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.     

A simple implementation of a normal mixture approach to differential gene expression in multiclass microarrays

MOTIVATION: An important problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. We provide a straightforward and easily implemented method for estimating the posterior probability that an individual gene is null. The problem can be expressed in a two-component mixture framework, using an empirical Bayes approach. Current methods of implementing this approach either have some limitations due to the minimal assumptions made or with more specific assumptions are computationally intensive. RESULTS: By converting to a z-score the value of the test statistic used to test the significance of each gene, we propose a simple two-component normal mixture that models adequately the distribution of this score. The usefulness of our approach is demonstrated on three real datasets.     

A mixture model for estimating the local false discovery rate in DNA microarray analysis

MOTIVATION: Statistical methods based on controlling the false discovery rate (FDR) or positive false discovery rate (pFDR) are now well established in identifying differentially expressed genes in DNA microarray. Several authors have recently raised the important issue that FDR or pFDR may give misleading inference when specific genes are of interest because they average the genes under consideration with genes that show stronger evidence for differential expression. The paper proposes a flexible and robust mixture model for estimating the local FDR which quantifies how plausible each specific gene expresses differentially. RESULTS: We develop a special mixture model tailored to multiple testing by requiring the P-value distribution for the differentially expressed genes to be stochastically smaller than the P-value distribution for the non-differentially expressed genes. A smoothing mechanism is built in. The proposed model gives robust estimation of local FDR for any reasonable underlying P-value distributions. It also provides a single framework for estimating the proportion of differentially expressed genes, pFDR, negative predictive values, sensitivity and specificity. A cervical cancer study shows that the local FDR gives more specific and relevant quantification of the evidence for differential expression that can be substantially different from pFDR. AVAILABILITY: An R function implementing the proposed model is available at http://www.geocities.com/jg_liao/software     

Identifying differentially expressed genes in meta-analysis via Bayesian model-based clustering

A Bayesian model-based clustering approach is proposed for identifying differentially expressed genes in meta-analysis. A Bayesian hierarchical model is used as a scientific tool for combining information from different studies, and a mixture prior is used to separate differentially expressed genes from non-differentially expressed genes. Posterior estimation of the parameters and missing observations are done by using a simple Markov chain Monte Carlo method. From the estimated mixture model, useful measure of significance of a test such as the Bayesian false discovery rate (FDR), the local FDR (Efron et al., 2001), and the integration-driven discovery rate (IDR; Choi et al., 2003) can be easily computed. The model-based approach is also compared with commonly used permutation methods, and it is shown that the model-based approach is superior to the permutation methods when there are excessive under-expressed genes compared to over-expressed genes or vice versa. The proposed method is applied to four publicly available prostate cancer gene expression data sets and simulated data sets.     

A global approach to identify differentially expressed genes in cDNA (two-color) microarray experiments

MOTIVATION: Currently most of the methods for identifying differentially expressed genes fall into the category of so called single-gene-analysis, performing hypothesis testing on a gene-by-gene basis. In a single-gene-analysis approach, estimating the variability of each gene is required to determine whether a gene is differentially expressed or not. Poor accuracy of variability estimation makes it difficult to identify genes with small fold-changes unless a very large number of replicate experiments are performed. RESULTS: We propose a method that can avoid the difficult task of estimating variability for each gene, while reliably identifying a group of differentially expressed genes with low false discovery rates, even when the fold-changes are very small. In this article, a new characterization of differentially expressed genes is established based on a theorem about the distribution of ranks of genes sorted by (log) ratios within each array. This characterization of differentially expressed genes based on rank is an example of all-gene-analysis instead of single gene analysis. We apply the method to a cDNA microarray dataset and many low fold-changed genes (as low as 1.3 fold-changes) are reliably identified without carrying out hypothesis testing on a gene-by-gene basis. The false discovery rate is estimated in two different ways reflecting the variability from all the genes without the complications related to multiple hypothesis testing. We also provide some comparisons between our approach and single-gene-analysis based methods. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Identifying differentially expressed genes using false discovery rate controlling procedures

