Tail posterior probability for inference in pairwise and multiclass gene expression data

We consider the problem of identifying differentially expressed genes in microarray data in a Bayesian framework with a noninformative prior distribution on the parameter quantifying differential expression. We introduce a new rule, tail posterior probability, based on the posterior distribution of the standardized difference, to identify genes differentially expressed between two conditions, and we derive a frequentist estimator of the false discovery rate associated with this rule. We compare it to other Bayesian rules in the considered settings. We show how the tail posterior probability can be extended to testing a compound null hypothesis against a class of specific alternatives in multiclass data.     

Quantitative trait nucleotide analysis using Bayesian model selection

Although much attention has been given to statistical genetic methods for the initial localization and fine mapping of quantitative trait loci (QTLs), little methodological work has been done to date on the problem of statistically identifying the most likely functional polymorphisms using sequence data. In this paper we provide a general statistical genetic framework, called Bayesian quantitative trait nucleotide (BQTN) analysis, for assessing the likely functional status of genetic variants. The approach requires the initial enumeration of all genetic variants in a set of resequenced individuals. These polymorphisms are then typed in a large number of individuals (potentially in families), and marker variation is related to quantitative phenotypic variation using Bayesian model selection and averaging. For each sequence variant a posterior probability of effect is obtained and can be used to prioritize additional molecular functional experiments. An example of this quantitative nucleotide analysis is provided using the GAW12 simulated data. The results show that the BQTN method may be useful for choosing the most likely functional variants within a gene (or set of genes). We also include instructions on how to use our computer program, SOLAR, for association analysis and BQTN analysis.     

Bayesian normalization and identification for differential gene expression data.

Commonly accepted intensity-dependent normalization in spotted microarray studies takes account of measurement errors in the differential expression ratio but ignores measurement errors in the total intensity, although the definitions imply the same measurement error components are involved in both statistics. Furthermore, identification of differentially expressed genes is usually considered separately following normalization, which is statistically problematic. By incorporating the measurement errors in both total intensities and differential expression ratios, we propose a measurement-error model for intensity-dependent normalization and identification of differentially expressed genes. This model is also flexible enough to incorporate intra-array and inter-array effects. A Bayesian framework is proposed for the analysis of the proposed measurement-error model to avoid the potential risk of using the common two-step procedure. We also propose a Bayesian identification of differentially expressed genes to control the false discovery rate instead of the ad hoc thresholding of the posterior odds ratio. The simulation study and an application to real microarray data demonstrate promising results.     

Bayesian Parameter Inference and Model Selection by Population Annealing in Systems Biology

Parameter inference and model selection are very important for mathematical modeling in systems biology. Bayesian statistics can be used to conduct both parameter inference and model selection. Especially, the framework named approximate Bayesian computation is often used for parameter inference and model selection in systems biology. However, Monte Carlo methods needs to be used to compute Bayesian posterior distributions. In addition, the posterior distributions of parameters are sometimes almost uniform or very similar to their prior distributions. In such cases, it is difficult to choose one specific value of parameter with high credibility as the representative value of the distribution. To overcome the problems, we introduced one of the population Monte Carlo algorithms, population annealing. Although population annealing is usually used in statistical mechanics, we showed that population annealing can be used to compute Bayesian posterior distributions in the approximate Bayesian computation framework. To deal with un-identifiability of the representative values of parameters, we proposed to run the simulations with the parameter ensemble sampled from the posterior distribution, named “posterior parameter ensemble”. We showed that population annealing is an efficient and convenient algorithm to generate posterior parameter ensemble. We also showed that the simulations with the posterior parameter ensemble can, not only reproduce the data used for parameter inference, but also capture and predict the data which was not used for parameter inference. Lastly, we introduced the marginal likelihood in the approximate Bayesian computation framework for Bayesian model selection. We showed that population annealing enables us to compute the marginal likelihood in the approximate Bayesian computation framework and conduct model selection depending on the Bayes factor.     

