Two-stage designs for experiments with a large number of hypotheses

MOTIVATION: When a large number of hypotheses are investigated the false discovery rate (FDR) is commonly applied in gene expression analysis or gene association studies. Conventional single-stage designs may lack power due to low sample sizes for the individual hypotheses. We propose two-stage designs where the first stage is used to screen the 'promising' hypotheses which are further investigated at the second stage with an increased sample size. A multiple test procedure based on sequential individual P-values is proposed to control the FDR for the case of independent normal distributions with known variance. RESULTS: The power of optimal two-stage designs is impressively larger than the power of the corresponding single-stage design with equal costs. Extensions to the case of unknown variances and correlated test statistics are investigated by simulations. Moreover, it is shown that the simple multiple test procedure using first stage data for screening purposes and deriving the test decisions only from second stage data is a very powerful option.     

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.     

High throughput screening of co-expressed gene pairs with controlled false discovery rate (FDR) and minimum acceptable strength (MAS).

Many exploratory microarray data analysis tools such as gene clustering and relevance networks rely on detecting pairwise gene co-expression. Traditional screening of pairwise co-expression either controls biological significance or statistical significance, but not both. The former approach does not provide stochastic error control, and the later approach screens many co-expressions with excessively low correlation. We have designed and implemented a statistically sound two-stage co-expression detection algorithm that controls both statistical significance (false discovery rate, FDR) and biological significance (minimum acceptable strength, MAS) of the discovered co-expressions. Based on estimation of pairwise gene correlation, the algorithm provides an initial co-expression discovery that controls only FDR, which is then followed by a second stage co-expression discovery which controls both FDR and MAS. It also computes and thresholds the set of FDR p-values for each correlation that satisfied the MAS criterion. Using simulated data, we validated asymptotic null distributions of the Pearson and Kendall correlation coefficients and the two-stage error-control procedure; we also compared our two-stage test procedure with another two-stage test procedure using the receiver operating characteristic (ROC) curve. We then used yeast galactose metabolism data to illustrate the advantage of our method for clustering genes and constructing a relevance network. The method has been implemented in an R package "GeneNT" that is freely available from the Comprehensive R Archive Network (CRAN): www.cran.r-project.org/.     

Improving power of genome-wide association studies with weighted false discovery rate control and prioritized subset analysis

The issue of large-scale testing has caught much attention with the advent of high-throughput technologies. In genomic studies, researchers are often confronted with a large number of tests. To make simultaneous inference for the many tests, the false discovery rate (FDR) control provides a practical balance between the number of true positives and the number of false positives. However, when few hypotheses are truly non-null, controlling the FDR may not provide additional advantages over controlling the family-wise error rate (e.g., the Bonferroni correction). To facilitate discoveries from a study, weighting tests according to prior information is a promising strategy. A 'weighted FDR control' (WEI) and a 'prioritized subset analysis' (PSA) have caught much attention. In this work, we compare the two weighting schemes with systematic simulation studies and demonstrate their use with a genome-wide association study (GWAS) on type 1 diabetes provided by the Wellcome Trust Case Control Consortium. The PSA and the WEI both can increase power when the prior is informative. With accurate and precise prioritization, the PSA can especially create substantial power improvements over the commonly-used whole-genome single-step FDR adjustment (i.e., the traditional un-weighted FDR control). When the prior is uninformative (true disease susceptibility regions are not prioritized), the power loss of the PSA and the WEI is almost negligible. However, a caution is that the overall FDR of the PSA can be slightly inflated if the prioritization is not accurate and precise. Our study highlights the merits of using information from mounting genetic studies, and provides insights to choose an appropriate weighting scheme to FDR control on GWAS.     

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.     

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.     

Screening and replication using the same data set: testing strategies for family-based studies in which all probands are affected

For genome-wide association studies in family-based designs, we propose a powerful two-stage testing strategy that can be applied in situations in which parent-offspring trio data are available and all offspring are affected with the trait or disease under study. In the first step of the testing strategy, we construct estimators of genetic effect size in the completely ascertained sample of affected offspring and their parents that are statistically independent of the family-based association/transmission disequilibrium tests (FBATs/TDTs) that are calculated in the second step of the testing strategy. For each marker, the genetic effect is estimated (without requiring an estimate of the SNP allele frequency) and the conditional power of the corresponding FBAT/TDT is computed. Based on the power estimates, a weighted Bonferroni procedure assigns an individually adjusted significance level to each SNP. In the second stage, the SNPs are tested with the FBAT/TDT statistic at the individually adjusted significance levels. Using simulation studies for scenarios with up to 1,000,000 SNPs, varying allele frequencies and genetic effect sizes, the power of the strategy is compared with standard methodology (e.g., FBATs/TDTs with Bonferroni correction). In all considered situations, the proposed testing strategy demonstrates substantial power increases over the standard approach, even when the true genetic model is unknown and must be selected based on the conditional power estimates. The practical relevance of our methodology is illustrated by an application to a genome-wide association study for childhood asthma, in which we detect two markers meeting genome-wide significance that would not have been detected using standard methodology.     

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.     