MOTIVATION: DNA microarrays have recently been used for the purpose of monitoring expression levels of thousands of genes simultaneously and identifying those genes that are differentially expressed. The probability that a false identification (type I error) is committed can increase sharply when the number of tested genes gets large. Correlation between the test statistics attributed to gene co-regulation and dependency in the measurement errors of the gene expression levels further complicates the problem. In this paper we address this very large multiplicity problem by adopting the false discovery rate (FDR) controlling approach. In order to address the dependency problem, we present three resampling-based FDR controlling procedures, that account for the test statistics distribution, and compare their performance to that of the na?ve application of the linear step-up procedure in Benjamini and Hochberg (1995). The procedures are studied using simulated microarray data, and their performance is examined relative to their ease of implementation. RESULTS: Comparative simulation analysis shows that all four FDR controlling procedures control the FDR at the desired level, and retain substantially more power then the family-wise error rate controlling procedures. In terms of power, using resampling of the marginal distribution of each test statistics substantially improves the performance over the na?ve one. The highest power is achieved, at the expense of a more sophisticated algorithm, by the resampling-based procedures that resample the joint distribution of the test statistics and estimate the level of FDR control. AVAILABILITY: An R program that adjusts p-values using FDR controlling procedures is freely available over the Internet at www.math.tau.ac.il/~ybenja.     

Identification of differentially expressed genes and false discovery rate in microarray studies

PURPOSE OF REVIEW: To highlight the development in microarray data analysis for the identification of differentially expressed genes, particularly via control of false discovery rate. RECENT FINDINGS: The emergence of high-throughput technology such as microarrays raises two fundamental statistical issues: multiplicity and sensitivity. We focus on the biological problem of identifying differentially expressed genes. First, multiplicity arises due to testing tens of thousands of hypotheses, rendering the standard P value meaningless. Second, known optimal single-test procedures such as the t-test perform poorly in the context of highly multiple tests. The standard approach of dealing with multiplicity is too conservative in the microarray context. The false discovery rate concept is fast becoming the key statistical assessment tool replacing the P value. We review the false discovery rate approach and argue that it is more sensible for microarray data. We also discuss some methods to take into account additional information from the microarrays to improve the false discovery rate. SUMMARY: There is growing consensus on how to analyse microarray data using the false discovery rate framework in place of the classical P value. Further research is needed on the preprocessing of the raw data, such as the normalization step and filtering, and on finding the most sensitive test procedure.     

Mixture models for gene expression experiments with two species

Cross-species research in drug development is novel and challenging. A bivariate mixture model utilizing information across two species was proposed to solve the fundamental problem of identifying differentially expressed genes in microarray experiments in order to potentially improve the understanding of translation between preclinical and clinical studies for drug development. The proposed approach models the joint distribution of treatment effects estimated from independent linear models. The mixture model posits up to nine components, four of which include groups in which genes are differentially expressed in both species. A comprehensive simulation to evaluate the model performance and one application on a real world data set, a mouse and human type II diabetes experiment, suggest that the proposed model, though highly structured, can handle various configurations of differential gene expression and is practically useful on identifying differentially expressed genes, especially when the magnitude of differential expression due to different treatment intervention is weak. In the mouse and human application, the proposed mixture model was able to eliminate unimportant genes and identify a list of genes that were differentially expressed in both species and could be potential gene targets for drug development.     

False discovery rate, sensitivity and sample size for microarray studies

MOTIVATION: In microarray data studies most researchers are keenly aware of the potentially high rate of false positives and the need to control it. One key statistical shift is the move away from the well-known P-value to false discovery rate (FDR). Less discussion perhaps has been spent on the sensitivity or the associated false negative rate (FNR). The purpose of this paper is to explain in simple ways why the shift from P-value to FDR for statistical assessment of microarray data is necessary, to elucidate the determining factors of FDR and, for a two-sample comparative study, to discuss its control via sample size at the design stage. RESULTS: We use a mixture model, involving differentially expressed (DE) and non-DE genes, that captures the most common problem of finding DE genes. Factors determining FDR are (1) the proportion of truly differentially expressed genes, (2) the distribution of the true differences, (3) measurement variability and (4) sample size. Many current small microarray studies are plagued with large FDR, but controlling FDR alone can lead to unacceptably large FNR. In evaluating a design of a microarray study, sensitivity or FNR curves should be computed routinely together with FDR curves. Under certain assumptions, the FDR and FNR curves coincide, thus simplifying the choice of sample size for controlling the FDR and FNR jointly.     

