Gaussian graphical models with applications to omics analyses.
Ontology highlight
ABSTRACT:
SUBMITTER: Shutta KH
PROVIDER: S-EPMC9672860 | biostudies-literature | 2022 Nov
REPOSITORIES: biostudies-literature
Publications
Gaussian graphical models with applications to omics analyses.
Shutta Katherine H KH De Vito Roberta R Scholtens Denise M DM Balasubramanian Raji R
Statistics in medicine 20220926 25
Gaussian graphical models (GGMs) provide a framework for modeling conditional dependencies in multivariate data. In this tutorial, we provide an overview of GGM theory and a demonstration of various GGM tools in R. The mathematical foundations of GGMs are introduced with the goal of enabling the researcher to draw practical conclusions by interpreting model results. Background literature is presented, emphasizing methods recently developed for high-dimensional applications such as genomics, prot ...[more]
Similar Datasets
Project description:Gaussian Graphical Models (GGMs) are tools to infer dependencies between biological variables. Popular applications are the reconstruction of gene, protein, and metabolite association networks. GGMs are an exploratory research tool that can be useful to discover interesting relations between genes (functional clusters) or to identify therapeutically interesting genes, but do not necessarily infer a network in the mechanistic sense. Although GGMs are well investigated from a theoretical and applied perspective, important extensions are not well known within the biological community. GGMs assume, for instance, multivariate normal distributed data. If this assumption is violated Mixed Graphical Models (MGMs) can be the better choice. In this review, we provide the theoretical foundations of GGMs, present extensions such as MGMs or multi-class GGMs, and illustrate how those methods can provide insight in biological mechanisms. We summarize several applications and present user-friendly estimation software. This article is part of a Special Issue entitled: Transcriptional Profiles and Regulatory Gene Networks edited by Dr. Dr. Federico Manuel Giorgi and Dr. Shaun Mahony.
| S-EPMC7166149 | biostudies-literature
Project description:Gaussian graphical models are widely used to represent conditional dependence among random variables. In this paper, we propose a novel estimator for data arising from a group of Gaussian graphical models that are themselves dependent. A motivating example is that of modeling gene expression collected on multiple tissues from the same individual: here the multivariate outcome is affected by dependencies acting not only at the level of the specific tissues, but also at the level of the whole body; existing methods that assume independence among graphs are not applicable in this case. To estimate multiple dependent graphs, we decompose the problem into two graphical layers: the systemic layer, which affects all outcomes and thereby induces cross-graph dependence, and the category-specific layer, which represents graph-specific variation. We propose a graphical EM technique that estimates both layers jointly, establish estimation consistency and selection sparsistency of the proposed estimator, and confirm by simulation that the EM method is superior to a simple one-step method. We apply our technique to mouse genomics data and obtain biologically plausible results.
| S-EPMC5640885 | biostudies-literature
Project description:The task of estimating a Gaussian graphical model in the high-dimensional setting is considered. The graphical lasso, which involves maximizing the Gaussian log likelihood subject to a lasso penalty, is a well-studied approach for this task. A surprising connection between the graphical lasso and hierarchical clustering is introduced: the graphical lasso in effect performs a two-step procedure, in which (1) single linkage hierarchical clustering is performed on the variables in order to identify connected components, and then (2) a penalized log likelihood is maximized on the subset of variables within each connected component. Thus, the graphical lasso determines the connected components of the estimated network via single linkage clustering. The single linkage clustering is known to perform poorly in certain finite-sample settings. Therefore, the cluster graphical lasso, which involves clustering the features using an alternative to single linkage clustering, and then performing the graphical lasso on the subset of variables within each cluster, is proposed. Model selection consistency for this technique is established, and its improved performance relative to the graphical lasso is demonstrated in a simulation study, as well as in applications to a university webpage and a gene expression data sets.