An adaptive single-step FDR procedure with applications to DNA microarray analysis

The use of multiple hypothesis testing procedures has been receiving a lot of attention recently by statisticians in DNA microarray analysis. The traditional FWER controlling procedures are not very useful in this situation since the experiments are exploratory by nature and researchers are more interested in controlling the rate of false positives rather than controlling the probability of making a single erroneous decision. This has led to increased use of FDR (False Discovery Rate) controlling procedures. Genovese and Wasserman proposed a single-step FDR procedure that is an asymptotic approximation to the original Benjamini and Hochberg stepwise procedure. In this paper, we modify the Genovese-Wasserman procedure to force the FDR control closer to the level alpha in the independence setting. Assuming that the data comes from a mixture of two normals, we also propose to make this procedure adaptive by first estimating the parameters using the EM algorithm and then using these estimated parameters into the above modification of the Genovese-Wasserman procedure. We compare this procedure with the original Benjamini-Hochberg and the SAM thresholding procedures. The FDR control and other properties of this adaptive procedure are verified numerically.     

Detecting dyads of related individuals in large collections of DNA-profiles by controlling the false discovery rate

The search for pairs (dyads) of related individuals in large databases of DNA-profiles has become an increasingly important inference tool in ecology. However, the many, partly dependent, pairwise comparisons introduce statistical issues. We show that the false discovery rate (FDR) procedure is well suited to control for the proportion of false positives, i.e. dyads consisting of unrelated individuals, which under normal circumstances would have been labelled as related individuals. We verify the behaviour of the standard FDR procedure by simulation, demonstrating that the FDR procedure works satisfactory in spite of the many dependent pairwise comparisons involved in an exhaustive database screening. A computer program that implements this method is available online. In addition, we propose to implement a second stage in the procedure, in which additional independent genetic markers are used to identify the false positives. We demonstrate the application of the approach in an analysis of a DNA database consisting of 3300 individual minke whales (Balaenoptera acutorostrata) each typed at ten microsatellite loci. Applying the standard procedure with an FDR of 50% led to the identification of 74 putative dyads of 1st- or 2nd-order relatives. However, introducing the second step, which involved additional genotypes at 15 microsatellite loci, revealed that only 21 of the putative dyads can be claimed with high certainty to be true dyads.     

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.     

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 weighted FDR procedure under discrete and heterogeneous null distributions

Multiple testing (MT) with false discovery rate (FDR) control has been widely conducted in the “discrete paradigm” where p-values have discrete and heterogeneous null distributions. However, in this scenario existing FDR procedures often lose some power and may yield unreliable inference, and for this scenario there does not seem to be an FDR procedure that partitions hypotheses into groups, employs data-adaptive weights and is nonasymptotically conservative. We propose a weighted p-value-based FDR procedure, “weighted FDR (wFDR) procedure” for short, for MT in the discrete paradigm that efficiently adapts to both heterogeneity and discreteness of p-value distributions. We theoretically justify the nonasymptotic conservativeness of the wFDR procedure under independence, and show via simulation studies that, for MT based on p-values of binomial test or Fisher's exact test, it is more powerful than six other procedures. The wFDR procedure is applied to two examples based on discrete data, a drug safety study, and a differential methylation study, where it makes more discoveries than two existing methods.     

Genomewide weighted hypothesis testing in family-based association studies, with an application to a 100K scan

For genomewide association (GWA) studies in family-based designs, we propose a novel two-stage strategy that weighs the association P values with the use of independently estimated weights. The association information contained in the family sample is partitioned into two orthogonal components--namely, the between-family information and the within-family information. The between-family component is used in the first (i.e., screening) stage to obtain a relative ranking of all the markers. The within-family component is used in the second (i.e., testing) stage in the framework of the standard family-based association test, and the resulting P values are weighted using the estimated marker ranking from the screening step. The approach is appealing, in that it ensures that all the markers are tested in the testing step and, at the same time, also uses information from the screening step. Through simulation studies, we show that testing all the markers is more powerful than testing only the most promising ones from the screening step, which was the method suggested by Van Steen et al. A comparison with a population-based approach shows that the approach achieves comparable power. In the presence of a reasonable level of population stratification, our approach is only slightly affected in terms of power and, since it is a family-based method, is completely robust to spurious effects. An application to a 100K scan in the Framingham Heart Study illustrates the practical advantages of our approach. The proposed method is of general applicability; it extends to any setting in which prior, independent ranking of hypotheses is available.     

On the False Discovery Rate and Expected Type I Errors

The paper is concerned with expected type I errors of some stepwise multiple test procedures based on independent p‐values controlling the so‐called false discovery rate (FDR). We derive an asymptotic result for the supremum of the expected type I error rate(EER) when the number of hypotheses tends to infinity. Among others, it will be shown that when the original Benjamini‐Hochberg step‐up procedure controls the FDR at level α, its EER may approach a value being slightly larger than α/4 when the number of hypotheses increases. Moreover, we derive some least favourable parameter configuration results, some bounds for the FDR and the EER as well as easily computable formulae for the familywise error rate (FWER) of two FDR‐controlling procedures. Finally, we discuss some undesirable properties of the FDR concept, especially the problem of cheating.     

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.     

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.     

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.