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PubMed2005

Quantitative trait Loci analysis using the false discovery rate.

False discovery rate control has become an essential tool in any study that has a very large multiplicity problem. False discovery rate-controlling procedures have also been found to be very effective in QTL analysis, ensuring reproducible results with few falsely discovered linkages and offering increased power to discover QTL, although their acceptance has been slower than in microarray analysis, for example. The reason is partly because the methodological aspects of applying the false discovery rate to QTL mapping are not well developed. Our aim in this work is to lay a solid foundation for the use of the false discovery rate in QTL mapping. We review the false discovery rate criterion, the appropriate interpretation of the FDR, and alternative formulations of the FDR that appeared in the statistical and genetics literature. We discuss important features of the FDR approach, some stemming from new developments in FDR theory and methodology, which deem it especially useful in linkage analysis. We review false discovery rate-controlling procedures--the BH, the resampling procedure, and the adaptive two-stage procedure-and discuss the validity of these procedures in single- and multiple-trait QTL mapping. Finally we argue that the control of the false discovery rate has an important role in suggesting, indicating the significance of, and confirming QTL and present guidelines for its use.

Chromosome Mapping↗

PubMed2005

Power and type I error rate of false discovery rate approaches in genome-wide association studies.

In genome-wide genetic studies with a large number of markers, balancing the type I error rate and power is a challenging issue. Recently proposed false discovery rate (FDR) approaches are promising solutions to this problem. Using the 100 simulated datasets of a genome-wide marker map spaced about 3 cM and phenotypes from the Genetic Analysis Workshop 14, we studied the type I error rate and power of Storey's FDR approach, and compared it to the traditional Bonferroni procedure. We confirmed that Storey's FDR approach had a strong control of FDR. We found that Storey's FDR approach only provided weak control of family-wise error rate (FWER). For these simulated datasets, Storey's FDR approach only had slightly higher power than the Bonferroni procedure. In conclusion, Storey's FDR approach is more powerful than the Bonferroni procedure if strong control of FDR or weak control of FWER is desired. Storey's FDR approach has little power advantage over the Bonferroni procedure if there is low linkage disequilibrium among the markers. Further evaluation of the type I error rate and power of the FDR approaches for higher linkage disequilibrium and for haplotype analyses is warranted.

False Positive Reactions↗

PubMed2005

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.

Bayes Theorem↗

PubMed2006

Some comments on instability of false discovery rate estimation.

Some extended false discovery rate (FDR) controlling multiple testing procedures rely heavily on empirical estimates of the FDR constructed from gene expression data. Such estimates are also used as performance indicators when comparing different methods for microarray data analysis. The present communication shows that the variance of the proposed estimators may be intolerably high, the correlation structure of microarray data being the main cause of their instability.

Algorithms↗

PubMed2005

Multidimensional local false discovery rate for microarray studies.

MOTIVATION: The false discovery rate (fdr) is a key tool for statistical assessment of differential expression (DE) in microarray studies. Overall control of the fdr alone, however, is not sufficient to address the problem of genes with small variance, which generally suffer from a disproportionally high rate of false positives. It is desirable to have an fdr-controlling procedure that automatically accounts for gene variability. METHODS: We generalize the local fdr as a function of multiple statistics, combining a common test statistic for assessing DE with its standard error information. We use a non-parametric mixture model for DE and non-DE genes to describe the observed multi-dimensional statistics, and estimate the distribution for non-DE genes via the permutation method. We demonstrate this fdr2d approach for simulated and real microarray data. RESULTS: The fdr2d allows objective assessment of DE as a function of gene variability. We also show that the fdr2d performs better than commonly used modified test statistics. AVAILABILITY: An R-package OCplus containing functions for computing fdr2d() and other operating characteristics of microarray data is available at http://www.meb.ki.se/~yudpaw.

Algorithms↗

PubMed2005

Bias in the estimation of false discovery rate in microarray studies.

MOTIVATION: The false discovery rate (FDR) provides a key statistical assessment for microarray studies. Its value depends on the proportion pi(0) of non-differentially expressed (non-DE) genes. In most microarray studies, many genes have small effects not easily separable from non-DE genes. As a result, current methods often overestimate pi(0) and FDR, leading to unnecessary loss of power in the overall analysis. METHODS: For the common two-sample comparison we derive a natural mixture model of the test statistic and an explicit bias formula in the standard estimation of pi(0). We suggest an improved estimation of pi(0) based on the mixture model and describe a practical likelihood-based procedure for this purpose. RESULTS: The analysis shows that a large bias occurs when pi(0) is far from 1 and when the non-centrality parameters of the distribution of the test statistic are near zero. The theoretical result also explains substantial discrepancies between non-parametric and model-based estimates of pi(0). Simulation studies indicate mixture-model estimates are less biased than standard estimates. The method is applied to breast cancer and lymphoma data examples. AVAILABILITY: An R-package OCplus containing functions to compute pi(0) based on the mixture model, the resulting FDR and other operating characteristics of microarray data, is freely available at http://www.meb.ki.se/~yudpaw CONTACT: yudi.pawitan@meb.ki.se and alexander.ploner@meb.ki.se.

