The knockoff-based multiple testing setup of Barber & Candes (2015) for variable selection in multiple regression where sample size is as large as the number of explanatory variables is considered. The method of Benjamini & Hochberg (1995) based on ordinary least squares estimates of the regression coefficients is adjusted to the setup, transforming it to a valid p-value based false discovery rate controlling method not relying on any specific correlation structure of the explanatory variables. Simulations and real data applications show that our proposed method that is agnostic to π0, the proportion of unimportant explanatory variables, and a data-adaptive version of it that uses an estimate of π0 are powerful competitors of the false discovery rate controlling method in Barber & Candes (2015).
We tackle the problem of the estimation of a vector of means from a single vector-valued observation $y$. Whereas previous work reduces the size of the estimates for the largest (absolute) sample elements via shrinkage (like James-Stein) or biases estimated via empirical Bayes methodology, we take a novel approach. We adapt recent developments by Lee et al (2013) in post selection inference for the Lasso to the orthogonal setting, where sample elements have different underlying signal sizes. This is exactly the setup encountered when estimating many means. It is shown that other selection proc…
This paper is a review of the popular Benjamini Hochberg Method and other related useful methods of Multiple Hypothesis testing. This is written with the purpose of serving a short but complete easy to understand review of the main article with proper background. The paper titled 'Controlling the False Discovery Rate-a practical and powerful Approach to multiple Testing' by benjamini et. al.[1] proposes a new framework of controlling the False Discovery Rate in a Multiple Hypothesis testing problem. It has been claimed that the procedure proposed in the paper results in a substantial gain in p…
We address a common problem in large-scale data analysis, and especially the field of genetics, the huge-scale testing problem, where millions to billions of hypotheses are tested together creating a computational challenge to perform multiple hypotheses testing procedures. As a solution we propose an alternative algorithm to the well used Linear Step Up procedure of Benjamini and Hochberg (1995). Our algorithm requires linear time and does not require any p-value ordering. It permits separating huge-scale testing problems arbitrarily into computationally feasible sets or chunks. Results from …
Multiple testing with discrete p-values routinely arises in various scientific endeavors. However, procedures, including the false discovery rate (FDR) controlling Benjamini-Hochberg (BH) procedure, often used in such settings, being developed originally for p-values with continuous distributions, are too conservative, and so may not be as powerful as one would hope for. Therefore, improving the BH procedure by suitably adapting it to discrete p-values without losing its FDR control is currently an important path of research. This paper studies the FDR control of the BH procedure when it is ap…
The introduction of the false discovery rate (FDR) by Benjamini and Hochberg has spurred a great interest in developing methodologies to control the FDR in various settings. The majority of existing approaches, however, address the FDR control for the case where an appropriate univariate test statistic is available. Modern hypothesis testing and data integration applications, on the other hand, routinely involve multivariate test statistics. The goal, in such settings, is to combine the evidence for each hypothesis and achieve greater power, while controlling the number of false discoveries. T…
In 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 adapts to the amount of signal present in the data. 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 SABHA, the "structure-adaptive Benjamini-Hochberg algorithm", as a generalization of these adaptive testing methods. SABHA incorporates pri…
We discuss several approaches to defining power in studies designed around the Benjamini-Hochberg (BH) false discovery rate (FDR) procedure. We focus primarily on the \textit{average power} and the $λ$-\textit{power}, which are the expected true positive fraction and the probability that the true positive fraction exceeds $λ$, respectively. We prove results concerning strong consistency and asymptotic normality for the positive call fraction (PCF), the true positive fraction (TPF) and false discovery fraction (FDF). Convergence of their corresponding expected values, including a convergence re…
Since Benjamini and Hochberg introduced false discovery rate (FDR) in their seminal paper, this has become a very popular approach to the multiple comparisons problem. An increasingly popular topic within functional data analysis is local inference, i.e., the continuous statistical testing of a null hypothesis along the domain. The principal issue in this topic is the infinite amount of tested hypotheses, which can be seen as an extreme case of the multiple comparisons problem. In this paper we define and discuss the notion of false discovery rate in a very general functional data setting. Mor…
Efron et al. (2001) proposed empirical Bayes formulation of the frequentist Benjamini and Hochbergs False Discovery Rate method (Benjamini and Hochberg,1995). This article attempts to unify the `two cultures' using concepts of comparison density and distribution function. We have also shown how almost all of the existing local fdr methods can be viewed as proposing various model specification for comparison density - unifies the vast literature of false discovery methods under one concept and notation.
