Inferring network structure in non-normal and mixed discrete-continuous genomic data

Learning Bayesian Networks: A Copula Approach for Mixed-Type Data

Estimating dependence relationships between variables is a crucial issue in many applied domains and in particular psychology. When several variables are entertained, these can be organized into a network which encodes their set of conditional dependence relations. Typically however, the underlying network structure is completely unknown or can be partially drawn only; accordingly it should be learned from the available data, a process known as structure learning. In addition, data arising from social and psychological studies are often of different types, as they can include categorical, discrete and continuous measurements. In this paper, we develop a novel Bayesian methodology for structure learning of directed networks which applies to mixed data, i.e., possibly containing continuous, discrete, ordinal and binary variables simultaneously. Whenever available, our method can easily incorporate known dependence structures among variables represented by paths or edge directions that can be postulated in advance based on the specific problem under consideration. We evaluate the proposed method through extensive simulation studies, with appreciable performances in comparison with current state-of-the-art alternative methods. Finally, we apply our methodology to well-being data from a social survey promoted by the United Nations, and mental health data collected from a cohort of medical students. R code implementing the proposed methodology is available at https://github.com/FedeCastelletti/bayes_networks_mixed_data .

From DNA human sequence to the chromatin higher order organisation and its biological meaning: Using biomolecular interaction networks to understand the influence of structural variation on spatial genome organisation and its functional effect

The three-dimensional structure of the human genome has been proven to have a significant functional impact on gene expression. The high-order spatial chromatin is organised first by looping mediated by multiple protein factors, and then it is further formed into larger structures of topologically associated domains (TADs) or chromatin contact domains (CCDs), followed by A/B compartments and finally the chromosomal territories (CTs). The genetic variation observed in human population influences the multi-scale structures, posing a question regarding the functional impact of structural variants reflected by the variability of the genes expression patterns. The current methods of evaluating the functional effect include eQTLs analysis which uses statistical testing of influence of variants on spatially close genes. Rarely, non-coding DNA sequence changes are evaluated by their impact on the biomolecular interaction network (BIN) reflecting the cellular interactome that can be analysed by the classical graph-theoretic algorithms. Therefore, in the second part of the review, we introduce the concept of BIN, i.e. a meta-network model of the complete molecular interactome developed by integrating various biological networks. The BIN meta-network model includes DNA-protein binding by the plethora of protein factors as well as chromatin interactions, therefore allowing connection of genomics with the downstream biomolecular processes present in a cell. As an illustration, we scrutinise the chromatin interactions mediated by the CTCF protein detected in a ChIA-PET experiment in the human lymphoblastoid cell line GM12878. In the corresponding BIN meta-network the DNA spatial proximity is represented as a graph model, combined with the Proteins-Interaction Network (PIN) of human proteome using the Gene Association Network (GAN). Furthermore, we enriched the BIN with the signalling and metabolic pathways and Gene Ontology (GO) terms to assert its functional context. Finally, we mapped the Single Nucleotide Polymorphisms (SNPs) from the GWAS studies and identified the chromatin mutational hot-spots associated with a significant enrichment of SNPs related to autoimmune diseases. Afterwards, we mapped Structural Variants (SVs) from healthy individuals of 1000 Genomes Project and identified an interesting example of the missing protein complex associated with protein Q6GYQ0 due to a deletion on chromosome 14. Such an analysis using the meta-network BIN model is therefore helpful in evaluating the influence of genetic variation on spatial organisation of the genome and its functional effect in a cell.

Bayesian graphical models for modern biological applications

Graphical models are powerful tools that are regularly used to investigate complex dependence structures in high-throughput biomedical datasets. They allow for holistic, systems-level view of the various biological processes, for intuitive and rigorous understanding and interpretations. In the context of large networks, Bayesian approaches are particularly suitable because it encourages sparsity of the graphs, incorporate prior information, and most importantly account for uncertainty in the graph structure. These features are particularly important in applications with limited sample size, including genomics and imaging studies. In this paper, we review several recently developed techniques for the analysis of large networks under non-standard settings, including but not limited to, multiple graphs for data observed from multiple related subgroups, graphical regression approaches used for the analysis of networks that change with covariates, and other complex sampling and structural settings. We also illustrate the practical utility of some of these methods using examples in cancer genomics and neuroimaging.

Bayesian Variable Selection for Gaussian Copula Regression Models

We develop a novel Bayesian method to select important predictors in regression models with multiple responses of diverse types. A sparse Gaussian copula regression model is used to account for the multivariate dependencies between any combination of discrete and/or continuous responses and their association with a set of predictors. We utilize the parameter expansion for data augmentation strategy to construct a Markov chain Monte Carlo algorithm for the estimation of the parameters and the latent variables of the model. Based on a centered parametrization of the Gaussian latent variables, we design a fixed-dimensional proposal distribution to update jointly the latent binary vectors of important predictors and the corresponding non-zero regression coefficients. For Gaussian responses and for outcomes that can be modeled as a dependent version of a Gaussian response, this proposal leads to a Metropolis-Hastings step that allows an efficient exploration of the predictors' model space. The proposed strategy is tested on simulated data and applied to real data sets in which the responses consist of low-intensity counts, binary, ordinal and continuous variables.

Using Bayesian Latent Gaussian Graphical Models to Infer Symptom Associations in Verbal Autopsies

Learning dependence relationships among variables of mixed types provides insights in a variety of scientific settings and is a well-studied problem in statistics. Existing methods, however, typically rely on copious, high quality data to accurately learn associations. In this paper, we develop a method for scientific settings where learning dependence structure is essential, but data are sparse and have a high fraction of missing values. Specifically, our work is motivated by survey-based cause of death assessments known as verbal autopsies (VAs). We propose a Bayesian approach to characterize dependence relationships using a latent Gaussian graphical model that incorporates informative priors on the marginal distributions of the variables. We demonstrate such information can improve estimation of the dependence structure, especially in settings with little training data. We show that our method can be integrated into existing probabilistic cause-of-death assignment algorithms and improves model performance while recovering dependence patterns between symptoms that can inform efficient questionnaire design in future data collection.