A Bayesian mixture model for metaanalysis of microarray studies

The increased availability of microarray data has been calling for statistical methods to integrate findings across studies. A common goal of microarray analysis is to determine differentially expressed genes between two conditions, such as treatment vs control. A recent Bayesian metaanalysis model used a prior distribution for the mean log-expression ratios that was a mixture of two normal distributions. This model centered the prior distribution of differential expression at zero, and separated genes into two groups only: expressed and nonexpressed. Here, we introduce a Bayesian three-component truncated normal mixture prior model that more flexibly assigns prior distributions to the differentially expressed genes and produces three groups of genes: up and downregulated, and nonexpressed. We found in simulations of two and five studies that the three-component model outperformed the two-component model using three comparison measures. When analyzing biological data of Bacillus subtilis, we found that the three-component model discovered more genes and omitted fewer genes for the same levels of posterior probability of differential expression than the two-component model, and discovered more genes for fixed thresholds of Bayesian false discovery. We assumed that the data sets were produced from the same microarray platform and were prescaled.     

Systems genetics approach reveals candidate genes for parasite resistance from quantitative trait loci studies in agricultural species

A systems genetics approach combining pathway analysis of quantitative trait loci (QTL) and gene expression information has provided strong evidence for common pathways associated with genetic resistance to internal parasites. Gene data, collected from published QTL regions in sheep, cattle, mice, rats and humans, and microarray data from sheep, were converted to human Entrez Gene IDs and compared to the KEGG pathway database. Selection of pathways from QTL data was based on a selection index that ensured that the selected pathways were in all species and the majority of the projects overall and within species. Pathways with either up- and down-regulated genes, primarily up-regulated genes or primarily down-regulated genes, were selected from gene expression data. After comparing the data sets independently, the pathways from each data set were compared and the common set of pathways and genes was identified. Comparisons within data sets identified 21 pathways from QTL data and 66 pathways from gene expression data. Both selected sets were enriched with pathways involved in immune functions, disease and cell responses to signals. The analysis identified 14 pathways that were common between QTL and gene expression data, and four directly associated with IFNγ or MHCII, with 31 common genes, including three MHCII genes. In conclusion, a systems genetics approach combining data from multiple QTL and gene expression projects led to the discovery of common pathways associated with genetic resistance to internal parasites. This systems genetics approach may prove significant for the discovery of candidate genes for many other multifactorial, economically important traits.     

Pathway Analysis of GWAS Provides New Insights into Genetic Susceptibility to 3 Inflammatory Diseases

Although the introduction of genome-wide association studies (GWAS) have greatly increased the number of genes associated with common diseases, only a small proportion of the predicted genetic contribution has so far been elucidated. Studying the cumulative variation of polymorphisms in multiple genes acting in functional pathways may provide a complementary approach to the more common single SNP association approach in understanding genetic determinants of common disease. We developed a novel pathway-based method to assess the combined contribution of multiple genetic variants acting within canonical biological pathways and applied it to data from 14,000 UK individuals with 7 common diseases. We tested inflammatory pathways for association with Crohn''s disease (CD), rheumatoid arthritis (RA) and type 1 diabetes (T1D) with 4 non-inflammatory diseases as controls. Using a variable selection algorithm, we identified variants responsible for the pathway association and evaluated their use for disease prediction using a 10 fold cross-validation framework in order to calculate out-of-sample area under the Receiver Operating Curve (AUC). The generalisability of these predictive models was tested on an independent birth cohort from Northern Finland. Multiple canonical inflammatory pathways showed highly significant associations (p 10⁻³–10⁻²⁰) with CD, T1D and RA. Variable selection identified on average a set of 205 SNPs (149 genes) for T1D, 350 SNPs (189 genes) for RA and 493 SNPs (277 genes) for CD. The pattern of polymorphisms at these SNPS were found to be highly predictive of T1D (91% AUC) and RA (85% AUC), and weakly predictive of CD (60% AUC). The predictive ability of the T1D model (without any parameter refitting) had good predictive ability (79% AUC) in the Finnish cohort. Our analysis suggests that genetic contribution to common inflammatory diseases operates through multiple genes interacting in functional pathways.     

