Abstract: Investigation of the genetic architecture of gene expression traits has aided interpretation of disease and trait-associated genetic variants; however, key aspects of expression quantitative trait loci (eQTL) study design and analysis remain understudied. We used extensive, empirically driven simulations to explore eQTL study design and the performance of various analysis strategies. Across multiple testing correction methods, false discoveries of genes with eQTLs (eGenes) were substantially inflated when false discovery rate (FDR) control was applied to all tests and only appropriately controlled using hierarchical procedures. All multiple testing correction procedures had low power and inflated FDR for eGenes whose causal SNPs had small allele frequencies using small sample sizes (e.g. frequency 25%). Overestimation of eQTL effect sizes, so-called 'Winner's Curse', was common in low and moderate power settings. To address this, we developed a bootstrap method (BootstrapQTL) that led to more accurate effect size estimation. These insights provide a foundation for future eQTL studies, especially those with sampling constraints and subtly different conditions.

Abstract: Modern scientific studies from many diverse areas of research abound with multiple hypothesis testing concerns. The false discovery rate (FDR) is one of the most commonly used approaches for measuring and controlling error rates when performing multiple tests. Adaptive FDRs rely on an estimate of the proportion of null hypotheses among all the hypotheses being tested. This proportion is typically estimated once for each collection of hypotheses. Here, we propose a regression framework to estimate the proportion of null hypotheses conditional on observed covariates. This may then be used as a multiplication factor with the Benjamini-Hochberg adjusted p-values, leading to a plug-in FDR estimator. We apply our method to a genome-wise association meta-analysis for body mass index. In our framework, we are able to use the sample sizes for the individual genomic loci and the minor allele frequencies as covariates. We further evaluate our approach via a number of simulation scenarios. We provide an implementation of this novel method for estimating the proportion of null hypotheses in a regression framework as part of the Bioconductor package swfdr.

Abstract: Verifying that a statistically significant result is scientifically meaningful is not only good scientific practice, it is a natural way to control the Type I error rate Here we introduce a novel extension of the p-value—a second-generation p-value (pδ)–that formally accounts for scientific relevance and leverages this natural Type I Error control The approach relies on a pre-specified interval null hypothesis that represents the collection of effect sizes that are scientifically uninteresting or are practically null The second-generation p-value is the proportion of data-supported hypotheses that are also null hypotheses As such, second-generation p-values indicate when the data are compatible with null hypotheses (pδ = 1), or with alternative hypotheses (pδ = 0), or when the data are inconclusive (0 < pδ < 1) Moreover, second-generation p-values provide a proper scientific adjustment for multiple comparisons and reduce false discovery rates This is an advance for environments rich in data, where traditional p-value adjustments are needlessly punitive Second-generation p-values promote transparency, rigor and reproducibility of scientific results by a priori specifying which candidate hypotheses are practically meaningful and by providing a more reliable statistical summary of when the data are compatible with alternative or null hypotheses

Abstract: Threshold selection is a critical issue for extreme value analysis with threshold-based approaches. Under suitable conditions, exceedances over a high threshold have been shown to follow the generalized Pareto distribution (GPD) asymptotically. In practice, however, the threshold must be chosen. If the chosen threshold is too low, the GPD approximation may not hold and bias can occur. If the threshold is chosen too high, reduced sample size increases the variance of parameter estimates. To process batch analyses, commonly used selection methods such as graphical diagnostics are subjective and cannot be automated. We develop an efficient technique to evaluate and apply the Anderson–Darling test to the sample of exceedances above a fixed threshold. In order to automate threshold selection, this test is used in conjunction with a recently developed stopping rule that controls the false discovery rate in ordered hypothesis testing. Previous attempts in this setting do not account for the issue of ordered multiple testing. The performance of the method is assessed in a large scale simulation study that mimics practical return level estimation. This procedure was repeated at hundreds of sites in the western US to generate return level maps of extreme precipitation.