| S-EPMC4307846 | biostudies-literature
Project description:The Cancer Genome Atlas (TCGA) provides a rich resource that can be used to understand how genes interact in cancer cells and has collected RNA-Seq gene expression data for many types of human cancer. However, mining the data to uncover the hidden gene-interaction patterns remains a challenge. Gaussian graphical model (GGM) is often used to learn genetic networks because it defines an undirected graphical structure, revealing the conditional dependences of genes. In this study, we focus on inferring gene interactions in 15 specific types of human cancer using RNA-Seq expression data and GGM with graphical lasso. We take advantage of the corresponding Kyoto Encyclopedia of Genes and Genomes pathway maps to define the subsets of related genes. RNA-Seq expression levels of the subsets of genes in solid cancerous tumor and normal tissues were extracted from TCGA. The gene expression data sets were cleaned and formatted, and the genetic network corresponding to each cancer type was then inferred using GGM with graphical lasso. The inferred networks reveal stable conditional dependences among the genes at the expression level and confirm the essential roles played by the genes that encode proteins involved in the two key signaling pathway phosphoinositide 3-kinase (PI3K)/AKT/mTOR and Ras/Raf/MEK/ERK in human carcinogenesis. These stable dependences elucidate the expression level interactions among the genes that are implicated in many different human cancers. The inferred genetic networks were examined to further identify and characterize a collection of gene interactions that are unique to cancer. The cross-cancer genetic interactions revealed from our study provide another set of knowledge for cancer biologists to propose strong hypotheses, so further biological investigations can be conducted effectively.
| S-EPMC6456846 | biostudies-literature
Project description:Despite major methodological developments, Bayesian inference in Gaussian graphical models remains challenging in high dimension due to the tremendous size of the model space. This article proposes a method to infer the marginal and conditional independence structures between variables by multiple testing, which bypasses the exploration of the model space. Specifically, we introduce closed-form Bayes factors under the Gaussian conjugate model to evaluate the null hypotheses of marginal and conditional independence between variables. Their computation for all pairs of variables is shown to be extremely efficient, thereby allowing us to address large problems with thousands of nodes as required by modern applications. Moreover, we derive exact tail probabilities from the null distributions of the Bayes factors. These allow the use of any multiplicity correction procedure to control error rates for incorrect edge inclusion. We demonstrate the proposed approach on various simulated examples as well as on a large gene expression data set from The Cancer Genome Atlas.
| S-EPMC6916355 | biostudies-literature
Project description:MotivationGaussian graphical models (GGMs) are network representations of random variables (as nodes) and their partial correlations (as edges). GGMs overcome the challenges of high-dimensional data analysis by using shrinkage methodologies. Therefore, they have become useful to reconstruct gene regulatory networks from gene-expression profiles. However, it is often ignored that the partial correlations are 'shrunk' and that they cannot be compared/assessed directly. Therefore, accurate (differential) network analyses need to account for the number of variables, the sample size, and also the shrinkage value, otherwise, the analysis and its biological interpretation would turn biased. To date, there are no appropriate methods to account for these factors and address these issues.ResultsWe derive the statistical properties of the partial correlation obtained with the Ledoit-Wolf shrinkage. Our result provides a toolbox for (differential) network analyses as (i) confidence intervals, (ii) a test for zero partial correlation (null-effects) and (iii) a test to compare partial correlations. Our novel (parametric) methods account for the number of variables, the sample size and the shrinkage values. Additionally, they are computationally fast, simple to implement and require only basic statistical knowledge. Our simulations show that the novel tests perform better than DiffNetFDR-a recently published alternative-in terms of the trade-off between true and false positives. The methods are demonstrated on synthetic data and two gene-expression datasets from Escherichia coli and Mus musculus.Availability and implementationThe R package with the methods and the R script with the analysis are available in https://github.com/V-Bernal/GeneNetTools.Supplementary informationSupplementary data are available at Bioinformatics online.
