In situ analysis of cross-hybridisation on microarrays and the inference of expression correlation

Background

When conducting multiple hypothesis tests, it is important to control the number of false positives, or the False Discovery Rate (FDR). However, there is a tradeoff between controlling FDR and maximizing power. Several methods have been proposed, such as the q-value method, to estimate the proportion of true null hypothesis among the tested hypotheses, and use this estimation in the control of FDR. These methods usually depend on the assumption that the test statistics are independent (or only weakly correlated). However, many types of data, for example microarray data, often contain large scale correlation structures. Our objective was to develop methods to control the FDR while maintaining a greater level of power in highly correlated datasets by improving the estimation of the proportion of null hypotheses.

Results

We showed that when strong correlation exists among the data, which is common in microarray datasets, the estimation of the proportion of null hypotheses could be highly variable resulting in a high level of variation in the FDR. Therefore, we developed a re-sampling strategy to reduce the variation by breaking the correlations between gene expression values, then using a conservative strategy of selecting the upper quartile of the re-sampling estimations to obtain a strong control of FDR.

Conclusion

With simulation studies and perturbations on actual microarray datasets, our method, compared to competing methods such as q-value, generated slightly biased estimates on the proportion of null hypotheses but with lower mean square errors. When selecting genes with controlling the same FDR level, our methods have on average a significantly lower false discovery rate in exchange for a minor reduction in the power.     

On correcting the overestimation of the permutation-based false discovery rate estimator

MOTIVATION: Recent attempts to account for multiple testing in the analysis of microarray data have focused on controlling the false discovery rate (FDR), which is defined as the expected percentage of the number of false positive genes among the claimed significant genes. As a consequence, the accuracy of the FDR estimators will be important for correctly controlling FDR. Xie et al. found that the standard permutation method of estimating FDR is biased and proposed to delete the predicted differentially expressed (DE) genes in the estimation of FDR for one-sample comparison. However, we notice that the formula of the FDR used in their paper is incorrect. This makes the comparison results reported in their paper unconvincing. Other problems with their method include the biased estimation of FDR caused by over- or under-deletion of DE genes in the estimation of FDR and by the implicit use of an unreasonable estimator of the true proportion of equivalently expressed (EE) genes. Due to the great importance of accurate FDR estimation in microarray data analysis, it is necessary to point out such problems and propose improved methods. RESULTS: Our results confirm that the standard permutation method overestimates the FDR. With the correct FDR formula, we show the method of Xie et al. always gives biased estimation of FDR: it overestimates when the number of claimed significant genes is small, and underestimates when the number of claimed significant genes is large. To overcome these problems, we propose two modifications. The simulation results show that our estimator gives more accurate estimation.     

ExactFDR: exact computation of false discovery rate estimate in case-control association studies

Genome-wide association studies require accurate and fast statistical methods to identify relevant signals from the background noise generated by a huge number of simultaneously tested hypotheses. It is now commonly accepted that exact computations of association probability value (P-value) are preferred to chi(2) and permutation-based approximations. Following the same principle, the ExactFDR software package improves speed and accuracy of the permutation-based false discovery rate (FDR) estimation method by replacing the permutation-based estimation of the null distribution by the generalization of the algorithm used for computing individual exact P-values. It provides a quick and accurate non-conservative estimator of the proportion of false positives in a given selection of markers, and is therefore an efficient and pragmatic tool for the analysis of genome-wide association studies.     

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.     

Wavelet thresholding with bayesian false discovery rate control

The false discovery rate (FDR) procedure has become a popular method for handling multiplicity in high-dimensional data. The definition of FDR has a natural Bayesian interpretation; it is the expected proportion of null hypotheses mistakenly rejected given a measure of evidence for their truth. In this article, we propose controlling the positive FDR using a Bayesian approach where the rejection rule is based on the posterior probabilities of the null hypotheses. Correspondence between Bayesian and frequentist measures of evidence in hypothesis testing has been studied in several contexts. Here we extend the comparison to multiple testing with control of the FDR and illustrate the procedure with an application to wavelet thresholding. The problem consists of recovering signal from noisy measurements. This involves extracting wavelet coefficients that result from true signal and can be formulated as a multiple hypotheses-testing problem. We use simulated examples to compare the performance of our approach to the Benjamini and Hochberg (1995, Journal of the Royal Statistical Society, Series B57, 289-300) procedure. We also illustrate the method with nuclear magnetic resonance spectral data from human brain.     

