Establishing Informative Prior for Gene Expression Variance from Public Databases

Identifying differential expressed genes across various conditions or genotypes is the most typical approach to studying the regulation of gene expression. An estimate of gene-specific variance is often needed for the assessment of statistical significance in most differential expression (DE) detection methods, including linear models (e.g., for transformed and normalized microarray data) and generalized linear models (e.g., for count data in RNAseq). Due to a common limit in sample size, the variance estimate is often unstable in small experiments. Shrinkage estimates using empirical Bayes methods have proven useful in improving the variance estimate, hence improving the detection of DE. The most widely used empirical Bayes methods borrow information across genes within the same experiments. In these methods, genes are considered exchangeable or exchangeable conditioning on expression level. We propose, with the increasing accumulation of expression data, borrowing information from historical data on the same gene can provide better estimate of gene-specific variance, thus further improve DE detection. Specifically, we show that the variation of gene expression is truly gene-specific and reproducible between different experiments. We present a new method to establish informative gene-specific prior on the variance of expression using existing public data, and illustrate how to shrink the variance estimate and detect DE. We demonstrate improvement in DE detection under our strategy compared to leading DE detection methods.     

Borrowing information across genes and experiments for improved error variance estimation in microarray data analysis

Statistical inference for microarray experiments usually involves the estimation of error variance for each gene. Because the sample size available for each gene is often low, the usual unbiased estimator of the error variance can be unreliable. Shrinkage methods, including empirical Bayes approaches that borrow information across genes to produce more stable estimates, have been developed in recent years. Because the same microarray platform is often used for at least several experiments to study similar biological systems, there is an opportunity to improve variance estimation further by borrowing information not only across genes but also across experiments. We propose a lognormal model for error variances that involves random gene effects and random experiment effects. Based on the model, we develop an empirical Bayes estimator of the error variance for each combination of gene and experiment and call this estimator BAGE because information is Borrowed Across Genes and Experiments. A permutation strategy is used to make inference about the differential expression status of each gene. Simulation studies with data generated from different probability models and real microarray data show that our method outperforms existing approaches.     

Statistical analysis of microarray data: a Bayesian approach

The potential of microarray data is enormous. It allows us to monitor the expression of thousands of genes simultaneously. A common task with microarray is to determine which genes are differentially expressed between two samples obtained under two different conditions. Recently, several statistical methods have been proposed to perform such a task when there are replicate samples under each condition. Two major problems arise with microarray data. The first one is that the number of replicates is very small (usually 2-10), leading to noisy point estimates. As a consequence, traditional statistics that are based on the means and standard deviations, e.g. t-statistic, are not suitable. The second problem is that the number of genes is usually very large (approximately 10,000), and one is faced with an extreme multiple testing problem. Most multiple testing adjustments are relatively conservative, especially when the number of replicates is small. In this paper we present an empirical Bayes analysis that handles both problems very well. Using different parametrizations, we develop four statistics that can be used to test hypotheses about the means and/or variances of the gene expression levels in both one- and two-sample problems. The methods are illustrated using experimental data with prior knowledge. In addition, we present the result of a simulation comparing our methods to well-known statistics and multiple testing adjustments.     

Identifying sites under positive selection with uncertain parameter estimates.

Codon-based substitution models are routinely used to measure selective pressures acting on protein-coding genes. To this effect, the nonsynonymous to synonymous rate ratio (dN/dS = omega) is estimated. The proportion of amino-acid sites potentially under positive selection, as indicated by omega > 1, is inferred by fitting a probability distribution where some sites are permitted to have omega > 1. These sites are then inferred by means of an empirical Bayes or by a Bayes empirical Bayes approach that, respectively, ignores or accounts for sampling errors in maximum-likelihood estimates of the distribution used to infer the proportion of sites with omega > 1. Here, we extend a previous full-Bayes approach to include models with high power and low false-positive rates when inferring sites under positive selection. We propose some heuristics to alleviate the computational burden, and show that (i) full Bayes can be superior to empirical Bayes when analyzing a small data set or small simulated data, (ii) full Bayes has only a small advantage over Bayes empirical Bayes with our small test data, and (iii) Bayesian methods appear relatively insensitive to mild misspecifications of the random process generating adaptive evolution in our simulations, but in practice can prove extremely sensitive to model specification. We suggest that the codon model used to detect amino acids under selection should be carefully selected, for instance using Akaike information criterion (AIC).     

