False Discovery Rate Level Research Articles

Multiple Testing with the Structure-Adaptive Benjamini–Hochberg Algorithm

SummaryIn multiple-testing problems, where a large number of hypotheses are tested simultaneously, false discovery rate (FDR) control can be achieved with the well-known Benjamini–Hochberg procedure, which a(0, 1]dapts to the amount of signal in the data, under certain distributional assumptions. Many modifications of this procedure have been proposed to improve power in scenarios where the hypotheses are organized into groups or into a hierarchy, as well as other structured settings. Here we introduce the ‘structure-adaptive Benjamini–Hochberg algorithm’ (SABHA) as a generalization of these adaptive testing methods. The SABHA method incorporates prior information about any predetermined type of structure in the pattern of locations of the signals and nulls within the list of hypotheses, to reweight the p-values in a data-adaptive way. This raises the power by making more discoveries in regions where signals appear to be more common. Our main theoretical result proves that the SABHA method controls the FDR at a level that is at most slightly higher than the target FDR level, as long as the adaptive weights are constrained sufficiently so as not to overfit too much to the data—interestingly, the excess FDR can be related to the Rademacher complexity or Gaussian width of the class from which we choose our data-adaptive weights. We apply this general framework to various structured settings, including ordered, grouped and low total variation structures, and obtain the bounds on the FDR for each specific setting. We also examine the empirical performance of the SABHA method on functional magnetic resonance imaging activity data and on gene–drug response data, as well as on simulated data.

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.

Re-assessment of multiple testing strategies for more efficient genome-wide association studies.

Although enormous costs have been dedicated to discovering relevant disease-related genetic variants, especially in genome-wide association studies (GWASs), only a small fraction of estimated heritability can be explained by these results. This is the so-called missing heritability problem. The conventional use of overly conservative multiple testing strategies based on controlling the familywise error rate (FWER), in particular with a genome-wide significance threshold of P <5 × 10-8, is one of the most important issues from a statistical perspective. To help resolve this problem, we performed comprehensive re-assessments of currently available strategies using recently published, extremely large-scale GWAS data sets of rheumatoid arthritis and schizophrenia (>50,000 subjects). The estimates of statistical power averaged for all disease-related genetic variants of the standard FWER-based strategy were only 0.09% for the rheumatoid arthritis data and 0.04% for the schizophrenia data. To design more efficient strategies, we also conducted an extensive comparison of multiple testing strategies by applying false discovery rate (FDR)-controlling procedures to these data sets and simulations, and found that the FDR-based procedures achieved higher power than the FWER-based strategy, even at a strict FDR level (e.g., FDR = 1%). We also discuss a useful alternative measure, namely "partial power," which is an averaged power for detecting the clinically and biologically meaningful genetic factors with the largest effects. Simulation results suggest that the FDR-based procedures can achieve sufficient partial power (>80%) for detecting these factors (odds ratios of >1.05) with 80,000 subjects, and thus this may be a useful measure for defining realistic objectives of future GWASs.

MULTIPLE TESTING OF LOCAL MAXIMA FOR DETECTION OF PEAKS IN RANDOM FIELDS.

A topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic Gaussian noise, where tests are performed at local maxima of the smoothed observed signals. The procedure generalizes the one-dimensional scheme of [31] to Euclidean domains of arbitrary dimension. Two methods are developed according to two different ways of computing p-values: (i) using the exact distribution of the height of local maxima, available explicitly when the noise field is isotropic [9, 10]; (ii) using an approximation to the overshoot distribution of local maxima above a pre-threshold, applicable when the exact distribution is unknown, such as when the stationary noise field is non-isotropic [9]. The algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency, with specific rates, as the search space and signal strength get large. The optimal smoothing bandwidth and optimal pre-threshold are obtained to achieve maximum power. Simulations show that FDR levels are maintained in non-asymptotic conditions. The methods are illustrated in the analysis of functional magnetic resonance images of the brain.

Progressive calibration and averaging for tandem mass spectrometry statistical confidence estimation: Why settle for a single decoy?

