FDR control by the BH procedure for two-sided correlated tests with implications to gene expression data analysis

The multiple testing problem attributed to gene expression analysis is challenging not only by its size, but also by possible dependence between the expression levels of different genes resulting from coregulations of the genes. Furthermore, the measurement errors of these expression levels may be dependent as well since they are subjected to several technical factors. Multiple testing of such data faces the challenge of correlated test statistics. In such a case, the control of the False Discovery Rate (FDR) is not straightforward, and thus demands new approaches and solutions that will address multiplicity while accounting for this dependency. This paper investigates the effects of dependency between bormal test statistics on FDR control in two-sided testing, using the linear step-up procedure (BH) of Benjamini and Hochberg (1995). The case of two multiple hypotheses is examined first. A simulation study offers primary insight into the behavior of the FDR subjected to different levels of correlation and distance between null and alternative means. A theoretical analysis follows in order to obtain explicit upper bounds to the FDR. These results are then extended to more than two multiple tests, thereby offering a better perspective on the effect of the proportion of false null hypotheses, as well as the structure of the test statistics correlation matrix. An example from gene expression data analysis is presented.     

Bayesian models based on test statistics for multiple hypothesis testing problems

Motivation: We propose a Bayesian method for the problem ofmultiple hypothesis testing that is routinely encountered inbioinformatics research, such as the differential gene expressionanalysis. Our algorithm is based on modeling the distributionsof test statistics under both null and alternative hypotheses.We substantially reduce the complexity of the process of definingposterior model probabilities by modeling the test statisticsdirectly instead of modeling the full data. Computationally,we apply a Bayesian FDR approach to control the number of rejectionsof null hypotheses. To check if our model assumptions for thetest statistics are valid for various bioinformatics experiments,we also propose a simple graphical model-assessment tool. Results: Using extensive simulations, we demonstrate the performanceof our models and the utility of the model-assessment tool.In the end, we apply the proposed methodology to an siRNA screeningand a gene expression experiment. Contact: yuanji{at}mdanderson.org Supplementary information: Supplementary data are availableat Bioinformatics online. Associate Editor: Chris Stoeckert     

An investigation on performance of Significance Analysis of Microarray (SAM) for the comparisons of several treatments with one control in the presence of small-variance genes

One of multiple testing problems in drug finding experiments is the comparison of several treatments with one control. In this paper we discuss a particular situation of such an experiment, i.e., a microarray setting, where the many-to-one comparisons need to be addressed for thousands of genes simultaneously. For a gene-specific analysis, Dunnett's single step procedure is considered within gene tests, while the FDR controlling procedures such as Significance Analysis of Microarrays (SAM) and Benjamini and Hochberg (BH) False Discovery Rate (FDR) adjustment are applied to control the error rate across genes. The method is applied to a microarray experiment with four treatment groups (three microarrays in each group) and 16,998 genes. Simulation studies are conducted to investigate the performance of the SAM method and the BH-FDR procedure with regard to controlling the FDR, and to investigate the effect of small-variance genes on the FDR in the SAM procedure.     

Statistical Detection of EEG Synchrony Using Empirical Bayesian Inference

There is growing interest in understanding how the brain utilizes synchronized oscillatory activity to integrate information across functionally connected regions. Computing phase-locking values (PLV) between EEG signals is a popular method for quantifying such synchronizations and elucidating their role in cognitive tasks. However, high-dimensionality in PLV data incurs a serious multiple testing problem. Standard multiple testing methods in neuroimaging research (e.g., false discovery rate, FDR) suffer severe loss of power, because they fail to exploit complex dependence structure between hypotheses that vary in spectral, temporal and spatial dimension. Previously, we showed that a hierarchical FDR and optimal discovery procedures could be effectively applied for PLV analysis to provide better power than FDR. In this article, we revisit the multiple comparison problem from a new Empirical Bayes perspective and propose the application of the local FDR method (locFDR; Efron, 2001) for PLV synchrony analysis to compute FDR as a posterior probability that an observed statistic belongs to a null hypothesis. We demonstrate the application of Efron''s Empirical Bayes approach for PLV synchrony analysis for the first time. We use simulations to validate the specificity and sensitivity of locFDR and a real EEG dataset from a visual search study for experimental validation. We also compare locFDR with hierarchical FDR and optimal discovery procedures in both simulation and experimental analyses. Our simulation results showed that the locFDR can effectively control false positives without compromising on the power of PLV synchrony inference. Our results from the application locFDR on experiment data detected more significant discoveries than our previously proposed methods whereas the standard FDR method failed to detect any significant discoveries.     

