural hierarchal structure of genes related to a given pathway as a group of interacting genes to conduct selection of both pathways and genes. We posit the problem in a hierarchical structured variable selection (HSVS) framework to analyze the corresponding gene expression data. HSVS methods conduct simultaneous variable selection at the pathway (group level) and the gene (within-group) level. To adapt to the overlapping group structure present in the pathway–gene hierarchy of the data, we developed an overlap-HSVS method that introduces latent partial effect variables that partition the marginal effect of the covariates and corresponding weights for a proportional shrinkage of the partial effects. Combining gene expression data with prior pathway information from the KEGG databases, we identified several gene–pathway combinations that are significantly associated with clinical outcomes of multiple myeloma. Biological discoveries support this relationship for the pathways and the corresponding genes we identified.