Estimating the false discovery rate (FDR) among a list of tandem mass spectrum identifications is mostly done through target-decoy competition (TDC). Here we offer two new methods that can use an arbitrarily small number of additional randomly drawn decoy databases to improve TDC. Specifically, "Partial Calibration" utilizes a new meta-scoring scheme that allows us to gradually benefit from the increase in the number of identifications calibration yields and "Averaged TDC" (a-TDC) reduces the liberal bias of TDC for small FDR values and its variability throughout. Combining a-TDC with "Progressive Calibration" (PC), which attempts to find the "right" number of decoys required for calibration we see substantial impact in real datasets: when analyzing the Plasmodium falciparum data it typically yields almost the entire 17% increase in discoveries that "full calibration" yields (at FDR level 0.05) using 60 times fewer decoys. Our methods are further validated using a novel realistic simulation scheme and importantly, they apply more generally to the problem of controlling the FDR among discoveries from searching an incomplete database.

BAYESIAN MULTIPLE TESTING UNDER SPARSITY FOR POLYNOMIAL-TAILED DISTRIBUTIONS

This paper considers Bayesian multiple testing under sparsity for polynomial-tailed distributions satisfying a monotone likelihood ratio property. Included in this class of distributions are the Student's t, the Pareto, and many other distributions. We prove some general asymptotic optimality results under fixed and random thresholding. As examples of these general results, we establish the Bayesian asymptotic optimality of several multiple testing procedures in the literature for appropriately chosen false discovery rate levels. We also show by simulation that the Benjamini-Hochberg procedure with a false discovery rate level different from the asymptotically optimal one can lead to high Bayes risk.

Epigenetic Mediators Between Childhood Socioeconomic Disadvantage and Mid-Life Body Mass Index: The New England Family Study.

Childhood socioeconomic disadvantage is associated with adulthood obesity risk; however, epigenetic mechanisms are poorly understood. This work's objective was to evaluate whether associations of childhood socioeconomic disadvantage with adulthood body mass index (BMI) are mediated by DNA methylation. Participants were 141 men and women from the New England Family Study, prospectively followed prenatally through a mean age of 47 years. Epigenomewide DNA methylation was evaluated in peripheral blood and adipose tissue obtained at adulthood, using the Infinium HumanMethylation450K BeadChip. Childhood socioeconomic status (SES) at age 7 years was assessed directly from parents' reports. Offspring adiposity was directly assessed using BMI at a mean age of 47 years. Associations of SES, DNA methylation, and BMI were estimated using least square estimators. Statistical mediation analyses were performed using joint significance test and bootstrapping. Of CpG sites significant at the 25% false discovery rate level in epigenomewide methylation BMI analyses, 91 sites in men and 71 sites in women were additionally significant for SES-methylation associations (p < .001) in adipose tissue. Many involved genes biologically relevant for development of obesity, including fatty acid synthase, transmembrane protein 88, signal transducer and activator of transcription 3, and neuritin 1. There was no evidence of epigenetic mediation in peripheral blood leukocytes. DNA methylation at specific genes may be mediators of associations between childhood socioeconomic disadvantage and mid-life BMI in adipose tissue. Findings motivate continued efforts to study if and how childhood socioeconomic disadvantage is biologically embedded at the level of the epigenome in regions etiologically relevant for adiposity.

Abstract P3-07-52: Identification of serum biomarkers associated with Akt inhibitor MK-2206-induced toxicity in a pre-surgical breast cancer (BC) trial