Towards accurate estimation of the proportion of true null hypotheses in multiple testing

BACKGROUND: Biomedical researchers are now often faced with situations where it is necessary to test a large number of hypotheses simultaneously, eg, in comparative gene expression studies using high-throughput microarray technology. To properly control false positive errors the FDR (false discovery rate) approach has become widely used in multiple testing. The accurate estimation of FDR requires the proportion of true null hypotheses being accurately estimated. To date many methods for estimating this quantity have been proposed. Typically when a new method is introduced, some simulations are carried out to show the improved accuracy of the new method. However, the simulations are often very limited to covering only a few points in the parameter space. RESULTS: Here I have carried out extensive in silico experiments to compare some commonly used methods for estimating the proportion of true null hypotheses. The coverage of these simulations is unprecedented thorough over the parameter space compared to typical simulation studies in the literature. Thus this work enables us to draw conclusions globally as to the performance of these different methods. It was found that a very simple method gives the most accurate estimation in a dominantly large area of the parameter space. Given its simplicity and its overall superior accuracy I recommend its use as the first choice for estimating the proportion of true null hypotheses in multiple testing.     

Estimation of false discovery rates in multiple testing: application to gene microarray data

Testing for significance with gene expression data from DNA microarray experiments involves simultaneous comparisons of hundreds or thousands of genes. If R denotes the number of rejections (declared significant genes) and V denotes the number of false rejections, then V/R, if R > 0, is the proportion of false rejected hypotheses. This paper proposes a model for the distribution of the number of rejections and the conditional distribution of V given R, V / R. Under the independence assumption, the distribution of R is a convolution of two binomials and the distribution of V / R has a noncentral hypergeometric distribution. Under an equicorrelated model, the distributions are more complex and are also derived. Five false discovery rate probability error measures are considered: FDR = E(V/R), pFDR = E(V/R / R > 0) (positive FDR), cFDR = E(V/R / R = r) (conditional FDR), mFDR = E(V)/E(R) (marginal FDR), and eFDR = E(V)/r (empirical FDR). The pFDR, cFDR, and mFDR are shown to be equivalent under the Bayesian framework, in which the number of true null hypotheses is modeled as a random variable. We present a parametric and a bootstrap procedure to estimate the FDRs. Monte Carlo simulations were conducted to evaluate the performance of these two methods. The bootstrap procedure appears to perform reasonably well, even when the alternative hypotheses are correlated (rho = .25). An example from a toxicogenomic microarray experiment is presented for illustration.     

False discovery rate control in two-stage designs

ABSTRACT: BACKGROUND: For gene expression or gene association studies with a large number of hypotheses the number of measurements per marker in a conventional single-stage design is often low due to limited resources. Two-stage designs have been proposed where in a first stage promising hypotheses are identified and further investigated in the second stage with larger sample sizes. For two types of two-stage designs proposed in the literature we derive multiple testing procedures controlling the False Discovery Rate (FDR) demonstrating FDR control by simulations: designs where a fixed number of top-ranked hypotheses are selected and designs where the selection in the interim analysis is based on an FDR threshold. In contrast to earlier approaches which use only the second-stage data in the hypothesis tests (pilot approach), the proposed testing procedures are based on the pooled data from both stages (integrated approach). Results: For both selection rules the multiple testing procedures control the FDR in the considered simulation scenarios. This holds for the case of independent observations across hypotheses as well as for certain correlation structures. Additionally, we show that in scenarios with small effect sizes the testing procedures based on the pooled data from both stages can give a considerable improvement in power compared to tests based on the second-stage data only. Conclusion: The proposed hypothesis tests provide a tool for FDR control for the considered two-stage designs. Comparing the integrated approaches for both selection rules with the corresponding pilot approaches showed an advantage of the integrated approach in many simulation scenarios.     