Abstract: Motivated by applications in genomics, we consider in this paper global and multiple testing for the comparisons of two high-dimensional linear regression models. A procedure for testing the equality of the two regression vectors globally is proposed and shown to be particularly powerful against sparse alternatives. We then introduce a multiple testing procedure for identifying unequal coordinates while controlling the false discovery rate and false discovery proportion. Theoretical justifications are provided to guarantee the validity of the proposed tests and optimality results are established under sparsity assumptions on the regression coefficients. The proposed testing procedures are easy to implement. Numerical properties of the procedures are investigated through simulation and data analysis. The results show that the proposed tests maintain the desired error rates under the null and have good power under the alternative at moderate sample sizes. The procedures are applied to the Framingham Offspring study to investigate the interactions between smoking and cardiovascular related genetic mutations important for an inflammation marker.

Abstract: High-dimensional logistic regression is widely used in analyzing data with binary outcomes. In this paper, global testing and large-scale multiple testing for the regression coefficients are considered in both single- and two-regression settings. A test statistic for testing the global null hypothesis is constructed using a generalized low-dimensional projection for bias correction and its asymptotic null distribution is derived. A lower bound for the global testing is established, which shows that the proposed test is asymptotically minimax optimal over some sparsity range. For testing the individual coefficients simultaneously, multiple testing procedures are proposed and shown to control the false discovery rate (FDR) and falsely discovered variables (FDV) asymptotically. Simulation studies are carried out to examine the numerical performance of the proposed tests and their superiority over existing methods. The testing procedures are also illustrated by analyzing a data set of a metabolomics study that investigates the association between fecal metabolites and pediatric Crohn's disease and the effects of treatment on such associations.

Abstract: The tree-based scan statistic is a statistical data mining tool that has been used for signal detection with a self-controlled design in vaccine safety studies. This disproportionality statistic adjusts for multiple testing in evaluation of thousands of potential adverse events. However, many drug safety questions are not well suited for self-controlled analysis. We propose a method that combines tree-based scan statistics with propensity score-matched analysis of new initiator cohorts, a robust design for investigations of drug safety. We conducted plasmode simulations to evaluate performance. In multiple realistic scenarios, tree-based scan statistics in cohorts that were propensity score matched to adjust for confounding outperformed tree-based scan statistics in unmatched cohorts. In scenarios where confounding moved point estimates away from the null, adjusted analyses recovered the prespecified type 1 error while unadjusted analyses inflated type 1 error. In scenarios where confounding moved point estimates toward the null, adjusted analyses preserved power, whereas unadjusted analyses greatly reduced power. Although complete adjustment of true confounders had the best performance, matching on a moderately mis-specified propensity score substantially improved type 1 error and power compared with no adjustment. When there was true elevation in risk of an adverse event, there were often co-occurring signals for clinically related concepts. TreeScan with propensity score matching shows promise as a method for screening and prioritization of potential adverse events. It should be followed by clinical review and safety studies specifically designed to quantify the magnitude of effect, with confounding control targeted to the outcome of interest.

Abstract: We present false discovery rate (FDR) smoothing, an empirical-Bayes method for exploiting spatial structure in large multiple-testing problems. FDR smoothing automatically finds spatially localized regions of significant test statistics. It then relaxes the threshold of statistical significance within these regions, and tightens it elsewhere, in a manner that controls the overall false discovery rate at a given level. This results in increased power and cleaner spatial separation of signals from noise. The approach requires solving a nonstandard high-dimensional optimization problem, for which an efficient augmented-Lagrangian algorithm is presented. In simulation studies, FDR smoothing exhibits state-of-the-art performance at modest computational cost. In particular, it is shown to be far more robust than existing methods for spatially dependent multiple testing. We also apply the method to a dataset from an fMRI experiment on spatial working memory, where it detects patterns that are much more b...