Bayesian network and nonparametric heteroscedastic regression for nonlinear modeling of genetic network

We propose a new statistical method for constructing a genetic network from microarray gene expression data by using a Bayesian network. An essential point of Bayesian network construction is the estimation of the conditional distribution of each random variable. We consider fitting nonparametric regression models with heterogeneous error variances to the microarray gene expression data to capture the nonlinear structures between genes. Selecting the optimal graph, which gives the best representation of the system among genes, is still a problem to be solved. We theoretically derive a new graph selection criterion from Bayes approach in general situations. The proposed method includes previous methods based on Bayesian networks. We demonstrate the effectiveness of the proposed method through the analysis of Saccharomyces cerevisiae gene expression data newly obtained by disrupting 100 genes.     

A scalable method for integration and functional analysis of multiple microarray datasets

MOTIVATION: The diverse microarray datasets that have become available over the past several years represent a rich opportunity and challenge for biological data mining. Many supervised and unsupervised methods have been developed for the analysis of individual microarray datasets. However, integrated analysis of multiple datasets can provide a broader insight into genetic regulation of specific biological pathways under a variety of conditions. RESULTS: To aid in the analysis of such large compendia of microarray experiments, we present Microarray Experiment Functional Integration Technology (MEFIT), a scalable Bayesian framework for predicting functional relationships from integrated microarray datasets. Furthermore, MEFIT predicts these functional relationships within the context of specific biological processes. All results are provided in the context of one or more specific biological functions, which can be provided by a biologist or drawn automatically from catalogs such as the Gene Ontology (GO). Using MEFIT, we integrated 40 Saccharomyces cerevisiae microarray datasets spanning 712 unique conditions. In tests based on 110 biological functions drawn from the GO biological process ontology, MEFIT provided a 5% or greater performance increase for 54 functions, with a 5% or more decrease in performance in only two functions.     

Iterative Bayesian Model Averaging: a method for the application of survival analysis to high-dimensional microarray data

Background  

Microarray technology is increasingly used to identify potential biomarkers for cancer prognostics and diagnostics. Previously, we have developed the iterative Bayesian Model Averaging (BMA) algorithm for use in classification. Here, we extend the iterative BMA algorithm for application to survival analysis on high-dimensional microarray data. The main goal in applying survival analysis to microarray data is to determine a highly predictive model of patients' time to event (such as death, relapse, or metastasis) using a small number of selected genes. Our multivariate procedure combines the effectiveness of multiple contending models by calculating the weighted average of their posterior probability distributions. Our results demonstrate that our iterative BMA algorithm for survival analysis achieves high prediction accuracy while consistently selecting a small and cost-effective number of predictor genes.     

Bayesian model averaging: development of an improved multi-class, gene selection and classification tool for microarray data

MOTIVATION: Selecting a small number of relevant genes for accurate classification of samples is essential for the development of diagnostic tests. We present the Bayesian model averaging (BMA) method for gene selection and classification of microarray data. Typical gene selection and classification procedures ignore model uncertainty and use a single set of relevant genes (model) to predict the class. BMA accounts for the uncertainty about the best set to choose by averaging over multiple models (sets of potentially overlapping relevant genes). RESULTS: We have shown that BMA selects smaller numbers of relevant genes (compared with other methods) and achieves a high prediction accuracy on three microarray datasets. Our BMA algorithm is applicable to microarray datasets with any number of classes, and outputs posterior probabilities for the selected genes and models. Our selected models typically consist of only a few genes. The combination of high accuracy, small numbers of genes and posterior probabilities for the predictions should make BMA a powerful tool for developing diagnostics from expression data. AVAILABILITY: The source codes and datasets used are available from our Supplementary website.     