Screening for partial conjunction hypotheses

SUMMARY: We consider the problem of testing for partial conjunction of hypothesis, which argues that at least u out of n tested hypotheses are false. It offers an in-between approach to the testing of the conjunction of null hypotheses against the alternative that at least one is not, and the testing of the disjunction of null hypotheses against the alternative that all hypotheses are not null. We suggest powerful test statistics for testing such a partial conjunction hypothesis that are valid under dependence between the test statistics as well as under independence. We then address the problem of testing many partial conjunction hypotheses simultaneously using the false discovery rate (FDR) approach. We prove that if the FDR controlling procedure in Benjamini and Hochberg (1995, Journal of the Royal Statistical Society, Series B 57, 289-300) is used for this purpose the FDR is controlled under various dependency structures. Moreover, we can screen at all levels simultaneously in order to display the findings on a superimposed map and still control an appropriate FDR measure. We apply the method to examples from microarray analysis and functional magnetic resonance imaging (fMRI), two application areas where the need for partial conjunction analysis has been identified.     

Evaluating statistical methods using plasmode data sets in the age of massive public databases: an illustration using false discovery rates

Plasmode is a term coined several years ago to describe data sets that are derived from real data but for which some truth is known. Omic techniques, most especially microarray and genomewide association studies, have catalyzed a new zeitgeist of data sharing that is making data and data sets publicly available on an unprecedented scale. Coupling such data resources with a science of plasmode use would allow statistical methodologists to vet proposed techniques empirically (as opposed to only theoretically) and with data that are by definition realistic and representative. We illustrate the technique of empirical statistics by consideration of a common task when analyzing high dimensional data: the simultaneous testing of hundreds or thousands of hypotheses to determine which, if any, show statistical significance warranting follow-on research. The now-common practice of multiple testing in high dimensional experiment (HDE) settings has generated new methods for detecting statistically significant results. Although such methods have heretofore been subject to comparative performance analysis using simulated data, simulating data that realistically reflect data from an actual HDE remains a challenge. We describe a simulation procedure using actual data from an HDE where some truth regarding parameters of interest is known. We use the procedure to compare estimates for the proportion of true null hypotheses, the false discovery rate (FDR), and a local version of FDR obtained from 15 different statistical methods.     

Linear Mixed Model Selection for False Discovery Rate Control in Microarray Data Analysis

Summary In a microarray experiment, one experimental design is used to obtain expression measures for all genes. One popular analysis method involves fitting the same linear mixed model for each gene, obtaining gene‐specific p‐values for tests of interest involving fixed effects, and then choosing a threshold for significance that is intended to control false discovery rate (FDR) at a desired level. When one or more random factors have zero variance components for some genes, the standard practice of fitting the same full linear mixed model for all genes can result in failure to control FDR. We propose a new method that combines results from the fit of full and selected linear mixed models to identify differentially expressed genes and provide FDR control at target levels when the true underlying random effects structure varies across genes.     

Controlling False Discoveries in Multidimensional Directional Decisions,with Applications to Gene Expression Data on Ordered Categories

Summary Microarray gene expression studies over ordered categories are routinely conducted to gain insights into biological functions of genes and the underlying biological processes. Some common experiments are time‐course/dose‐response experiments where a tissue or cell line is exposed to different doses and/or durations of time to a chemical. A goal of such studies is to identify gene expression patterns/profiles over the ordered categories. This problem can be formulated as a multiple testing problem where for each gene the null hypothesis of no difference between the successive mean gene expressions is tested and further directional decisions are made if it is rejected. Much of the existing multiple testing procedures are devised for controlling the usual false discovery rate (FDR) rather than the mixed directional FDR (mdFDR), the expected proportion of Type I and directional errors among all rejections. Benjamini and Yekutieli (2005, Journal of the American Statistical Association 100, 71–93) proved that an augmentation of the usual Benjamini–Hochberg (BH) procedure can control the mdFDR while testing simple null hypotheses against two‐sided alternatives in terms of one‐dimensional parameters. In this article, we consider the problem of controlling the mdFDR involving multidimensional parameters. To deal with this problem, we develop a procedure extending that of Benjamini and Yekutieli based on the Bonferroni test for each gene. A proof is given for its mdFDR control when the underlying test statistics are independent across the genes. The results of a simulation study evaluating its performance under independence as well as under dependence of the underlying test statistics across the genes relative to other relevant procedures are reported. Finally, the proposed methodology is applied to a time‐course microarray data obtained by Lobenhofer et al. (2002, Molecular Endocrinology 16, 1215–1229). We identified several important cell‐cycle genes, such as DNA replication/repair gene MCM4 and replication factor subunit C2, which were not identified by the previous analyses of the same data by Lobenhofer et al. (2002) and Peddada et al. (2003, Bioinformatics 19, 834–841). Although some of our findings overlap with previous findings, we identify several other genes that complement the results of Lobenhofer et al. (2002) .     