Bayesian classification and non-Bayesian label estimation via EM algorithm to identify differentially expressed genes: a comparative study

Gene classification problem is studied considering the ratio of gene expression levels, X, in two-channel microarrays and a non-observed categorical variable indicating how differentially expressed the gene is: non differentially expressed, down-regulated or up-regulated. Supposing X from a mixture of Gamma distributions, two methods are proposed and results are compared. The first method is based on an hierarchical Bayesian model. The conditional predictive probability of a gene to belong to each group is calculated and the gene is assigned to the group for which this conditional probability is higher. The second method uses EM algorithm to estimate the most likely group label for each gene, that is, to assign the gene to the group which contains it with the higher estimated probability.     

Surrogate variable analysis using partial least squares (SVA-PLS) in gene expression studies

MOTIVATION: In a typical gene expression profiling study, our prime objective is to identify the genes that are differentially expressed between the samples from two different tissue types. Commonly, standard analysis of variance (ANOVA)/regression is implemented to identify the relative effects of these genes over the two types of samples from their respective arrays of expression levels. But, this technique becomes fundamentally flawed when there are unaccounted sources of variability in these arrays (latent variables attributable to different biological, environmental or other factors relevant in the context). These factors distort the true picture of differential gene expression between the two tissue types and introduce spurious signals of expression heterogeneity. As a result, many genes which are actually differentially expressed are not detected, whereas many others are falsely identified as positives. Moreover, these distortions can be different for different genes. Thus, it is also not possible to get rid of these variations by simple array normalizations. This both-way error can lead to a serious loss in sensitivity and specificity, thereby causing a severe inefficiency in the underlying multiple testing problem. In this work, we attempt to identify the hidden effects of the underlying latent factors in a gene expression profiling study by partial least squares (PLS) and apply ANCOVA technique with the PLS-identified signatures of these hidden effects as covariates, in order to identify the genes that are truly differentially expressed between the two concerned tissue types. RESULTS: We compare the performance of our method SVA-PLS with standard ANOVA and a relatively recent technique of surrogate variable analysis (SVA), on a wide variety of simulation settings (incorporating different effects of the hidden variable, under situations with varying signal intensities and gene groupings). In all settings, our method yields the highest sensitivity while maintaining relatively reasonable values for the specificity, false discovery rate and false non-discovery rate. Application of our method to gene expression profiling for acute megakaryoblastic leukemia shows that our method detects an additional six genes, that are missed by both the standard ANOVA method as well as SVA, but may be relevant to this disease, as can be seen from mining the existing literature.     

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.     

Estimating effect sizes of differentially expressed genes for power and sample-size assessments in microarray experiments

Summary In microarray screening for differentially expressed genes using multiple testing, assessment of power or sample size is of particular importance to ensure that few relevant genes are removed from further consideration prematurely. In this assessment, adequate estimation of the effect sizes of differentially expressed genes is crucial because of its substantial impact on power and sample‐size estimates. However, conventional methods using top genes with largest observed effect sizes would be subject to overestimation due to random variation. In this article, we propose a simple estimation method based on hierarchical mixture models with a nonparametric prior distribution to accommodate random variation and possible large diversity of effect sizes across differential genes, separated from nuisance, nondifferential genes. Based on empirical Bayes estimates of effect sizes, the power and false discovery rate (FDR) can be estimated to monitor them simultaneously in gene screening. We also propose a power index that concerns selection of top genes with largest effect sizes, called partial power. This new power index could provide a practical compromise for the difficulty in achieving high levels of usual overall power as confronted in many microarray experiments. Applications to two real datasets from cancer clinical studies are provided.     

How many replicates of arrays are required to detect gene expression changes in microarray experiments? A mixture model approach

Background  

It has been recognized that replicates of arrays (or spots) may be necessary for reliably detecting differentially expressed genes in microarray experiments. However, the often-asked question of how many replicates are required has barely been addressed in the literature. In general, the answer depends on several factors: a given magnitude of expression change, a desired statistical power (that is, probability) to detect it, a specified Type I error rate, and the statistical method being used to detect the change. Here, we discuss how to calculate the number of replicates in the context of applying a nonparametric statistical method, the normal mixture model approach, to detect changes in gene expression.     

Bayesian hierarchical model for identifying changes in gene expression from microarray experiments.