Resampling-based empirical Bayes multiple testing procedures for controlling generalized tail probability and expected value error rates: focus on the false discovery rate and simulation study

This article proposes resampling-based empirical Bayes multiple testing procedures for controlling a broad class of Type I error rates, defined as generalized tail probability (gTP) error rates, gTP (q,g) = Pr(g (V(n),S(n)) > q), and generalized expected value (gEV) error rates, gEV (g) = E [g (V(n),S(n))], for arbitrary functions g (V(n),S(n)) of the numbers of false positives V(n) and true positives S(n). Of particular interest are error rates based on the proportion g (V(n),S(n)) = V(n) /(V(n) + S(n)) of Type I errors among the rejected hypotheses, such as the false discovery rate (FDR), FDR = E [V(n) /(V(n) + S(n))]. The proposed procedures offer several advantages over existing methods. They provide Type I error control for general data generating distributions, with arbitrary dependence structures among variables. Gains in power are achieved by deriving rejection regions based on guessed sets of true null hypotheses and null test statistics randomly sampled from joint distributions that account for the dependence structure of the data. The Type I error and power properties of an FDR-controlling version of the resampling-based empirical Bayes approach are investigated and compared to those of widely-used FDR-controlling linear step-up procedures in a simulation study. The Type I error and power trade-off achieved by the empirical Bayes procedures under a variety of testing scenarios allows this approach to be competitive with or outperform the Storey and Tibshirani (2003) linear step-up procedure, as an alternative to the classical Benjamini and Hochberg (1995) procedure.     

A moment-based method for estimating the proportion of true null hypotheses and its application to microarray gene expression data

Due to advances in experimental technologies, it is feasible to collect measurements for a large number of variables. When these variables are simultaneously screened by a statistical test, it is necessary to consider the adjustment for multiple hypothesis testing. The false discovery rate has been proposed and widely used to address this issue. A related problem is the estimation of the proportion of true null hypotheses. The long-standing difficulty to this problem is the identifiability of the nonparametric model. In this study, we propose a moment-based method coupled with sample splitting for estimating this proportion. If the p values from the alternative hypothesis are homogeneously distributed, then the proposed method will solve the identifiability and give its optimal performances. When the p values from the alternative hypothesis are heterogeneously distributed, we propose to approximate this mixture distribution so that the identifiability can be achieved. Theoretical aspects of the approximation error are discussed. The proposed estimation method is completely nonparametric and simple with an explicit formula. Simulation studies show the favorable performances of the proposed method when it is compared to the other existing methods. Two microarray gene expression data sets are considered for applications.     

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.     

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.     

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.     

Assigning significance to peptides identified by tandem mass spectrometry using decoy databases

Automated methods for assigning peptides to observed tandem mass spectra typically return a list of peptide-spectrum matches, ranked according to an arbitrary score. In this article, we describe methods for converting these arbitrary scores into more useful statistical significance measures. These methods employ a decoy sequence database as a model of the null hypothesis, and use false discovery rate (FDR) analysis to correct for multiple testing. We first describe a simple FDR inference method and then describe how estimating and taking into account the percentage of incorrectly identified spectra in the entire data set can lead to increased statistical power.     

Construction of null statistics in permutation-based multiple testing for multi-factorial microarray experiments

MOTIVATION: The parametric F-test has been widely used in the analysis of factorial microarray experiments to assess treatment effects. However, the normality assumption is often untenable for microarray experiments with small replications. Therefore, permutation-based methods are called for help to assess the statistical significance. The distribution of the F-statistics across all the genes on the array can be regarded as a mixture distribution with a proportion of statistics generated from the null distribution of no differential gene expression whereas the other proportion of statistics generated from the alternative distribution of genes differentially expressed. This results in the fact that the permutation distribution of the F-statistics may not approximate well to the true null distribution of the F-statistics. Therefore, the construction of a proper null statistic to better approximate the null distribution of F-statistic is of great importance to the permutation-based multiple testing in microarray data analysis. RESULTS: In this paper, we extend the ideas of constructing null statistics based on pairwise differences to neglect the treatment effects from the two-sample comparison problem to the multifactorial balanced or unbalanced microarray experiments. A null statistic based on a subpartition method is proposed and its distribution is employed to approximate the null distribution of the F-statistic. The proposed null statistic is able to accommodate unbalance in the design and is also corrected for the undue correlation between its numerator and denominator. In the simulation studies and real biological data analysis, the number of true positives and the false discovery rate (FDR) of the proposed null statistic are compared with those of the permutated version of the F-statistic. It has been shown that our proposed method has a better control of the FDRs and a higher power than the standard permutation method to detect differentially expressed genes because of the better approximated tail probabilities.     