Empirical Bayes analysis of single nucleotide polymorphisms

Background  

An important goal of whole-genome studies concerned with single nucleotide polymorphisms (SNPs) is the identification of SNPs associated with a covariate of interest such as the case-control status or the type of cancer. Since these studies often comprise the genotypes of hundreds of thousands of SNPs, methods are required that can cope with the corresponding multiple testing problem. For the analysis of gene expression data, approaches such as the empirical Bayes analysis of microarrays have been developed particularly for the detection of genes associated with the response. However, the empirical Bayes analysis of microarrays has only been suggested for binary responses when considering expression values, i.e. continuous predictors.     

Empirical Bayes estimation of gene-specific effects in micro-array research

Micro-array technology allows investigators the opportunity to measure expression levels of thousands of genes simultaneously. However, investigators are also faced with the challenge of simultaneous estimation of gene expression differences for thousands of genes with very small sample sizes. Traditional estimators of differences between treatment means (ordinary least squares estimators or OLS) are not the best estimators if interest is in estimation of gene expression differences for an ensemble of genes. In the case that gene expression differences are regarded as exchangeable samples from a common population, estimators are available that result in much smaller average mean-square error across the population of gene expression difference estimates. We have simulated the application of such an estimator, namely an empirical Bayes (EB) estimator of random effects in a hierarchical linear model (normal-normal). Simulation results revealed mean-square error as low as 0.05 times the mean-square error of OLS estimators (i.e., the difference between treatment means). We applied the analysis to an example dataset as a demonstration of the shrinkage of EB estimators and of the reduction in mean-square error, i.e., increase in precision, associated with EB estimators in this analysis. The method described here is available in software that is available at .     

Use of within-array replicate spots for assessing differential expression in microarray experiments

MOTIVATION: Spotted arrays are often printed with probes in duplicate or triplicate, but current methods for assessing differential expression are not able to make full use of the resulting information. The usual practice is to average the duplicate or triplicate results for each probe before assessing differential expression. This results in the loss of valuable information about genewise variability. RESULTS: A method is proposed for extracting more information from within-array replicate spots in microarray experiments by estimating the strength of the correlation between them. The method involves fitting separate linear models to the expression data for each gene but with a common value for the between-replicate correlation. The method greatly improves the precision with which the genewise variances are estimated and thereby improves inference methods designed to identify differentially expressed genes. The method may be combined with empirical Bayes methods for moderating the genewise variances between genes. The method is validated using data from a microarray experiment involving calibration and ratio control spots in conjunction with spiked-in RNA. Comparing results for calibration and ratio control spots shows that the common correlation method results in substantially better discrimination of differentially expressed genes from those which are not. The spike-in experiment also confirms that the results may be further improved by empirical Bayes smoothing of the variances when the sample size is small. AVAILABILITY: The methodology is implemented in the limma software package for R, available from the CRAN repository http://www.r-project.org     

tigaR: integrative significance analysis of temporal differential gene expression induced by genomic abnormalities

Background

To determine which changes in the host cell genome are crucial for cervical carcinogenesis, a longitudinal in vitro model system of HPV-transformed keratinocytes was profiled in a genome-wide manner. Four cell lines affected with either HPV16 or HPV18 were assayed at 8 sequential time points for gene expression (mRNA) and gene copy number (DNA) using high-resolution microarrays. Available methods for temporal differential expression analysis are not designed for integrative genomic studies.

Results

Here, we present a method that allows for the identification of differential gene expression associated with DNA copy number changes over time. The temporal variation in gene expression is described by a generalized linear mixed model employing low-rank thin-plate splines. Model parameters are estimated with an empirical Bayes procedure, which exploits integrated nested Laplace approximation for fast computation. Iteratively, posteriors of hyperparameters and model parameters are estimated. The empirical Bayes procedure shrinks multiple dispersion-related parameters. Shrinkage leads to more stable estimates of the model parameters, better control of false positives and improvement of reproducibility. In addition, to make estimates of the DNA copy number more stable, model parameters are also estimated in a multivariate way using triplets of features, imposing a spatial prior for the copy number effect.

Conclusion

With the proposed method for analysis of time-course multilevel molecular data, more profound insight may be gained through the identification of temporal differential expression induced by DNA copy number abnormalities. In particular, in the analysis of an integrative oncogenomics study with a time-course set-up our method finds genes previously reported to be involved in cervical carcinogenesis. Furthermore, the proposed method yields improvements in sensitivity, specificity and reproducibility compared to existing methods. Finally, the proposed method is able to handle count (RNAseq) data from time course experiments as is shown on a real data set.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-327) contains supplementary material, which is available to authorized users.     