Abstract Background: The PI3K/Akt/mTOR pathway is an important oncogenic driver in BC. A major hurdle in clinical Akt inhibitor development has been dose-limiting toxicities, such as rash. To facilitate the risk assessment of Akt inhibitor associated toxicity, we hypothesize that circulating biomarkers can be identified in proteins secreted by the tumor or tumor microenvironment and systemic response after treatment. Exosomes are small membrane bound vesicles containing proteins, mRNA, miRNA, and lipids that are secreted from host cells and remain viable after long-term storage of blood. In this study, we focused on identifying biomarkers associated with drug rash from serum exosomes in BC patients treated with the Akt inhibitor MK-2206. Methods: In an open-label pre-surgical trial, 2 doses of weekly MK2206 were administered to patients (pts) with stage I-III invasive BC: first at day -9 and second at day -2 from surgery. Sera were collected before and after MK2206. 200 μL of serum was used to isolate total exosomes by precipitation and centrifugation, followed by trypsin digestion and multiplexing labeling analysis. The Orbitrap mass spectrometer was used to acquire LC-MS/MS data. 1,053 unique proteins were identified from the uniProt database. Maximum false discovery rate level (FDR) for predictive biomarkers was controlled at 26% (q&amp;lt;0.26). Analysis was conducted on pre-MK-2206 and post-MK-2206 treated sera from pts to develop a protein signature associated with rash and identify candidate biomarkers of MK-2206-associated rash. Results: The study was discontinued after 12 pts were enrolled due to toxicity. Notably, an acneiform/maculopapular rash was observed in 5 pts. Unsupervised principal component analysis on the pre-MK-2206 specimens and the entire set of 1,053 proteins demonstrated that 4 of the 5 pts with rash formed a distinct cluster. 30 proteins were differentially expressed in pre-MK-2206 samples from pts who developed rash vs. no rash (q&amp;lt;0.26), with ≥1.5 fold difference in expression level in those with rash after MK-2206. Ingenuity pathway analysis revealed statistically significant over-representation of pathways involved in lipid metabolism (including MALRD1, AWAT2), nucleic acid synthesis (PPAT, ADSLL1), and protein synthesis (PPIB). 45 proteins were significantly different in post-MK-2206 samples (q&amp;lt;0.285). Lipid metabolism was the most significantly over-represented pathway in post-MK-2206 samples. Conclusions: We demonstrated that mass spectrometry-based proteomic analysis of patient-derived serum exosomes is a promising approach to study drug-induced toxicity. We found significant changes of circulating proteins before and after MK-2206. Increased expression of different proteins involved in lipid metabolism appears to predict skin toxicity, commonly seen with PI3K/Akt pathway inhibitors. Since the PI3K/Akt signaling pathway plays a role in physiological regulation of lipid metabolism, lipid metabolic profiles of BC patients might be important for predicting the risk and controlling toxicity induced by Akt inhibitors. These toxicity-associated biomarkers should be validated and then assessed prospectively in clinical trials. Citation Format: Mundi PS, Chen E, Sparano J, Andreopoulou E, Taback B, Wiechmann L, Feldman S, Ananthakrishnan P, Hibshoosh H, Connolly E, Crew K, Maurer M, Hershman DL, Kalinsky K. Identification of serum biomarkers associated with Akt inhibitor MK-2206-induced toxicity in a pre-surgical breast cancer (BC) trial. [abstract]. In: Proceedings of the Thirty-Eighth Annual CTRC-AACR San Antonio Breast Cancer Symposium: 2015 Dec 8-12; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2016;76(4 Suppl):Abstract nr P3-07-52.

Tract Orientation and Angular Dispersion Deviation Indicator (TOADDI): A framework for single-subject analysis in diffusion tensor imaging

The purpose of this work is to develop a framework for single-subject analysis of diffusion tensor imaging (DTI) data. This framework is termed Tract Orientation and Angular Dispersion Deviation Indicator (TOADDI) because it is capable of testing whether an individual tract as represented by the major eigenvector of the diffusion tensor and its corresponding angular dispersion are significantly different from a group of tracts on a voxel-by-voxel basis. This work develops two complementary statistical tests based on the elliptical cone of uncertainty, which is a model of uncertainty or dispersion of the major eigenvector of the diffusion tensor. The orientation deviation test examines whether the major eigenvector from a single subject is within the average elliptical cone of uncertainty formed by a collection of elliptical cones of uncertainty. The shape deviation test is based on the two-tailed Wilcoxon–Mann–Whitney two-sample test between the normalized shape measures (area and circumference) of the elliptical cones of uncertainty of the single subject against a group of controls. The False Discovery Rate (FDR) and False Non-discovery Rate (FNR) were incorporated in the orientation deviation test. The shape deviation test uses FDR only. TOADDI was found to be numerically accurate and statistically effective. Clinical data from two Traumatic Brain Injury (TBI) patients and one non-TBI subject were tested against the data obtained from a group of 45 non-TBI controls to illustrate the application of the proposed framework in single-subject analysis. The frontal portion of the superior longitudinal fasciculus seemed to be implicated in both tests (orientation and shape) as significantly different from that of the control group. The TBI patients and the single non-TBI subject were well separated under the shape deviation test at the chosen FDR level of 0.0005. TOADDI is a simple but novel geometrically based statistical framework for analyzing DTI data. TOADDI may be found useful in single-subject, graph-theoretic and group analyses of DTI data or DTI-based tractography techniques.