Benjamini and Hochberg (1995) proposed the false discovery rate (FDR) as an alternative to the family-wise error rate in multiple testing problems, and proposed a procedure to control the FDR. For discrete data this procedure may be highly conservative. We investigate alternative, more powerful, procedures that exploit the discreteness of the tests and have FDR levels closer in magnitude to the desired nominal level. Moreover, we develop a novel step-down procedure that dominates the step-down procedure of Benjamini and Liu (1999) for discrete data. We consider an application to pharmacovigila…
Controlling the false discovery rate in high-dimensional linear models using model-X knockoffs and $p$-values
Southwestern University of Finance and Economics（西南财经大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Temple University（天普大学）；Tongji University（同济大学）
The generalized linear models (GLM) have been widely used in practice to model non-Gaussian response variables. When the number of explanatory features is relatively large, scientific researchers are of interest to perform controlled feature selection in order to simplify the downstream analysis. This paper introduces a new framework for feature selection in GLMs that can achieve false discovery rate (FDR) control in two asymptotic regimes. The key step is to construct a mirror statistic to measure the importance of each feature, which is based upon two (asymptotically) independent estimates o…
We introduce a new class of methods for finite-sample false discovery rate (FDR) control in multiple testing problems with dependent test statistics where the dependence is fully or partially known. Our approach separately calibrates a data-dependent p-value rejection threshold for each hypothesis, relaxing or tightening the threshold as appropriate to target exact FDR control. In addition to our general framework we propose a concrete algorithm, the dependence-adjusted Benjamini-Hochberg (dBH) procedure, which adaptively thresholds the q-value for each hypothesis. Under positive regression de…
Closed testing procedures are classically used for familywise error rate (FWER) control, but they can also be used to obtain simultaneous confidence bounds for the false discovery proportion (FDP) in all subsets of the hypotheses. In this paper we investigate the special case of closed testing with Simes local tests. We construct a novel fast and exact shortcut which we use to investigate the power of this method when the number of hypotheses goes to infinity. We show that, if a minimal amount of signal is present, the average power to detect false hypotheses at any desired FDP level does not …
University of Southern California（南加州大学）；Stanford University（斯坦福大学）
Multiple hypothesis testing is a core problem in statistical inference and arises in almost every scientific field. Given a set of null hypotheses $\mathcal{H}(n) = (H_1,\dotsc, H_n)$, Benjamini and Hochberg introduced the false discovery rate (FDR), which is the expected proportion of false positives among rejected null hypotheses, and proposed a testing procedure that controls FDR below a pre-assigned significance level. Nowadays FDR is the criterion of choice for large scale multiple hypothesis testing. In this paper we consider the problem of controlling FDR in an "online manner". Concrete…
Multiple hypotheses testing is a core problem in statistical inference and arises in almost every scientific field. Given a sequence of null hypotheses $\mathcal{H}(n) = (H_1,..., H_n)$, Benjamini and Hochberg \cite{benjamini1995controlling} introduced the false discovery rate (FDR) criterion, which is the expected proportion of false positives among rejected null hypotheses, and proposed a testing procedure that controls FDR below a pre-assigned significance level. They also proposed a different criterion, called mFDR, which does not control a property of the realized set of tests; rather it …
In this article, we propose a generalized weighted version of the well-known Benjamini-Hochberg (BH) procedure. The rigorous weighting scheme used by our method enables it to encode structural information from simultaneous multi-way classification as well as hierarchical partitioning of hypotheses into groups, with provisions to accommodate overlapping groups. The method is proven to control the False Discovery Rate (FDR) when the p-values involved are Positively Regression Dependent on the Subset (PRDS) of null p-values. A data-adaptive version of the method is proposed. Simulations show that…
One important partition of algorithms for controlling the false discovery rate (FDR) in multiple testing is into offline and online algorithms. The first generally achieve significantly higher power of discovery, while the latter allow making decisions sequentially as well as adaptively formulating hypotheses based on past observations. Using existing methodology, it is unclear how one could trade off the benefits of these two broad families of algorithms, all the while preserving their formal FDR guarantees. To this end, we introduce $\text{Batch}_{\text{BH}}$ and $\text{Batch}_{\text{St-BH}}…