Recent developments in microarrays technology enable researchers to study simultaneously the expression of thousands of genes from one cell line or tissue sample. This new technology is often used to assess changes in mRNA expression upon a specified transfection for a cell line in order to identify target genes. For such experiments, the range of differential expression is moderate, and teasing out the modified genes is challenging and calls for detailed modeling. The aim of this paper is to propose a methodological framework for studies that investigate differential gene expression through microarrays technology that is based on a fully Bayesian mixture approach (Richardson and Green, 1997). A case study that investigated those genes that were differentially expressed in two cell lines (normal and modified by a gene transfection) is provided to illustrate the performance and usefulness of this approach.     

Identification of differentially expressed gene categories in microarray studies using nonparametric multivariate analysis

MOTIVATION: The field of microarray data analysis is shifting emphasis from methods for identifying differentially expressed genes to methods for identifying differentially expressed gene categories. The latter approaches utilize a priori information about genes to group genes into categories and enhance the interpretation of experiments aimed at identifying expression differences across treatments. While almost all of the existing approaches for identifying differentially expressed gene categories are practically useful, they suffer from a variety of drawbacks. Perhaps most notably, many popular tools are based exclusively on gene-specific statistics that cannot detect many types of multivariate expression change. RESULTS: We have developed a nonparametric multivariate method for identifying gene categories whose multivariate expression distribution differs across two or more conditions. We illustrate our approach and compare its performance to several existing procedures via the analysis of a real data set and a unique data-based simulation study designed to capture the challenges and complexities of practical data analysis. We show that our method has good power for differentiating between differentially expressed and non-differentially expressed gene categories, and we utilize a resampling based strategy for controlling the false discovery rate when testing multiple categories. AVAILABILITY: R code (www.r-project.org) for implementing our approach is available from the first author by request.     

Power and sample size estimation in microarray studies

Background  

Before conducting a microarray experiment, one important issue that needs to be determined is the number of arrays required in order to have adequate power to identify differentially expressed genes. This paper discusses some crucial issues in the problem formulation, parameter specifications, and approaches that are commonly proposed for sample size estimation in microarray experiments. Common methods for sample size estimation are formulated as the minimum sample size necessary to achieve a specified sensitivity (proportion of detected truly differentially expressed genes) on average at a specified false discovery rate (FDR) level and specified expected proportion (π ₁) of the true differentially expression genes in the array. Unfortunately, the probability of detecting the specified sensitivity in such a formulation can be low. We formulate the sample size problem as the number of arrays needed to achieve a specified sensitivity with 95% probability at the specified significance level. A permutation method using a small pilot dataset to estimate sample size is proposed. This method accounts for correlation and effect size heterogeneity among genes.     

An empirical bayes adjustment to increase the sensitivity of detecting differentially expressed genes in microarray experiments

MOTIVATION: Detection of differentially expressed genes is one of the major goals of microarray experiments. Pairwise comparison for each gene is not appropriate without controlling the overall (experimentwise) type 1 error rate. Dudoit et al. have advocated use of permutation-based step-down P-value adjustments to correct the observed significance levels for the individual (i.e. for each gene) two sample t-tests. RESULTS: In this paper, we consider an ANOVA formulation of the gene expression levels corresponding to multiple tissue types. We provide resampling-based step-down adjustments to correct the observed significance levels for the individual ANOVA t-tests for each gene and for each pair of tissue type comparisons. More importantly, we introduce a novel empirical Bayes adjustment to the t-test statistics that can be incorporated into the step-down procedure. Using simulated data, we show that the empirical Bayes adjustment improved the sensitivity of detecting differentially expressed genes up to 16%, while maintaining a high level of specificity. This adjustment also reduces the false non-discovery rate to some degree at the cost of a modest increase in the false discovery rate. We illustrate our approach using a human colon cancer dataset consisting of oligonucleotide arrays of normal, adenoma and carcinoma cells. The number of genes with differential expression level declared statistically significant was about 50 when comparing normal to adenoma cells and about five when comparing adenoma to carcinoma cells. This list includes genes previously known to be associated with colon cancer as well as some novel genes. AVAILABILITY: R code for the empirical Bayes adjustment and step-down P-value calculation via resampling are available from the supplementary web-site. Supplementary information: http://www.mathstat.gsu.edu/~matsnd/EB/supp.htm