Effective feature selection framework for cluster analysis of microarray data

The microarray technique has become a standard means in simultaneously examining expression of all genes measured in different circumstances. As microarray data are typically characterized by high dimensional features with a small number of samples, feature selection needs to be incorporated to identify a subset of genes that are meaningful for biological interpretation and accountable for the sample variation. In this article, we present a simple, yet effective feature selection framework suitable for two-dimensional microarray data. Our correlation-based, nonparametric approach allows compact representation of class-specific properties with a small number of genes. We evaluated our method using publicly available experimental data and obtained favorable results.     

Bayesian Network Expansion Identifies New ROS and Biofilm Regulators

Signaling and regulatory pathways that guide gene expression have only been partially defined for most organisms. However, given the increasing number of microarray measurements, it may be possible to reconstruct such pathways and uncover missing connections directly from experimental data. Using a compendium of microarray gene expression data obtained from Escherichia coli, we constructed a series of Bayesian network models for the reactive oxygen species (ROS) pathway as defined by EcoCyc. A consensus Bayesian network model was generated using those networks sharing the top recovered score. This microarray-based network only partially agreed with the known ROS pathway curated from the literature and databases. A top network was then expanded to predict genes that could enhance the Bayesian network model using an algorithm we termed ‘BN+1’. This expansion procedure predicted many stress-related genes (e.g., dusB and uspE), and their possible interactions with other ROS pathway genes. A term enrichment method discovered that biofilm-associated microarray data usually contained high expression levels of both uspE and gadX. The predicted involvement of gene uspE in the ROS pathway and interactions between uspE and gadX were confirmed experimentally using E. coli reporter strains. Genes gadX and uspE showed a feedback relationship in regulating each other''s expression. Both genes were verified to regulate biofilm formation through gene knockout experiments. These data suggest that the BN+1 expansion method can faithfully uncover hidden or unknown genes for a selected pathway with significant biological roles. The presently reported BN+1 expansion method is a generalized approach applicable to the characterization and expansion of other biological pathways and living systems.     

Global gene expression profiling in Escherichia coli K12. The effects of leucine-responsive regulatory protein

Leucine-responsive regulatory protein (Lrp) is a global regulatory protein that affects the expression of multiple genes and operons in bacteria. Although the physiological purpose of Lrp-mediated gene regulation remains unclear, it has been suggested that it functions to coordinate cellular metabolism with the nutritional state of the environment. The results of gene expression profiles between otherwise isogenic lrp(+) and lrp(-) strains of Escherichia coli support this suggestion. The newly discovered Lrp-regulated genes reported here are involved either in small molecule or macromolecule synthesis or degradation, or in small molecule transport and environmental stress responses. Although many of these regulatory effects are direct, others are indirect consequences of Lrp-mediated changes in the expression levels of other global regulatory proteins. Because computational methods to analyze and interpret high dimensional DNA microarray data are still an early stage, much of the emphasis of this work is directed toward the development of methods to identify differentially expressed genes with a high level of confidence. In particular, we describe a Bayesian statistical framework for a posterior estimate of the standard deviation of gene measurements based on a limited number of replications. We also describe an algorithm to compute a posterior estimate of differential expression for each gene based on the experiment-wide global false positive and false negative level for a DNA microarray data set. This allows the experimenter to compute posterior probabilities of differential expression for each individual differential gene expression measurement.     

Quantifying evidence for candidate gene polymorphisms: Bayesian analysis combining sequence-specific and quantitative trait loci colocation information

We calculate posterior probabilities for candidate genes as a function of genomic location. Posterior probabilities for quantitative trait loci (QTL) presence in a small interval are calculated using a Bayesian model-selection approach based on the Bayesian information criterion (BIC) and used to combine QTL colocation information with sequence-specific evidence, e.g., from differential expression and/or association studies. Our method takes into account uncertainty in estimation of number and locations of QTL and estimated map position. Posterior probabilities for QTL presence were calculated for simulated data with n = 100, 300, and 1200 QTL progeny and compared with interval mapping and composite-interval mapping. Candidate genes that mapped to QTL regions had substantially larger posterior probabilities. Among candidates with a given Bayes factor, those that map near a QTL are more promising for further investigation with association studies and functional testing or for use in marker-aided selection. The BIC is shown to correspond very closely to Bayes factors for linear models with a nearly noninformative Zellner prior for the simulated QTL data with n > or = 100. It is shown how to modify the BIC to use a subjective prior for the QTL effects.     