An extended sequential goodness-of-fit multiple testing method for discrete data.

The sequential goodness-of-fit (SGoF) multiple testing method has recently been proposed as an alternative to the familywise error rate- and the false discovery rate-controlling procedures in high-dimensional problems. For discrete data, the SGoF method may be very conservative. In this paper, we introduce an alternative SGoF-type procedure that takes into account the discreteness of the test statistics. Like the original SGoF, our new method provides weak control of the false discovery rate/familywise error rate but attains false discovery rate levels closer to the desired nominal level, and thus it is more powerful. We study the performance of this method in a simulation study and illustrate its application to a real pharmacovigilance data set.

A Robust Approach to Identifying Differential Circulating miRNAs in Breast Cancer

This article proposes and explores a robust approach to identifying differential circulating miRNAs in the plasma of patients with breast cancer. The proposed approach, developed in the framework of the M-estimation, is used to provide protection against potential outliers in miRNA expression data. As the study involves multiple comparisonswith a large number of circulating miRNAs, robust multiple tests are adopted at a given level of false discovery rate (FDR). Also, due to the uncertainties in the underlying distributions of the miRNA expression data sets, the p-values of the multiple tests are approximated using a permutation method. The empirical properties of the proposed robust tests are studied in simulations. An application is provided using miRNA expression data from a breast cancer study.

Efficient Stratified Testing Procedure for a False Discovery Rate

The false discovery rate (FDR) has become a popular error measure in the large-scale simultaneous testing. When data are collected from heterogenous sources and form grouped hypotheses testing, it may be beneficial to use the distinct feature of groups to conduct the multiple hypotheses testing. We propose a stratified testing procedure that uses different FDR levels according to the stratification features based on p-values. Our proposed method is easy to implement in practice. Simulations studies show that the proposed method produces more efficient testing results. The stratified testing procedure minimizes the overall false negative rate (FNR) level, while controlling the overall FDR. An example from a type II diabetes mice study further illustrates the practical advantages of this new approach.

RNA-Seq using two populations reveals genes and alleles controlling wood traits and growth in Eucalyptus nitens.

Eucalyptus nitens is a perennial forest tree species grown mainly for kraft pulp production in many parts of the world. Kraft pulp yield (KPY) is a key determinant of plantation profitability and increasing the KPY of trees grown in plantations is a major breeding objective. To speed up the breeding process, molecular markers that can predict KPY are desirable. To achieve this goal, we carried out RNA-Seq studies on trees at extremes of KPY in two different trials to identify genes and alleles whose expression correlated with KPY. KPY is positively correlated with growth measured as diameter at breast height (DBH) in both trials. In total, six RNA bulks from two treatments were sequenced on an Illumina HiSeq platform. At 5% false discovery rate level, 3953 transcripts showed differential expression in the same direction in both trials; 2551 (65%) were down-regulated and 1402 (35%) were up-regulated in low KPY samples. The genes up-regulated in low KPY trees were largely involved in biotic and abiotic stress response reflecting the low growth among low KPY trees. Genes down-regulated in low KPY trees mainly belonged to gene categories involved in wood formation and growth. Differential allelic expression was observed in 2103 SNPs (in 1068 genes) and of these 640 SNPs (30%) occurred in 313 unique genes that were also differentially expressed. These SNPs may represent the cis-acting regulatory variants that influence total gene expression. In addition we also identified 196 genes which had Ka/Ks ratios greater than 1.5, suggesting that these genes are under positive selection. Candidate genes and alleles identified in this study will provide a valuable resource for future association studies aimed at identifying molecular markers for KPY and growth.