Multiple hypothesis testing is a central topic in statistics, but despite abundant work on the false discovery rate (FDR) and the corresponding Type-II error concept known as the false non-discovery rate (FNR), a fine-grained understanding of the fundamental limits of multiple testing has not been developed. Our main contribution is to derive a precise non-asymptotic tradeoff between FNR and FDR for a variant of the generalized Gaussian sequence model. Our analysis is flexible enough to permit analyses of settings where the problem parameters vary with the number of hypotheses $n$, including v…
Online testing procedures aim to control the extent of false discoveries over a sequence of hypothesis tests, allowing for the possibility that early-stage test results influence the choice of hypotheses to be tested in later stages. Typically, online methods assume that a permanent decision regarding the current test (reject or not reject) must be made before advancing to the next test. We instead assume that each hypothesis requires an immediate preliminary decision, but also allows us to update that decision until a preset deadline. Roughly speaking, this lets us apply a Benjamini-Hochberg-…
We propose a general and flexible procedure for testing multiple hypotheses about sequential (or streaming) data that simultaneously controls both the false discovery rate (FDR) and false nondiscovery rate (FNR) under minimal assumptions about the data streams which may differ in distribution, dimension, and be dependent. All that is needed is a test statistic for each data stream that controls the conventional type I and II error probabilities, and no information or assumptions are required about the joint distribution of the statistics or data streams. The procedure can be used with sequenti…
Applying Benjamini and Hochberg (B-H) method to multiple Student's $t$ tests is a popular technique in gene selection in microarray data analysis. Because of the non-normality of the population, the true p-values of the hypothesis tests are typically unknown. Hence, it is common to use the standard normal distribution N(0,1), Student's $t$ distribution $t_{n-1}$ or the bootstrap method to estimate the p-values. In this paper, we first study N(0,1) and $t_{n-1}$ calibrations. We prove that, when the population has the finite 4-th moment and the dimension $m$ and the sample size $n$ satisfy $\lo…
MicroRNAs (miRNAs) are small non-coding RNAs that function as regulators of gene expression. In recent years, there has been a tremendous and growing interest among researchers to investigate the role of miRNAs in normal cellular as well as in disease processes. Thus to investigate the role of miRNAs in oral cancer, we analyse the expression levels of miRNAs to identify miRNAs with statistically significant differential expression in cancer tissues. In this article, we propose a novel Bayesian hierarchical model of miRNA expression data. Compelling evidences have demonstrated that the transcri…
Bringing Closure to False Discovery Rate Control: A General Principle for Multiple Testing
Carnegie Mellon University（卡内基梅隆大学）；Ca’ Foscari University of Venice（威尼斯福斯卡里宫大学）；University of Bremen（不来梅大学）；University of Twente（特温特大学）；Centrum Wiskunde & Informatica（荷兰数学与计算机科学研究中心）；Leiden University Medical Center（莱顿大学医学中心）
Modern biomedical research frequently involves testing multiple related hypotheses, while maintaining control over a suitable error rate. In many applications the false discovery rate (FDR), which is the expected proportion of false positives among the rejected hypotheses, has become the standard error criterion. Procedures that control the FDR, such as the well-known Benjamini-Hochberg procedure, assume that all p-values are available to be tested at a single time point. However, this ignores the sequential nature of many biomedical experiments, where a sequence of hypotheses is tested withou…
We consider large-scale studies in which thousands of significance tests are performed simultaneously. In some of these studies, the multiple testing procedure can be severely biased by latent confounding factors such as batch effects and unmeasured covariates that correlate with both primary variable(s) of interest (e.g. treatment variable, phenotype) and the outcome. Over the past decade, many statistical methods have been proposed to adjust for the confounders in hypothesis testing. We unify these methods in the same framework, generalize them to include multiple primary variables and multi…
Everywhere Valid Bounds on False Discovery Proportions in Conformal Inference
Department of Statistics, Stanford University（斯坦福大学统计学系）；Department of Statistics and Data Science, University of Pennsylvania（宾夕法尼亚大学统计学与数据科学系）；Department of Mathematics, Stanford University（斯坦福大学数学系）
The False Discovery Rate (FDR) is a commonly used type I error rate in multiple testing problems. It is defined as the expected False Discovery Proportion (FDP), that is, the expected fraction of false positives among rejected hypotheses. When the hypotheses are independent, the Benjamini-Hochberg procedure achieves FDR control at any pre-specified level. By construction, FDR control offers no guarantee in terms of power, or type II error. A number of alternative procedures have been developed, including plug-in procedures that aim at gaining power by incorporating an estimate of the proportio…