Bayesian meta-analysis models for microarray data: a comparative study

Background

With the growing abundance of microarray data, statistical methods are increasingly needed to integrate results across studies. Two common approaches for meta-analysis of microarrays include either combining gene expression measures across studies or combining summaries such as p-values, probabilities or ranks. Here, we compare two Bayesian meta-analysis models that are analogous to these methods.

Results

Two Bayesian meta-analysis models for microarray data have recently been introduced. The first model combines standardized gene expression measures across studies into an overall mean, accounting for inter-study variability, while the second combines probabilities of differential expression without combining expression values. Both models produce the gene-specific posterior probability of differential expression, which is the basis for inference. Since the standardized expression integration model includes inter-study variability, it may improve accuracy of results versus the probability integration model. However, due to the small number of studies typical in microarray meta-analyses, the variability between studies is challenging to estimate. The probability integration model eliminates the need to model variability between studies, and thus its implementation is more straightforward. We found in simulations of two and five studies that combining probabilities outperformed combining standardized gene expression measures for three comparison values: the percent of true discovered genes in meta-analysis versus individual studies; the percent of true genes omitted in meta-analysis versus separate studies, and the number of true discovered genes for fixed levels of Bayesian false discovery. We identified similar results when pooling two independent studies of Bacillus subtilis. We assumed that each study was produced from the same microarray platform with only two conditions: a treatment and control, and that the data sets were pre-scaled.

Conclusion

The Bayesian meta-analysis model that combines probabilities across studies does not aggregate gene expression measures, thus an inter-study variability parameter is not included in the model. This results in a simpler modeling approach than aggregating expression measures, which accounts for variability across studies. The probability integration model identified more true discovered genes and fewer true omitted genes than combining expression measures, for our data sets.     

Symmetric and asymmetric multi-modality biclustering analysis for microarray data matrix

Machine learning techniques offer a viable approach to cluster discovery from microarray data, which involves identifying and classifying biologically relevant groups in genes and conditions. It has been recognized that genes (whether or not they belong to the same gene group) may be co-expressed via a variety of pathways. Therefore, they can be adequately described by a diversity of coherence models. In fact, it is known that a gene may participate in multiple pathways that may or may not be co-active under all conditions. It is therefore biologically meaningful to simultaneously divide genes into functional groups and conditions into co-active categories--leading to the so-called biclustering analysis. For this, we have proposed a comprehensive set of coherence models to cope with various plausible regulation processes. Furthermore, a multivariate biclustering analysis based on fusion of different coherence models appears to be promising because the expression level of genes from the same group may follow more than one coherence models. The simulation studies further confirm that the proposed framework enjoys the advantage of high prediction performance.     

Gene selection: a Bayesian variable selection approach

Selection of significant genes via expression patterns is an important problem in microarray experiments. Owing to small sample size and the large number of variables (genes), the selection process can be unstable. This paper proposes a hierarchical Bayesian model for gene (variable) selection. We employ latent variables to specialize the model to a regression setting and uses a Bayesian mixture prior to perform the variable selection. We control the size of the model by assigning a prior distribution over the dimension (number of significant genes) of the model. The posterior distributions of the parameters are not in explicit form and we need to use a combination of truncated sampling and Markov Chain Monte Carlo (MCMC) based computation techniques to simulate the parameters from the posteriors. The Bayesian model is flexible enough to identify significant genes as well as to perform future predictions. The method is applied to cancer classification via cDNA microarrays where the genes BRCA1 and BRCA2 are associated with a hereditary disposition to breast cancer, and the method is used to identify a set of significant genes. The method is also applied successfully to the leukemia data. SUPPLEMENTARY INFORMATION: http://stat.tamu.edu/people/faculty/bmallick.html.