Multiple quantitative trait loci mapping with cofactors and application of alternative variants of the false discovery rate in an enlarged granddaughter design

The experimental power of a granddaughter design to detect quantitative trait loci (QTL) in dairy cattle is often limited by the availability of progeny-tested sires, by the ignoring of already identified QTL in the statistical analysis, and by the application of stringent experimentwise significance levels. This study describes an experiment that addressed these points. A large granddaughter design was set up that included sires from two countries (Germany and France), resulting in almost 2000 sires. The animals were genotyped for markers on nine different chromosomes. The QTL analysis was done for six traits separately using a multimarker regression that included putative QTL on other chromosomes as cofactors in the model. Different variants of the false discovery rate (FDR) were applied. Two of them accounted for the proportion of truly null hypotheses, which were estimated to be 0.28 and 0.3, respectively, and were therefore tailored to the experiment. A total of 25 QTL could be mapped when cofactors were included in the model-7 more than without cofactors. Controlling the FDR at 0.05 revealed 31 QTL for the two FDR methods that accounted for the proportion of truly null hypotheses. The relatively high power of this study can be attributed to the size of the experiment, to the QTL analysis with cofactors, and to the application of an appropriate FDR.     

Controlling the proportion of false positives in multiple dependent tests

Genome scan mapping experiments involve multiple tests of significance. Thus, controlling the error rate in such experiments is important. Simple extension of classical concepts results in attempts to control the genomewise error rate (GWER), i.e., the probability of even a single false positive among all tests. This results in very stringent comparisonwise error rates (CWER) and, consequently, low experimental power. We here present an approach based on controlling the proportion of false positives (PFP) among all positive test results. The CWER needed to attain a desired PFP level does not depend on the correlation among the tests or on the number of tests as in other approaches. To estimate the PFP it is necessary to estimate the proportion of true null hypotheses. Here we show how this can be estimated directly from experimental results. The PFP approach is similar to the false discovery rate (FDR) and positive false discovery rate (pFDR) approaches. For a fixed CWER, we have estimated PFP, FDR, pFDR, and GWER through simulation under a variety of models to illustrate practical and philosophical similarities and differences among the methods.     

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.     

Multiple testing with discrete data: Proportion of true null hypotheses and two adaptive FDR procedures

We consider multiple testing with false discovery rate (FDR) control when p values have discrete and heterogeneous null distributions. We propose a new estimator of the proportion of true null hypotheses and demonstrate that it is less upwardly biased than Storey's estimator and two other estimators. The new estimator induces two adaptive procedures, that is, an adaptive Benjamini–Hochberg (BH) procedure and an adaptive Benjamini–Hochberg–Heyse (BHH) procedure. We prove that the adaptive BH (aBH) procedure is conservative nonasymptotically. Through simulation studies, we show that these procedures are usually more powerful than their nonadaptive counterparts and that the adaptive BHH procedure is usually more powerful than the aBH procedure and a procedure based on randomized p‐value. The adaptive procedures are applied to a study of HIV vaccine efficacy, where they identify more differentially polymorphic positions than the BH procedure at the same FDR level.     

Improving false discovery rate estimation

MOTIVATION: Recent attempts to account for multiple testing in the analysis of microarray data have focused on controlling the false discovery rate (FDR). However, rigorous control of the FDR at a preselected level is often impractical. Consequently, it has been suggested to use the q-value as an estimate of the proportion of false discoveries among a set of significant findings. However, such an interpretation of the q-value may be unwarranted considering that the q-value is based on an unstable estimator of the positive FDR (pFDR). Another method proposes estimating the FDR by modeling p-values as arising from a beta-uniform mixture (BUM) distribution. Unfortunately, the BUM approach is reliable only in settings where the assumed model accurately represents the actual distribution of p-values. METHODS: A method called the spacings LOESS histogram (SPLOSH) is proposed for estimating the conditional FDR (cFDR), the expected proportion of false positives conditioned on having k 'significant' findings. SPLOSH is designed to be more stable than the q-value and applicable in a wider variety of settings than BUM. RESULTS: In a simulation study and data analysis example, SPLOSH exhibits the desired characteristics relative to the q-value and BUM. AVAILABILITY: The Web site www.stjuderesearch.org/statistics/splosh.html has links to freely available S-plus code to implement the proposed procedure.     

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.     

Testing Jumps via False Discovery Rate Control

Many recently developed nonparametric jump tests can be viewed as multiple hypothesis testing problems. For such multiple hypothesis tests, it is well known that controlling type I error often makes a large proportion of erroneous rejections, and such situation becomes even worse when the jump occurrence is a rare event. To obtain more reliable results, we aim to control the false discovery rate (FDR), an efficient compound error measure for erroneous rejections in multiple testing problems. We perform the test via the Barndorff-Nielsen and Shephard (BNS) test statistic, and control the FDR with the Benjamini and Hochberg (BH) procedure. We provide asymptotic results for the FDR control. From simulations, we examine relevant theoretical results and demonstrate the advantages of controlling the FDR. The hybrid approach is then applied to empirical analysis on two benchmark stock indices with high frequency data.