Fusing time series expression data through hybrid aggregation and hierarchical merge

SUMMARY: A novel integration approach targeting the combination of multi-experiment time series expression data is proposed. A recursive hybrid aggregation algorithm is initially employed to extract a set of genes, which are eventually of interest for the biological phenomenon under study. Next, a hierarchical merge procedure is specifically developed for the purpose of fusing together the multiple-experiment expression pro.les of the selected genes. This employs dynamic time warping alignment techniques in order to account adequately for the potential phase shift between the different experiments. We subsequently demonstrate that the resulting gene expression pro.les consistently re.ect the behavior of the original expression pro.les in the different experiments. SUPPLEMENTARY INFORMATION: Supplementary data are available athttp://www.tu-plovdiv.bg/Container/bi/DataIntegration/     

Weighted analysis of microarray gene expression using maximum-likelihood

MOTIVATION: The numerical values of gene expression measured using microarrays are usually presented to the biological end-user as summary statistics of spot pixel data, such as the spot mean, median and mode. Much of the subsequent data analysis reported in the literature, however, uses only one of these spot statistics. This results in sub-optimal estimates of gene expression levels and a need for improvement in quantitative spot variation surveillance. RESULTS: This paper develops a maximum-likelihood method for estimating gene expression using spot mean, variance and pixel number values available from typical microarray scanners. It employs a hierarchical model of variation between and within microarray spots. The hierarchical maximum-likelihood estimate (MLE) is shown to be a more efficient estimator of the mean than the 'conventional' estimate using solely the spot mean values (i.e. without spot variance data). Furthermore, under the assumptions of our model, the spot mean and spot variance are shown to be sufficient statistics that do not require the use of all pixel data.The hierarchical MLE method is applied to data from both Monte Carlo (MC) simulations and a two-channel dye-swapped spotted microarray experiment. The MC simulations show that the hierarchical MLE method leads to improved detection of differential gene expression particularly when 'outlier' spots are present on the arrays. Compared with the conventional method, the MLE method applied to data from the microarray experiment leads to an increase in the number of differentially expressed genes detected for low cut-off P-values of interest.     

Using linkage genome scans to improve power of association in genome scans

Scanning the genome for association between markers and complex diseases typically requires testing hundreds of thousands of genetic polymorphisms. Testing such a large number of hypotheses exacerbates the trade-off between power to detect meaningful associations and the chance of making false discoveries. Even before the full genome is scanned, investigators often favor certain regions on the basis of the results of prior investigations, such as previous linkage scans. The remaining regions of the genome are investigated simultaneously because genotyping is relatively inexpensive compared with the cost of recruiting participants for a genetic study and because prior evidence is rarely sufficient to rule out these regions as harboring genes with variation of conferring liability (liability genes). However, the multiple testing inherent in broad genomic searches diminishes power to detect association, even for genes falling in regions of the genome favored a priori. Multiple testing problems of this nature are well suited for application of the false-discovery rate (FDR) principle, which can improve power. To enhance power further, a new FDR approach is proposed that involves weighting the hypotheses on the basis of prior data. We present a method for using linkage data to weight the association P values. Our investigations reveal that if the linkage study is informative, the procedure improves power considerably. Remarkably, the loss in power is small, even when the linkage study is uninformative. For a class of genetic models, we calculate the sample size required to obtain useful prior information from a linkage study. This inquiry reveals that, among genetic models that are seemingly equal in genetic information, some are much more promising than others for this mode of analysis.     