Empirical Multidimensional Space for Scoring Peptide Spectrum Matches in Shotgun Proteomics

Data-dependent tandem mass spectrometry (MS/MS) is one of the main techniques for protein identification in shotgun proteomics. In a typical LC-MS/MS workflow, peptide product ion mass spectra (MS/MS spectra) are compared with those derived theoretically from a protein sequence database. Scoring of these matches results in peptide identifications. A set of peptide identifications is characterized by false discovery rate (FDR), which determines the fraction of false identifications in the set. The total number of peptides targeted for fragmentation is in the range of 10,000 to 20,000 for a several-hour LC-MS/MS run. Typically, <50% of these MS/MS spectra result in peptide-spectrum matches (PSMs). A small fraction of PSMs pass the preset FDR level (commonly 1%) giving a list of identified proteins, yet a large number of correct PSMs corresponding to the peptides originally present in the sample are left behind in the "grey area" below the identity threshold. Following the numerous efforts to recover these correct PSMs, here we investigate the utility of a scoring scheme based on the multiple PSM descriptors available from the experimental data. These descriptors include retention time, deviation between experimental and theoretical mass, number of missed cleavages upon in-solution protein digestion, precursor ion fraction (PIF), PSM count per sequence, potential modifications, median fragment mass error, (13)C isotope mass difference, charge states, and number of PSMs per protein. The proposed scheme utilizes a set of metrics obtained for the corresponding distributions of each of the descriptors. We found that the proposed PSM scoring algorithm differentiates equally or more efficiently between correct and incorrect identifications compared with existing postsearch validation approaches.

False discovery rate control under Archimedean copula

We are concerned with the false discovery rate (FDR) of the linear step-up test $\varphi^{LSU}$ considered by Benjamini and Hochberg (1995). It is well known that $\varphi^{LSU}$ controls the FDR at level $m_{0}q/m$ if the joint distribution of $p$-values is multivariate totally positive of order $2$. In this, $m$ denotes the total number of hypotheses, $m_{0}$ the number of true null hypotheses, and $q$ the nominal FDR level. Under the assumption of an Archimedean $p$-value copula with completely monotone generator, we derive a sharper upper bound for the FDR of $\varphi^{LSU}$ as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean $p$-value copulae allows us to increase the power of $\varphi^{LSU}$ by pre-estimating the copula parameter and adjusting $q$. Based on the lower bound, a sufficient condition is obtained under which the FDR of $\varphi^{LSU}$ is exactly equal to $m_{0}q/m$, as in the case of stochastically independent $p$-values. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by drawing a connection between infinite sequences of exchangeable $p$-values and Archimedean copulae with completely monotone generators. Our theoretical results are applied to important copula families, including Clayton copulae and Gumbel-Hougaard copulae.

Genome-wide temporal responses of human neuroblastoma SH-SY5Y cells to MPP+ neurotoxicity

The development of a stable and reliable dopaminergic (DAergic) neuronal injury model is particularly necessary for studying the pathogenesis of Parkinson’s disease (PD) and developing therapeutic strategies. As SH-SY5Y cells express the dopamine transporter (DAT), a protein expressed in DAergic neurons within the central nervous system, SH-SY5Y cells has been widely used in studies of mechanisms of 1-methyl-4-phenyl-pyridinium (MPP+)-induced neurotoxicity and the pathogenesis underlying MPP+-induced PD mimics. However, the genome-wide time dependent gene expression change of neuroblastoma SH-SY5Y cells by MPP+ toxicity has not been characterized. Here, we carried out a time series microarray experiment consisting of eight time points including time 0 (control) and 0.5, 1.5, 3, 6, 9, 12, and 24 h after MPP+ treatment using IIlumina human HT-12 expression v.4 bead array. The differentially expressed genes from time series microarray data were identified using the EDGE (Extraction of Differential Gene Expression) software package. In this study, 79 and 838 genes are considered significant at a 0.1% and 1% false-discovery rate level, respectively. Gene ontology (GO) analysis of these genes was carried out with the GOrilla web-server interactive software. GO terms such as the mitochondrial part (GO:0044429), the protein-DNA complex assembly (GO:0065004), and the nucleosome assembly (GO:0006334) had significant enrichment scores. Collectively, the information of time dependent gene expression might provide insight into the regulatory mechanisms induced by MPP+ toxicity in human neuroblastoma SH-SY5Y cells.

Improved detection of common variants associated with schizophrenia and bipolar disorder using pleiotropy-informed conditional false discovery rate.