Computer Simulation↗

PubMed2004

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

Algorithms↗

PubMedDate not supplied

A stochastic downhill search algorithm for estimating the local false discovery rate.

Screening for differential gene expression in microarray studies leads to difficult large-scale multiple testing problems. The local false discovery rate is a statistical concept for quantifying uncertainty in multiple testing. In this paper, we introduce a novel estimator for the local false discovery rate that is based on an algorithm which splits all genes into two groups, representing induced and noninduced genes, respectively. Starting from the full set of genes, we successively exclude genes until the gene-wise p-values of the remaining genes look like a typical sample from a uniform distribution. In comparison to other methods, our algorithm performs compatibly in detecting the shape of the local false discovery rate and has a smaller bias with respect to estimating the overall percentage of noninduced genes. Our algorithm is implemented in the Bioconductor compatible R package TWILIGHT version 1.0.1, which is available from http://compdiag.molgen.mpg.de/software or from the Bioconductor project at http://www.bioconductor.org.

Algorithms↗

PubMed2005

A note on using permutation-based false discovery rate estimates to compare different analysis methods for microarray data.

MOTIVATION: False discovery rate (FDR) is defined as the expected percentage of false positives among all the claimed positives. In practice, with the true FDR unknown, an estimated FDR can serve as a criterion to evaluate the performance of various statistical methods under the condition that the estimated FDR approximates the true FDR well, or at least, it does not improperly favor or disfavor any particular method. Permutation methods have become popular to estimate FDR in genomic studies. The purpose of this paper is 2-fold. First, we investigate theoretically and empirically whether the standard permutation-based FDR estimator is biased, and if so, whether the bias inappropriately favors or disfavors any method. Second, we propose a simple modification of the standard permutation to yield a better FDR estimator, which can in turn serve as a more fair criterion to evaluate various statistical methods. RESULTS: Both simulated and real data examples are used for illustration and comparison. Three commonly used test statistics, the sample mean, SAM statistic and Student's t-statistic, are considered. The results show that the standard permutation method overestimates FDR. The overestimation is the most severe for the sample mean statistic while the least for the t-statistic with the SAM-statistic lying between the two extremes, suggesting that one has to be cautious when using the standard permutation-based FDR estimates to evaluate various statistical methods. In addition, our proposed FDR estimation method is simple and outperforms the standard method.

Algorithms↗

PubMed2005

Rank-invariant resampling based estimation of false discovery rate for analysis of small sample microarray data.

BACKGROUND: The evaluation of statistical significance has become a critical process in identifying differentially expressed genes in microarray studies. Classical p-value adjustment methods for multiple comparisons such as family-wise error rate (FWER) have been found to be too conservative in analyzing large-screening microarray data, and the False Discovery Rate (FDR), the expected proportion of false positives among all positives, has been recently suggested as an alternative for controlling false positives. Several statistical approaches have been used to estimate and control FDR, but these may not provide reliable FDR estimation when applied to microarray data sets with a small number of replicates. RESULTS: We propose a rank-invariant resampling (RIR) based approach to FDR evaluation. Our proposed method generates a biologically relevant null distribution, which maintains similar variability to observed microarray data. We compare the performance of our RIR-based FDR estimation with that of four other popular methods. Our approach outperforms the other methods both in simulated and real microarray data. CONCLUSION: We found that the SAM's random shuffling and SPLOSH approaches were liberal and the other two theoretical methods were too conservative while our RIR approach provided more accurate FDR estimation than the other approaches.

Algorithms↗

PubMed2002

Application of the false discovery rate to quantitative trait loci interval mapping with multiple traits.

Controlling the false discovery rate (FDR) has been proposed as an alternative to controlling the genome-wise error rate (GWER) for detecting quantitative trait loci (QTL) in genome scans. The objective here was to implement FDR in the context of regression interval mapping for multiple traits. Data on five traits from an F2 swine breed cross were used. FDR was implemented using tests at every 1 cM (FDR1) and using tests with the highest test statistic for each marker interval (FDRm). For the latter, a method was developed to predict comparison-wise error rates. At low error rates, FDR1 behaved erratically; FDRm was more stable but gave similar significance thresholds and number of QTL detected. At the same error rate, methods to control FDR gave less stringent significance thresholds and more QTL detected than methods to control GWER. Although testing across traits had limited impact on FDR, single-trait testing was recommended because there is no theoretical reason to pool tests across traits for FDR. FDR based on FDRm was recommended for QTL detection in interval mapping because it provides significance tests that are meaningful, yet not overly stringent, such that a more complete picture of QTL is revealed.

Animals↗

PubMed2004

A simple procedure for estimating the false discovery rate.