Intensity-based hierarchical Bayes method improves testing for differentially expressed genes in microarray experiments

Background  

The small sample sizes often used for microarray experiments result in poor estimates of variance if each gene is considered independently. Yet accurately estimating variability of gene expression measurements in microarray experiments is essential for correctly identifying differentially expressed genes. Several recently developed methods for testing differential expression of genes utilize hierarchical Bayesian models to "pool" information from multiple genes. We have developed a statistical testing procedure that further improves upon current methods by incorporating the well-documented relationship between the absolute gene expression level and the variance of gene expression measurements into the general empirical Bayes framework.     

Differential expression analysis of multifactor RNA-Seq experiments with respect to biological variation

A flexible statistical framework is developed for the analysis of read counts from RNA-Seq gene expression studies. It provides the ability to analyse complex experiments involving multiple treatment conditions and blocking variables while still taking full account of biological variation. Biological variation between RNA samples is estimated separately from the technical variation associated with sequencing technologies. Novel empirical Bayes methods allow each gene to have its own specific variability, even when there are relatively few biological replicates from which to estimate such variability. The pipeline is implemented in the edgeR package of the Bioconductor project. A case study analysis of carcinoma data demonstrates the ability of generalized linear model methods (GLMs) to detect differential expression in a paired design, and even to detect tumour-specific expression changes. The case study demonstrates the need to allow for gene-specific variability, rather than assuming a common dispersion across genes or a fixed relationship between abundance and variability. Genewise dispersions de-prioritize genes with inconsistent results and allow the main analysis to focus on changes that are consistent between biological replicates. Parallel computational approaches are developed to make non-linear model fitting faster and more reliable, making the application of GLMs to genomic data more convenient and practical. Simulations demonstrate the ability of adjusted profile likelihood estimators to return accurate estimators of biological variability in complex situations. When variation is gene-specific, empirical Bayes estimators provide an advantageous compromise between the extremes of assuming common dispersion or separate genewise dispersion. The methods developed here can also be applied to count data arising from DNA-Seq applications, including ChIP-Seq for epigenetic marks and DNA methylation analyses.     

Empirical comparison of tests for differential expression on time-series microarray experiments

Methods for identifying differentially expressed genes were compared on time-series microarray data simulated from artificial gene networks. Select methods were further analyzed on existing immune response data of Boldrick et al. (2002, Proc. Natl. Acad. Sci. USA 99, 972-977). Based on the simulations, we recommend the ANOVA variants of Cui and Churchill. Efron and Tibshirani's empirical Bayes Wilcoxon rank sum test is recommended when the background cannot be effectively corrected. Our proposed GSVD-based differential expression method was shown to detect subtle changes. ANOVA combined with GSVD was consistent on background-normalized simulation data. GSVD with empirical Bayes was consistent without background correction. Based on the Boldrick et al. data, ANOVA is best suited to detect changes in temporal data, while GSVD and empirical Bayes effectively detect individual spikes or overall shifts, respectively. For methods tested on simulation data, lowess after background correction improved results. On simulation data without background correction, lowess decreased performance compared to median centering.     

A novel means of using gene clusters in a two-step empirical Bayes method for predicting classes of samples

MOTIVATION: The classification of samples using gene expression profiles is an important application in areas such as cancer research and environmental health studies. However, the classification is usually based on a small number of samples, and each sample is a long vector of thousands of gene expression levels. An important issue in parametric modeling for so many gene expression levels is the control of the number of nuisance parameters in the model. Large models often lead to intensive or even intractable computation, while small models may be inadequate for complex data.Methodology: We propose a two-step empirical Bayes classification method as a solution to this issue. At the first step, we use the model-based cluster algorithm with a non-traditional purpose of assigning gene expression levels to form abundance groups. At the second step, by assuming the same variance for all the genes in the same group, we substantially reduce the number of nuisance parameters in our statistical model. RESULTS: The proposed model is more parsimonious, which leads to efficient computation under an empirical Bayes estimation procedure. We consider two real examples and simulate data using our method. Desired low classification error rates are obtained even when a large number of genes are pre-selected for class prediction.     