NExUS: Bayesian simultaneous network estimation across unequal sample sizes

MOTIVATION

Network-based analyses of high-throughput genomics data provide a holistic, systems-level understanding of various biological mechanisms for a common population. However, when estimating multiple networks across heterogeneous sub-populations, varying sample sizes pose a challenge in the estimation and inference, as network differences may be driven by differences in power. We are particularly interested in addressing this challenge in the context of proteomic networks for related cancers, as the number of subjects available for rare cancer (sub-)types is often limited.

RESULTS

We develop NExUS (Network Estimation across Unequal Sample sizes), a Bayesian method that enables joint learning of multiple networks while avoiding artefactual relationship between sample size and network sparsity. We demonstrate through simulations that NExUS outperforms existing network estimation methods in this context, and apply it to learn network similarity and shared pathway activity for groups of cancers with related origins represented in The Cancer Genome Atlas (TCGA) proteomic data.

AVAILABILITY AND IMPLEMENTATION

The NExUS source code is freely available for download at https://github.com/priyamdas2/NExUS.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Hierarchical Normalized Completely Random Measures for Robust Graphical Modeling

Gaussian graphical models are useful tools for exploring network structures in multivariate normal data. In this paper we are interested in situations where data show departures from Gaussianity, therefore requiring alternative modeling distributions. The multivariate t-distribution, obtained by dividing each component of the data vector by a gamma random variable, is a straightforward generalization to accommodate deviations from normality such as heavy tails. Since different groups of variables may be contaminated to a different extent, Finegold and Drton (2014) introduced the Dirichlet t-distribution, where the divisors are clustered using a Dirichlet process. In this work, we consider a more general class of nonparametric distributions as the prior on the divisor terms, namely the class of normalized completely random measures (NormCRMs). To improve the effectiveness of the clustering, we propose modeling the dependence among the divisors through a nonparametric hierarchical structure, which allows for the sharing of parameters across the samples in the data set. This desirable feature enables us to cluster together different components of multivariate data in a parsimonious way. We demonstrate through simulations that this approach provides accurate graphical model inference, and apply it to a case study examining the dependence structure in radiomics data derived from The Cancer Imaging Atlas.

Associations between sexual habits, menstrual hygiene practices, demographics and the vaginal microbiome as revealed by Bayesian network analysis

The vaginal microbiome plays an influential role in several disease states in reproductive age women, including bacterial vaginosis (BV). While demographic characteristics are associated with differences in vaginal microbiome community structure, little is known about the influence of sexual and hygiene habits. Furthermore, associations between the vaginal microbiome and risk symptoms of bacterial vaginosis have not been fully elucidated. Using Bayesian network (BN) analysis of 16S rRNA gene sequence results, demographic and extensive questionnaire data, we describe both novel and previously documented associations between habits of women and their vaginal microbiome. The BN analysis approach shows promise in uncovering complex associations between disparate data types. Our findings based on this approach support published associations between specific microbiome members (e.g., Eggerthella, Gardnerella, Dialister, Sneathia and Ruminococcaceae), the Nugent score (a BV diagnostic) and vaginal pH (a risk symptom of BV). Additionally, we found that several microbiome members were directly connected to other risk symptoms of BV (such as vaginal discharge, odor, itch, irritation, and yeast infection) including L. jensenii, Corynebacteria, and Proteobacteria. No direct connections were found between the Nugent Score and risk symptoms of BV other than pH, indicating that the Nugent Score may not be the most useful criteria for assessment of clinical BV. We also found that demographics (i.e., age, ethnicity, previous pregnancy) were associated with the presence/absence of specific vaginal microbes. The resulting BN revealed several as-yet undocumented associations between birth control usage, menstrual hygiene practices and specific microbiome members. Many of these complex relationships were not identified using common analytical methods, i.e., ordination and PERMANOVA. While these associations require confirmatory follow-up study, our findings strongly suggest that future studies of the vaginal microbiome and vaginal pathologies should include detailed surveys of participants' sanitary, sexual and birth control habits, as these can act as confounders in the relationship between the microbiome and disease. Although the BN approach is powerful in revealing complex associations within multidimensional datasets, the need in some cases to discretize the data for use in BN analysis can result in loss of information. Future research is required to alleviate such limitations in constructing BN networks. Large sample sizes are also required in order to allow for the incorporation of a large number of variables (nodes) into the BN, particularly when studying associations between metadata and the microbiome. We believe that this approach is of great value, complementing other methods, to further our understanding of complex associations characteristic of microbiome research.

Rejoinder to statistical contributions to bioinformatics: Design, modelling, structure learning and Integration

We thank the discussants for their kind comments and their insightful analysis and discussion that has substantially added to the contribution of this issue. Overall, it seems the discussants have affirmed many of our primary points, and have also raised a number of other relevant and important issues that we did not emphasize in the paper. Several common threads emerged from these discussions, including the importance of software development, appropriate dissemination, and close collaboration with biomedical scientists and technology experts in order to ensure our work is relevant and impactful. Each discussant also mentioned other areas of bioinformatics that have been impacted by statistical researchers that we did not highlight in the original paper. In response, we will first summarize discuss these general themes, and then respond to specific comments of each discussant, and finally talk about the additional areas of bioinformatics impacted by statisticians that were mentioned by the reviewers.