Assessing reliability of gene clusters from gene expression data

The rapid development of microarray technologies has raised many challenging problems in experiment design and data analysis. Although many numerical algorithms have been successfully applied to analyze gene expression data, the effects of variations and uncertainties in measured gene expression levels across samples and experiments have been largely ignored in the literature. In this article, in the context of hierarchical clustering algorithms, we introduce a statistical resampling method to assess the reliability of gene clusters identified from any hierarchical clustering method. Using the clustering trees constructed from the resampled data, we can evaluate the confidence value for each node in the observed clustering tree. A majority-rule consensus tree can be obtained, showing clusters that only occur in a majority of the resampled trees. We illustrate our proposed methods with applications to two published data sets. Although the methods are discussed in the context of hierarchical clustering methods, they can be applied with other cluster-identification methods for gene expression data to assess the reliability of any gene cluster of interest. Electronic Publication     

A note on the false discovery rate and inconsistent comparisons between experiments

MOTIVATION: The false discovery rate (FDR) has been widely adopted to address the multiple comparisons issue in high-throughput experiments such as microarray gene-expression studies. However, while the FDR is quite useful as an approach to limit false discoveries within a single experiment, like other multiple comparison corrections it may be an inappropriate way to compare results across experiments. This article uses several examples based on gene-expression data to demonstrate the potential misinterpretations that can arise from using FDR to compare across experiments. Researchers should be aware of these pitfalls and wary of using FDR to compare experimental results. FDR should be augmented with other measures such as p-values and expression ratios. It is worth including standard error and variance information for meta-analyses and, if possible, the raw data for re-analyses. This is especially important for high-throughput studies because data are often re-used for different objectives, including comparing common elements across many experiments. No single error rate or data summary may be appropriate for all of the different objectives.     

Gene profiling for determining pluripotent genes in a time course microarray experiment

In microarray experiments, it is often of interest to identifygenes which have a prespecified gene expression profile withrespect to time. Methods available in the literature are, however,typically not stringent enough in identifying such genes, particularlywhen the profile requires equivalence of gene expression levelsat certain time points. In this paper, the authors introducea new methodology, called gene profiling, that uses simultaneousdifferential and equivalent gene expression level testing torank genes according to a prespecified gene expression profile.Gene profiling treats the vector of true gene expression levelsas a linear combination of appropriate vectors, for example,vectors that give the required criteria for the profile. Thisgene profile model is fitted to the data, and the resultingparameter estimates are summarized in a single test statisticthat is then used to rank the genes. The theoretical underpinningsof gene profiling (equivalence testing, intersection–uniontests) are discussed in this paper, and the gene profiling methodologyis applied to our motivating stem-cell experiment.     

More Specific Signal Detection in Functional Magnetic Resonance Imaging by False Discovery Rate Control for Hierarchically Structured Systems of Hypotheses

Signal detection in functional magnetic resonance imaging (fMRI) inherently involves the problem of testing a large number of hypotheses. A popular strategy to address this multiplicity is the control of the false discovery rate (FDR). In this work we consider the case where prior knowledge is available to partition the set of all hypotheses into disjoint subsets or families, e. g., by a-priori knowledge on the functionality of certain regions of interest. If the proportion of true null hypotheses differs between families, this structural information can be used to increase statistical power. We propose a two-stage multiple test procedure which first excludes those families from the analysis for which there is no strong evidence for containing true alternatives. We show control of the family-wise error rate at this first stage of testing. Then, at the second stage, we proceed to test the hypotheses within each non-excluded family and obtain asymptotic control of the FDR within each family at this second stage. Our main mathematical result is that this two-stage strategy implies asymptotic control of the FDR with respect to all hypotheses. In simulations we demonstrate the increased power of this new procedure in comparison with established procedures in situations with highly unbalanced families. Finally, we apply the proposed method to simulated and to real fMRI data.     

fdrtool: a versatile R package for estimating local and tail area-based false discovery rates

False discovery rate (FDR) methodologies are essential in the study of high-dimensional genomic and proteomic data. The R package 'fdrtool' facilitates such analyses by offering a comprehensive set of procedures for FDR estimation. Its distinctive features include: (i) many different types of test statistics are allowed as input data, such as P-values, z-scores, correlations and t-scores; (ii) simultaneously, both local FDR and tail area-based FDR values are estimated for all test statistics and (iii) empirical null models are fit where possible, thereby taking account of potential over- or underdispersion of the theoretical null. In addition, 'fdrtool' provides readily interpretable graphical output, and can be applied to very large scale (in the order of millions of hypotheses) multiple testing problems. Consequently, 'fdrtool' implements a flexible FDR estimation scheme that is unified across different test statistics and variants of FDR. AVAILABILITY: The program is freely available from the Comprehensive R Archive Network (http://cran.r-project.org/) under the terms of the GNU General Public License (version 3 or later). CONTACT: strimmer@uni-leipzig.de.