An empirical Bayesian approach for identifying differential coexpression in high-throughput experiments

A common goal of microarray and related high-throughput genomic experiments is to identify genes that vary across biological condition. Most often this is accomplished by identifying genes with changes in mean expression level, so called differentially expressed (DE) genes, and a number of effective methods for identifying DE genes have been developed. Although useful, these approaches do not accommodate other types of differential regulation. An important example concerns differential coexpression (DC). Investigations of this class of genes are hampered by the large cardinality of the space to be interrogated as well as by influential outliers. As a result, existing DC approaches are often underpowered, exceedingly prone to false discoveries, and/or computationally intractable for even a moderately large number of pairs. To address this, an empirical Bayesian approach for identifying DC gene pairs is developed. The approach provides a false discovery rate controlled list of significant DC gene pairs without sacrificing power. It is applicable within a single study as well as across multiple studies. Computations are greatly facilitated by a modification to the expectation-maximization algorithm and a procedural heuristic. Simulations suggest that the proposed approach outperforms existing methods in far less computational time; and case study results suggest that the approach will likely prove to be a useful complement to current DE methods in high-throughput genomic studies.     

Improved covariance matrix estimators for weighted analysis of microarray data.

Empirical Bayes models have been shown to be powerful tools for identifying differentially expressed genes from gene expression microarray data. An example is the WAME model, where a global covariance matrix accounts for array-to-array correlations as well as differing variances between arrays. However, the existing method for estimating the covariance matrix is very computationally intensive and the estimator is biased when data contains many regulated genes. In this paper, two new methods for estimating the covariance matrix are proposed. The first method is a direct application of the EM algorithm for fitting the multivariate t-distribution of the WAME model. In the second method, a prior distribution for the log fold-change is added to the WAME model, and a discrete approximation is used for this prior. Both methods are evaluated using simulated and real data. The first method shows equal performance compared to the existing method in terms of bias and variability, but is superior in terms of computer time. For large data sets (>15 arrays), the second method also shows superior computer run time. Moreover, for simulated data with regulated genes the second method greatly reduces the bias. With the proposed methods it is possible to apply the WAME model to large data sets with reasonable computer run times. The second method shows a small bias for simulated data, but appears to have a larger bias for real data with many regulated genes.     

Accounting for multiplicity in the evaluation of "signals" obtained by data mining from spontaneous report adverse event databases

Surveillance of drug products in the marketplace continues after approval, to identify rare potential toxicities that are unlikely to have been observed in the clinical trials carried out before approval. This surveillance accumulates large numbers of spontaneous reports of adverse events along with other information in spontaneous report databases. Recently developed empirical Bayes and Bayes methods provide a way to summarize the data in these databases, including a quantitative measure of the strength of the reporting association between the drugs and the events. Determining which of the particular drug-event associations, of which there may be many tens of thousands, are real reporting associations and which random noise presents a substantial problem of multiplicity because the resources available for medical and epidemiologic followup are limited. The issues are similar to those encountered with the evaluation of microarrays, but there are important differences. This report compares the application of a standard empirical Bayes approach with micorarray-inspired methods for controlling the False Discovery Rate, and a new Bayesian method for the resolution of the multiplicity problem to a relatively small database containing about 48,000 reports. The Bayesian approach appears to have attractive diagnostic properties in addition to being easy to interpret and implement computationally.     

Variance Component Estimation for Mixed Model Analysis of cDNA Microarray Data

Microarrays provide a valuable tool for the quantification of gene expression. Usually, however, there is a limited number of replicates leading to unsatisfying variance estimates in a gene‐wise mixed model analysis. As thousands of genes are available, it is desirable to combine information across genes. When more than two tissue types or treatments are to be compared it might be advisable to consider the array effect as random. Then information between arrays may be recovered, which can increase accuracy in estimation. We propose a method of variance component estimation across genes for a linear mixed model with two random effects. The method may be extended to models with more than two random effects. We assume that the variance components follow a log‐normal distribution. Assuming that the sums of squares from the gene‐wise analysis, given the true variance components, follow a scaled χ²‐distribution, we adopt an empirical Bayes approach. The variance components are estimated by the expectation of their posterior distribution. The new method is evaluated in a simulation study. Differentially expressed genes are more likely to be detected by tests based on these variance estimates than by tests based on gene‐wise variance estimates. This effect is most visible in studies with small array numbers. Analyzing a real data set on maize endosperm the method is shown to work well. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adjusting batch effects in microarray expression data using empirical Bayes methods

Non-biological experimental variation or "batch effects" are commonly observed across multiple batches of microarray experiments, often rendering the task of combining data from these batches difficult. The ability to combine microarray data sets is advantageous to researchers to increase statistical power to detect biological phenomena from studies where logistical considerations restrict sample size or in studies that require the sequential hybridization of arrays. In general, it is inappropriate to combine data sets without adjusting for batch effects. Methods have been proposed to filter batch effects from data, but these are often complicated and require large batch sizes ( > 25) to implement. Because the majority of microarray studies are conducted using much smaller sample sizes, existing methods are not sufficient. We propose parametric and non-parametric empirical Bayes frameworks for adjusting data for batch effects that is robust to outliers in small sample sizes and performs comparable to existing methods for large samples. We illustrate our methods using two example data sets and show that our methods are justifiable, easy to apply, and useful in practice. Software for our method is freely available at: http://biosun1.harvard.edu/complab/batch/.     