Abstract: While traditional multiple testing procedures prohibit adaptive analysis choices made by users, Goeman and Solari (2011) proposed a simultaneous inference framework that allows users such flexibility while preserving high-probability bounds on the false discovery proportion (FDP) of the chosen set. In this paper, we propose a new class of such simultaneous FDP bounds, tailored for nested sequences of rejection sets. While most existing simultaneous FDP bounds are based on closed testing using global null tests based on sorted p-values, we additionally consider the setting where side information can be leveraged to boost power, the variable selection setting where knockoff statistics can be used to order variables, and the online setting where decisions about rejections must be made as data arrives. Our finite-sample, closed form bounds are based on repurposing the FDP estimates from false discovery rate (FDR) controlling procedures designed for each of the above settings. These results establish a novel connection between the parallel literatures of simultaneous FDP bounds and FDR control methods, and use proof techniques employing martingales and filtrations that are new to both these literatures. We demonstrate the utility of our results by augmenting a recent knockoffs analysis of the UK Biobank dataset.

Abstract: 1.: The joint analysis of species’ evolutionary relatedness and their morphological evolution has offered much promise in understanding the processes that underpin the generation of biological diversity. 2.: Disparity through time (DTT) is a popular method that estimates the relative trait disparity within and between subclades, and compares this to the null hypothesis that trait values follow Brownian evolution along the time‐calibrated phylogenetic tree. To visualise the differences a confidence envelope is normally created by calculating, at every time point, the 97.5% minimum and 97.5% maximum disparity values from multiple simulations of the null model. The null hypothesis is rejected whenever the empirical DTT curve falls outside of this envelope, and these time periods may then be linked to events that may have sparked non‐random trait evolution. 3.: Here, simulated data are used to show this pointwise (ranking at each time point) method of envelope construction suffers from multiple testing and a poor, uncontrolled, false‐positive rate. As a consequence it cannot be recommended. Instead, each DTT curve can be given a single rank based upon their most extreme disparity value, relative to all other curves, and across all time points. Ordering curves this way leads to a test that avoids multiple testing, but still allows construction of a confidence envelope. The null hypothesis is rejected if the empirical DTT curve is ranked within the most extreme 5% ranked curves from the null model. Comparison of the rank envelope curve to the Morphological Disparity Index and Node Height tests shows it to have generally higher power to detect non‐Brownian trait evolution. An extension to allow simultaneous testing over multiple traits is also detailed. 4.: Overall the results suggest the new rank envelope test should be used in null model testing for DTT analyses. The rank envelope method can easily be adapted into recently developed posterior predictive simulation methods used in model selection analyses. More generally, the rank envelope test should be adopted whenever a null model produces a vector of correlated values and the user wants to determine where the empirical data are different to the null model.

Abstract: 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.

Abstract: Making accurate inference for gene regulatory networks, including inferring about pathway-by-pathway interactions, is an important and difficult task. Motivated by such genomic applications, we consider multiple testing for conditional dependence between subgroups of variables. Under a Gaussian graphical model framework, the problem is translated into simultaneous testing for a collection of submatrices of a high-dimensional precision matrix with each submatrix summarizing the dependence structure between two subgroups of variables.A novel multiple testing procedure is proposed and both theoretical and numerical properties of the procedure are investigated. Asymptotic null distribution of the test statistic for an individual hypothesis is established and the proposed multiple testing procedure is shown to asymptotically control the false discovery rate (FDR) and false discovery proportion (FDP) at the prespecified level under regularity conditions. Simulations show that the procedure works well in...

Abstract: Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges. This paper gives a selective overview on recent advance on high-dimensional factor models and their applications to statistics including Factor-Adjusted Robust Model selection (FarmSelect) and Factor-Adjusted Robust Multiple testing (FarmTest). We show that classical methods, especially principal component analysis (PCA), can be tailored to many new problems and provide powerful tools for statistical estimation and inference. We highlight PCA and its connections to matrix perturbation theory, robust statistics, random projection, false discovery rate, etc., and illustrate through several applications how insights from these fields yield solutions to modern challenges. We also present far-reaching connections between factor models and popular statistical learning problems, including network analysis and low-rank matrix recovery.