MOTIVATION: The most used criterion in microarray data analysis is nowadays the false discovery rate (FDR). In the framework of estimating procedures based on the marginal distribution of the P-values without any assumption on gene expression changes, estimators of the FDR are necessarily conservatively biased. Indeed, only an upper bound estimate can be obtained for the key quantity pi0, which is the probability for a gene to be unmodified. In this paper, we propose a novel family of estimators for pi0 that allows the calculation of FDR. RESULTS: The very simple method for estimating pi0 called LBE (Location Based Estimator) is presented together with results on its variability. Simulation results indicate that the proposed estimator performs well in finite sample and has the best mean square error in most of the cases as compared with the procedures QVALUE, BUM and SPLOSH. The different procedures are then applied to real datasets. AVAILABILITY: The R function LBE is available at http://ifr69.vjf.inserm.fr/lbe CONTACT: broet@vjf.inserm.fr.

Algorithms↗

PubMed2005

Local false discovery rate and minimum total error rate approaches to identifying interesting chromosomal regions.

The simultaneous testing of a large number of hypotheses in a genome scan, using individual thresholds for significance, inherently leads to inflated genome-wide false positive rates. There exist various approaches to approximating the correct genomewide p-values under various assumptions, either by way of asymptotics or simulations. We explore a philosophically different criterion, recently proposed in the literature, which controls the false discovery rate. The test statistics are assumed to arise from a mixture of distributions under the null and non-null hypotheses. We fit the mixture distribution using both a nonparametric approach and commingling analysis, and then apply the local false discovery rate to select cut-off points for regions to be declared interesting. Another criterion, the minimum total error, is also explored. Both criteria seem to be sensible alternatives to controlling the classical type I and type II error rates.

Chromosomes, Human↗

PubMed2005

Application of false discovery rate procedure to pairwise comparisons of refractive index of glass fragments.

This study was undertaken to apply a new method of controlling type I error when performing pairwise comparisons. The Benjamini and Hochberg false discovery rate (FDR) controlling procedures have proved to be very powerful tools in solving many practical problems but have not yet been applied to pairwise comparisons of refractive index of glass samples. Students t-test and Welch test (unequal variance Student's t-test) were applied to all possible pairwise comparisons. The comparisons were made on the basis of refractive index values of 72 glass fragments from different car windows and 69 different fragments from one windscreen. The type I error was controlled by the use of Benjamini and Hochberg false discovery rate (FDR) controlling procedure. To illustrate the importance of controlling type I error when using pairwise comparisons, results of pairwise comparisons with the FDR controlling procedure were compared to those comparisons made without any controlling procedure. Significantly fewer false negative results (false rejection of H0) were found during the use of FDR procedure in the comparison of refractive indices from the same windscreen than when comparing without controlling type I error. The results of application of FDR method were also compared to the results of other post-hoc tests, such as Tukey HSD test and Bonfferoni test. The FDR method has higher power than Bonferroni method and Tukey HSD method and control errors better than comparing without controlling type I error. The method of choice, for pairwise comparison of glass fragments on the basis of refractive index is Welch test with FDR adjustment. The method gave type I errors at level about 5.3% and type II error at about 3.7%.

Journal Article↗

PubMed2006

Robust estimation of the false discovery rate.

MOTIVATION: Presently available methods that use p-values to estimate or control the false discovery rate (FDR) implicitly assume that p-values are continuously distributed and based on two-sided tests. Therefore, it is difficult to reliably estimate the FDR when p-values are discrete or based on one-sided tests. RESULTS: A simple and robust method to estimate the FDR is proposed. The proposed method does not rely on implicit assumptions that tests are two-sided or yield continuously distributed p-values. The proposed method is proven to be conservative and have desirable large-sample properties. In addition, the proposed method was among the best performers across a series of 'real data simulations' comparing the performance of five currently available methods. AVAILABILITY: Libraries of S-plus and R routines to implement the method are freely available from www.stjuderesearch.org/depts/biostats.

Algorithms↗

PubMed2003

False discovery rate in linkage and association genome screens for complex disorders.

We explore the implications of the false discovery rate (FDR) controlling procedure in disease gene mapping. With the aid of simulations, we show how, under models commonly used, the simple step-down procedure introduced by Benjamini and Hochberg controls the FDR for the dependent tests on which linkage and association genome screens are based. This adaptive multiple comparison procedure may offer an important tool for mapping susceptibility genes for complex diseases.

Bipolar Disorder↗

PubMed2005

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.

Algorithms↗

PubMedDate not supplied

Bootstrapping of gene-expression data improves and controls the false discovery rate of differentially expressed genes.

The ordinary-, penalized-, and bootstrap t-test, least squares and best linear unbiased prediction were compared for their false discovery rates (FDR), i.e. the fraction of falsely discovered genes, which was empirically estimated in a duplicate of the data set. The bootstrap-t-test yielded up to 80% lower FDRs than the alternative statistics, and its FDR was always as good as or better than any of the alternatives. Generally, the predicted FDR from the bootstrapped P-values agreed well with their empirical estimates, except when the number of mRNA samples is smaller than 16. In a cancer data set, the bootstrap-t-test discovered 200 differentially regulated genes at a FDR of 2.6%, and in a knock-out gene expression experiment 10 genes were discovered at a FDR of 3.2%. It is argued that, in the case of microarray data, control of the FDR takes sufficient account of the multiple testing, whilst being less stringent than Bonferoni-type multiple testing corrections. Extensions of the bootstrap simulations to more complicated test-statistics are discussed.

Analysis of Variance↗