Testing differential expression in nonoverlapping gene pairs: a new perspective for the empirical Bayes method

The currently practiced methods of significance testing in microarray gene expression profiling are highly unstable and tend to be very low in power. These undesirable properties are due to the nature of multiple testing procedures, as well as extremely strong and long-ranged correlations between gene expression levels. In an earlier publication, we identified a special structure in gene expression data that produces a sequence of weakly dependent random variables. This structure, termed the delta-sequence, lies at the heart of a new methodology for selecting differentially expressed genes in nonoverlapping gene pairs. The proposed method has two distinct advantages: (1) it leads to dramatic gains in terms of the mean numbers of true and false discoveries, and in the stability of the results of testing; and (2) its outcomes are entirely free from the log-additive array-specific technical noise. We demonstrate the usefulness of this approach in conjunction with the nonparametric empirical Bayes method. The proposed modification of the empirical Bayes method leads to significant improvements in its performance. The new paradigm arising from the existence of the delta-sequence in biological data offers considerable scope for future developments in this area of methodological research.     

A Bayesian network classification methodology for gene expression data.

We present new techniques for the application of a Bayesian network learning framework to the problem of classifying gene expression data. The focus on classification permits us to develop techniques that address in several ways the complexities of learning Bayesian nets. Our classification model reduces the Bayesian network learning problem to the problem of learning multiple subnetworks, each consisting of a class label node and its set of parent genes. We argue that this classification model is more appropriate for the gene expression domain than are other structurally similar Bayesian network classification models, such as Naive Bayes and Tree Augmented Naive Bayes (TAN), because our model is consistent with prior domain experience suggesting that a relatively small number of genes, taken in different combinations, is required to predict most clinical classes of interest. Within this framework, we consider two different approaches to identifying parent sets which are supported by the gene expression observations and any other currently available evidence. One approach employs a simple greedy algorithm to search the universe of all genes; the second approach develops and applies a gene selection algorithm whose results are incorporated as a prior to enable an exhaustive search for parent sets over a restricted universe of genes. Two other significant contributions are the construction of classifiers from multiple, competing Bayesian network hypotheses and algorithmic methods for normalizing and binning gene expression data in the absence of prior expert knowledge. Our classifiers are developed under a cross validation regimen and then validated on corresponding out-of-sample test sets. The classifiers attain a classification rate in excess of 90% on out-of-sample test sets for two publicly available datasets. We present an extensive compilation of results reported in the literature for other classification methods run against these same two datasets. Our results are comparable to, or better than, any we have found reported for these two sets, when a train-test protocol as stringent as ours is followed.     

Bayesian robust inference for differential gene expression in microarrays with multiple samples

We consider the problem of identifying differentially expressed genes under different conditions using gene expression microarrays. Because of the many steps involved in the experimental process, from hybridization to image analysis, cDNA microarray data often contain outliers. For example, an outlying data value could occur because of scratches or dust on the surface, imperfections in the glass, or imperfections in the array production. We develop a robust Bayesian hierarchical model for testing for differential expression. Errors are modeled explicitly using a t-distribution, which accounts for outliers. The model includes an exchangeable prior for the variances, which allows different variances for the genes but still shrinks extreme empirical variances. Our model can be used for testing for differentially expressed genes among multiple samples, and it can distinguish between the different possible patterns of differential expression when there are three or more samples. Parameter estimation is carried out using a novel version of Markov chain Monte Carlo that is appropriate when the model puts mass on subspaces of the full parameter space. The method is illustrated using two publicly available gene expression data sets. We compare our method to six other baseline and commonly used techniques, namely the t-test, the Bonferroni-adjusted t-test, significance analysis of microarrays (SAM), Efron's empirical Bayes, and EBarrays in both its lognormal-normal and gamma-gamma forms. In an experiment with HIV data, our method performed better than these alternatives, on the basis of between-replicate agreement and disagreement.