Abstract: Thousands of papers using resting-state functional magnetic resonance imaging (RS-fMRI) have been published on brain disorders. Results in each paper may have survived correction for multiple comparison. However, since there have been no robust results from large scale meta-analysis, we do not know how many of published results are truly positives. The present meta-analytic work included 60 original studies, with 57 studies (4 datasets, 2266 participants) that used a between-group design and 3 studies (1 dataset, 107 participants) that employed a within-group design. To evaluate the effect size of brain disorders, a very large neuroimaging dataset ranging from neurological to psychiatric disorders together with healthy individuals have been analyzed. Parkinson9s disease off levodopa (PD-off) included 687 participants from 15 studies. PD on levodopa (PD-on) included 261 participants from 9 studies. Autism spectrum disorder (ASD) included 958 participants from 27 studies. The meta-analyses of a metric named amplitude of low frequency fluctuation (ALFF) showed that the effect size (Hedges9g) was 0.19 - 0.39 for the 4 datasets using between-group design and 0.46 for the dataset using within-group design. The effect size of PD-off, PD-on and ASD were 0.23, 0.39, and 0.19, respectively. Using the meta-analysis results as the robust results, the between-group design results of each study showed high false negative rates (median 99%), high false discovery rates (median 86%), and low accuracy (median 1%), regardless of whether stringent or liberal multiple comparison correction was used. The findings were similar for 4 RS-fMRI metrics including ALFF, regional homogeneity, and degree centrality, as well as for another widely used RS-fMRI metric namely seed-based functional connectivity. These observations suggest that multiple comparison correction does not control for false discoveries across multiple studies when the effect sizes are relatively small. Meta-analysis on un-thresholded t-maps is critical for the recovery of ground truth. We recommend that to achieve high reproducibility through meta-analysis, the neuroimaging research field should share raw data or, at minimum, provide un-thresholded statistical images.

Abstract: We propose a new adaptive sampling approach to multiple testing which aims to maximize statistical power while ensuring anytime false discovery control. We consider $n$ distributions whose means are partitioned by whether they are below or equal to a baseline (nulls), versus above the baseline (true positives). In addition, each distribution can be sequentially and repeatedly sampled. Using techniques from multi-armed bandits, we provide an algorithm that takes as few samples as possible to exceed a target true positive proportion (i.e. proportion of true positives discovered) while giving anytime control of the false discovery proportion (nulls predicted as true positives). Our sample complexity results match known information theoretic lower bounds and through simulations we show a substantial performance improvement over uniform sampling and an adaptive elimination style algorithm. Given the simplicity of the approach, and its sample efficiency, the method has promise for wide adoption in the biological sciences, clinical testing for drug discovery, and maximization of click through in A/B/n testing problems.

Abstract: 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 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.

Abstract: In many multiple testing problems, the individual null hypotheses (i) concern univariate parameters and (ii) are one-sided. In such problems, power gains can be obtained for bootstrap multiple testing procedures in scenarios where some of the parameters are ‘deep in the null’ by making certain adjustments to the null distribution under which to resample. In this paper, we compare a Bonferroni adjustment that is based on finite-sample considerations with certain ‘asymptotic’ adjustments previously suggested in the literature.

Abstract: Multivariate regression with high-dimensional covariates has many applications in genomic and genetic research, in which some covariates are expected to be associated with multiple responses. This paper considers joint testing for regression coefficients over multiple responses and develops simultaneous testing methods with false discovery rate control. The test statistic is based on inverse regression and bias-corrected group lasso estimates of the regression coefficients and is shown to have an asymptotic chi-squared null distribution. A row-wise multiple testing procedure is developed to identify the covariates associated with the responses. The procedure is shown to control the false discovery proportion and false discovery rate at a prespecified level asymptotically. Simulations demonstrate the gain in power, relative to entrywise testing, in detecting the covariates associated with the responses. The test is applied to an ovarian cancer dataset to identify the microRNA regulators that regulate protein expression.