| S-EPMC9665865 | biostudies-literature
Project description:MOTIVATION:One of the main goals in systems biology is to learn molecular regulatory networks from quantitative profile data. In particular, Gaussian graphical models (GGMs) are widely used network models in bioinformatics where variables (e.g. transcripts, metabolites or proteins) are represented by nodes, and pairs of nodes are connected with an edge according to their partial correlation. Reconstructing a GGM from data is a challenging task when the sample size is smaller than the number of variables. The main problem consists in finding the inverse of the covariance estimator which is ill-conditioned in this case. Shrinkage-based covariance estimators are a popular approach, producing an invertible 'shrunk' covariance. However, a proper significance test for the 'shrunk' partial correlation (i.e. the GGM edges) is an open challenge as a probability density including the shrinkage is unknown. In this article, we present (i) a geometric reformulation of the shrinkage-based GGM, and (ii) a probability density that naturally includes the shrinkage parameter. RESULTS:Our results show that the inference using this new 'shrunk' probability density is as accurate as Monte Carlo estimation (an unbiased non-parametric method) for any shrinkage value, while being computationally more efficient. We show on synthetic data how the novel test for significance allows an accurate control of the Type I error and outperforms the network reconstruction obtained by the widely used R package GeneNet. This is further highlighted in two gene expression datasets from stress response in Eschericha coli, and the effect of influenza infection in Mus musculus. AVAILABILITY AND IMPLEMENTATION:https://github.com/V-Bernal/GGM-Shrinkage. SUPPLEMENTARY INFORMATION:Supplementary data are available at Bioinformatics online.
| S-EPMC6901079 | biostudies-literature
Project description:Bayesian graphical models are a useful tool for understanding dependence relationships among many variables, particularly in situations with external prior information. In high-dimensional settings, the space of possible graphs becomes enormous, rendering even state-of-the-art Bayesian stochastic search computationally infeasible. We propose a deterministic alternative to estimate Gaussian and Gaussian copula graphical models using an Expectation Conditional Maximization (ECM) algorithm, extending the EM approach from Bayesian variable selection to graphical model estimation. We show that the ECM approach enables fast posterior exploration under a sequence of mixture priors, and can incorporate multiple sources of information.
| S-EPMC7540244 | biostudies-literature
Project description:BackgroundIn systems biology, it is important to reconstruct regulatory networks from quantitative molecular profiles. Gaussian graphical models (GGMs) are one of the most popular methods to this end. A GGM consists of nodes (representing the transcripts, metabolites or proteins) inter-connected by edges (reflecting their partial correlations). Learning the edges from quantitative molecular profiles is statistically challenging, as there are usually fewer samples than nodes ('high dimensional problem'). Shrinkage methods address this issue by learning a regularized GGM. However, it remains open to study how the shrinkage affects the final result and its interpretation.ResultsWe show that the shrinkage biases the partial correlation in a non-linear way. This bias does not only change the magnitudes of the partial correlations but also affects their order. Furthermore, it makes networks obtained from different experiments incomparable and hinders their biological interpretation. We propose a method, referred to as 'un-shrinking' the partial correlation, which corrects for this non-linear bias. Unlike traditional methods, which use a fixed shrinkage value, the new approach provides partial correlations that are closer to the actual (population) values and that are easier to interpret. This is demonstrated on two gene expression datasets from Escherichia coli and Mus musculus.ConclusionsGGMs are popular undirected graphical models based on partial correlations. The application of GGMs to reconstruct regulatory networks is commonly performed using shrinkage to overcome the 'high-dimensional problem'. Besides it advantages, we have identified that the shrinkage introduces a non-linear bias in the partial correlations. Ignoring this type of effects caused by the shrinkage can obscure the interpretation of the network, and impede the validation of earlier reported results.
| S-EPMC8424921 | biostudies-literature
Project description:For multivariate spatial Gaussian process (GP) models, customary specifications of cross-covariance functions do not exploit relational inter-variable graphs to ensure process-level conditional independence among the variables. This is undesirable, especially for highly multivariate settings, where popular cross-covariance functions such as the multivariate Matérn suffer from a "curse of dimensionality" as the number of parameters and floating point operations scale up in quadratic and cubic order, respectively, in the number of variables. We propose a class of multivariate "Graphical Gaussian Processes" using a general construction called "stitching" that crafts cross-covariance functions from graphs and ensures process-level conditional independence among variables. For the Matérn family of functions, stitching yields a multivariate GP whose univariate components are Matérn GPs, and conforms to process-level conditional independence as specified by the graphical model. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate the utility of the graphical Matérn GP to jointly model highly multivariate spatial data using simulation examples and an application to air-pollution modelling.
| S-EPMC9838617 | biostudies-literature