Several lines of evidence suggest that genome-wide association studies (GWAS) have the potential to explain more of the “missing heritability” of common complex phenotypes. However, reliable methods to identify a larger proportion of single nucleotide polymorphisms (SNPs) that impact disease risk are currently lacking. Here, we use a genetic pleiotropy-informed conditional false discovery rate (FDR) method on GWAS summary statistics data to identify new loci associated with schizophrenia (SCZ) and bipolar disorders (BD), two highly heritable disorders with significant missing heritability. Epidemiological and clinical evidence suggest similar disease characteristics and overlapping genes between SCZ and BD. Here, we computed conditional Q–Q curves of data from the Psychiatric Genome Consortium (SCZ; n = 9,379 cases and n = 7,736 controls; BD: n = 6,990 cases and n = 4,820 controls) to show enrichment of SNPs associated with SCZ as a function of association with BD and vice versa with a corresponding reduction in FDR. Applying the conditional FDR method, we identified 58 loci associated with SCZ and 35 loci associated with BD below the conditional FDR level of 0.05. Of these, 14 loci were associated with both SCZ and BD (conjunction FDR). Together, these findings show the feasibility of genetic pleiotropy-informed methods to improve gene discovery in SCZ and BD and indicate overlapping genetic mechanisms between these two disorders.

MULTIPLE TESTING OF LOCAL MAXIMA FOR DETECTION OF PEAKS IN CHIP-SEQ DATA.

A topological multiple testing approach to peak detection is proposed for the problem of detecting transcription factor binding sites in ChIP-Seq data. After kernel smoothing of the tag counts over the genome, the presence of a peak is tested at each observed local maximum, followed by multiple testing correction at the desired false discovery rate level. Valid p-values for candidate peaks are computed via Monte Carlo simulations of smoothed Poisson sequences, whose background Poisson rates are obtained via linear regression from a Control sample at two different scales. The proposed method identifies nearby binding sites that other methods do not.

Improved Detection of Common Variants Associated with Schizophrenia by Leveraging Pleiotropy with Cardiovascular-Disease Risk Factors

Several lines of evidence suggest that genome-wide association studies (GWASs) have the potential to explain more of the "missing heritability" of common complex phenotypes. However, reliable methods for identifying a larger proportion of SNPs are currently lacking. Here, we present a genetic-pleiotropy-informed method for improving gene discovery with the use of GWAS summary-statistics data. We applied this methodology to identify additional loci associated with schizophrenia (SCZ), a highly heritable disorder with significant missing heritability. Epidemiological and clinical studies suggest comorbidity between SCZ and cardiovascular-disease (CVD) risk factors, including systolic blood pressure, triglycerides, low- and high-density lipoprotein, body mass index, waist-to-hip ratio, and type 2 diabetes. Using stratified quantile-quantile plots, we show enrichment of SNPs associated with SCZ as a function of the association with several CVD risk factors and a corresponding reduction in false discovery rate (FDR). We validate this "pleiotropic enrichment" by demonstrating increased replication rate across independent SCZ substudies. Applying the stratified FDR method, we identified 25 loci associated with SCZ at a conditional FDR level of 0.01. Of these, ten loci are associated with both SCZ and CVD risk factors, mainly triglycerides and low- and high-density lipoproteins but also waist-to-hip ratio, systolic blood pressure, and body mass index. Together, these findings suggest the feasibility of using genetic-pleiotropy-informed methods for improving gene discovery in SCZ and identifying potential mechanistic relationships with various CVD risk factors.

Some optimality properties of FDR controlling rules under sparsity

False Discovery Rate (FDR) and the Bayes risk are two different statistical measures, which can be used to evaluate and compare multiple testing procedures. Recent results show that under sparsity FDR controlling procedures, like the popular Benjamini-Hochberg (BH) procedure, perform also very well in terms of the Bayes risk. In particular asymptotic Bayes optimality under sparsity (ABOS) of BH was shown previously for location and scale models based on log-concave densities. This article extends previous work to a substantially larger set of distributions of effect sizes under the alternative, where the alternative distribution of true signals does not change with the number of tests $m$, while the sample size $n$ slowly increases. ABOS of BH and the corresponding step-down procedure based on FDR levels proportional to $n^{-1/2}$ are proved. A simulation study shows that these asymptotic results are relevant already for relatively small values of $m$ and $n$. Apart from showing asymptotic optimality of BH, our results on the optimal FDR level provide a natural extension of the well known results on the significance levels of Bayesian tests.