Abstract: Efforts to develop more efficient multiple hypothesis testing procedures for false discovery rate (FDR) control have focused on incorporating an estimate of the proportion of true null hypotheses (such procedures are called adaptive) or exploiting heterogeneity across tests via some optimal weighting scheme. This paper combines these approaches using a weighted adaptive multiple decision function (WAMDF) framework. Optimal weights for a flexible random effects model are derived and a WAMDF that controls the FDR for arbitrary weighting schemes when test statistics are independent under the null hypotheses is given. Asymptotic and numerical assessment reveals that, under weak dependence, the proposed WAMDFs provide more efficient FDR control even if optimal weights are misspecified. The robustness and flexibility of the proposed methodology facilitates the development of more efficient, yet practical, FDR procedures for heterogeneous data. To illustrate, two different weighted adaptive FDR methods for heterogeneous sample sizes are developed and applied to data.

Abstract: This paper proposes a simple nonparametric test of the hypothesis of no measurement error in explanatory variables and of the hypothesis that measurement error, if there is any, does not distort a given object of interest We show that, under weak assumptions, both of these hypotheses are equivalent to certain restrictions on the joint distribution of an observable outcome and two observable variables that are related to the latent explanatory variable Existing nonparametric tests for conditional independence can be used to directly test these restrictions without having to solve for the distribution of unobservables In consequence, the test controls size under weak conditions and possesses power against a large class of nonclassical measurement error models, including many that are not identified If the test detects measurement error, a multiple hypothesis testing procedure allows the researcher to recover subpopulations that are free from measurement error Finally, we use the proposed methodology to study the reliability of administrative earnings records in the US, finding evidence for the presence of measurement error originating from young individuals with high earnings growth (in absolute terms)

Abstract: The measurement of statistical evidence is of considerable current interest in fields where statistical criteria are used to determine knowledge. The most commonly used approach to measuring such e...

Abstract: Economists are often interested in identifying effective policies or treatments together with subpopulations of individuals who respond positively (or with a sign that is expected) to these treatment interventions. In this paper, we propose an optimal false discovery rate controlling method that is especially useful for such one‐sided testing problems. The proposed procedure is optimal in the sense of minimizing the false non‐discovery rate while controlling the false discovery rate at a pre‐specified level; it uses a deconvolution method based on non‐parametric maximum likelihood estimation, which allows for a broader class of treatment effect distributions than existing methods do. The proposed test demonstrates good small‐sample performance in Monte Carlo simulations and it is applied to study the effect of attending a more selective high school in Romania. The application reveals strong evidence of treatment effect heterogeneity, in that students who marginally gain access to higher‐ranked schools are more likely to benefit if the higher‐ranked school has a relatively high admission score cut‐off – or, in other words, is more selective.

Abstract: Multiple testing problems are a staple of modern statistical analysis. The fundamental objective of multiple testing procedures is to reject as many false null hypotheses as possible (that is, maximize some notion of power), subject to controlling an overall measure of false discovery, like family-wise error rate (FWER) or false discovery rate (FDR). In this paper we formulate multiple testing of simple hypotheses as an infinite-dimensional optimization problem, seeking the most powerful rejection policy which guarantees strong control of the selected measure. In that sense, our approach is a generalization of the optimal Neyman-Pearson test for a single hypothesis. We show that for exchangeable hypotheses, for both FWER and FDR and relevant notions of power, these problems can be formulated as infinite linear programs and can in principle be solved for any number of hypotheses. We also characterize maximin rules for complex alternatives, and demonstrate that such rules can be found in practice, leading to improved practical procedures compared to existing alternatives. We derive explicit optimal tests for FWER or FDR control for three independent normal means. We find that the power gain over natural competitors is substantial in all settings examined. Finally, we apply our optimal maximin rule to subgroup analyses in systematic reviews from the Cochrane library, leading to an increase in the number of findings while guaranteeing strong FWER control against the one sided alternative.