This paper provides two general classes of multiple decision functions where each member of the first class strongly controls the family-wise error rate (FWER), while each member of the second class strongly controls the false discovery rate (FDR). These classes offer the possibility that an optimal multiple decision function with respect to a pre-specified criterion, such as the missed discovery rate (MDR), could be found within these classes. Such multiple decision functions can be utilized in multiple testing, specifically, but not limited to, the analysis of high-dimensional microarray data sets.
In many applications of multiple hypothesis testing where more than one false rejection can be tolerated, procedures controlling error rates measuring at least $k$ false rejections, instead of at least one, for some fixed $k\ge 1$ can potentially increase the ability of a procedure to detect false null hypotheses. The $k$-FWER, a generalized version of the usual familywise error rate (FWER), is such an error rate that has recently been introduced in the literature and procedures controlling it have been proposed. A further generalization of a result on the $k$-FWER is provided in this article.…
Often in multiple testing, the hypotheses appear in non-overlapping blocks with the associated $p$-values exhibiting dependence within but not between blocks. We consider adapting the Benjamini-Hochberg method for controlling the false discovery rate (FDR) and the Bonferroni method for controlling the familywise error rate (FWER) to such dependence structure without losing their ultimate controls over the FDR and FWER, respectively, in a non-asymptotic setting. We present variants of conventional adaptive Benjamini-Hochberg and Bonferroni methods with proofs of their respective controls over t…
The concept of $k$-FWER has received much attention lately as an appropriate error rate for multiple testing when one seeks to control at least $k$ false rejections, for some fixed $k\ge 1$. A less conservative notion, the $k$-FDR, has been introduced very recently by Sarkar [Ann. Statist. 34 (2006) 394--415], generalizing the false discovery rate of Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289--300]. In this article, we bring newer insight to the $k$-FDR considering a mixture model involving independent $p$-values before motivating the developments of some new procedures…
A classical approach for dealing with the multiple testing problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of at least one false rejection. In many applications, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of $k$ or more false rejections, which is called the $k$-FWER. In this article, a unified approach is presen…
Familywise error rate (FWER) has been a cornerstone in simultaneous inference for decades, and the classical Bonferroni method has been one of the most prominent frequentist approaches for controlling FWER. The present article studies the limiting behavior of Bonferroni FWER in a multiple testing problem as the number of hypotheses grows to infinity. We establish that in the equicorrelated normal setup with positive equicorrelation, Bonferroni FWER tends to zero asymptotically. We extend this result for generalized familywise error rates and to arbitrarily correlated setups.
Familywise error rate (FWER) has been a cornerstone in simultaneous inference for decades, and the classical Bonferroni method has been one of the most prominent frequentist approaches for controlling FWER. The present article studies the behavior of the FWER for Bonferroni procedure in a multiple testing problem. We establish upper bounds on FWER for Bonferroni method under the equicorrelated and general normal setups in nonasymptotic case.
Consider the multiple testing problem of testing null hypotheses $H_1,...,H_s$. A classical approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate ($\mathit{FWER}$), the probability of even one false rejection. But if $s$ is large, control of the $\mathit{FWER}$ is so stringent that the ability of a procedure that controls the $\mathit{FWER}$ to detect false null hypotheses is limited. It is therefore desirable to consider other measures of error control. This article considers two generalizations of the $\mathit{FWER}$. …
In a multiple testing problem where one is willing to tolerate a few false rejections, procedure controlling the familywise error rate (FWER) can potentially be improved in terms of its ability to detect false null hypotheses by generalizing it to control the $k$-FWER, the probability of falsely rejecting at least $k$ null hypotheses, for some fixed $k>1$. Simes' test for testing the intersection null hypothesis is generalized to control the $k$-FWER weakly, that is, under the intersection null hypothesis, and Hochberg's stepup procedure for simultaneous testing of the individual null hypothes…
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 …
Stepwise multiple testing procedures have attracted several statisticians for decades and are also quite popular with statistics users because of their technical simplicity. The Bonferroni procedure has been one of the earliest and most prominent testing rules for controlling the familywise error rate (FWER). A recent article established that the FWER for the Bonferroni method asymptotically (i.e., when the number of hypotheses becomes arbitrarily large) approaches zero under any positively equicorrelated multivariate normal framework. However, similar results for the limiting behaviors of FWE…
Results on the false discovery rate (FDR) and the false nondiscovery rate (FNR) are developed for single-step multiple testing procedures. In addition to verifying desirable properties of FDR and FNR as measures of error rates, these results extend previously known results, providing further insights, particularly under dependence, into the notions of FDR and FNR and related measures. First, considering fixed configurations of true and false null hypotheses, inequalities are obtained to explain how an FDR- or FNR-controlling single-step procedure, such as a Bonferroni or uSidák procedure, can …
Recent results concerning asymptotic Bayes-optimality under sparsity (ABOS) of multiple testing procedures are extended to fairly generally distributed effect sizes under the alternative. An asymptotic framework is considered where both the number of tests m and the sample size m go to infinity, while the fraction p of true alternatives converges to zero. It is shown that under mild restrictions on the loss function nontrivial asymptotic inference is possible only if n increases to infinity at least at the rate of log m. Based on this assumption precise conditions are given under which the Bon…
Hypothesis testing is a key part of empirical science and multiple testing as well as the combination of evidence from several tests are continued areas of research. In this article we consider the problem of combining the results of multiple hypothesis tests to i) test global hypotheses and ii) make marginal inference while controlling the k-FWER. We propose a new family of combination tests for joint hypotheses, called the 'Too Many, Too Improbable' (TMTI) statistics, which we show through simulation to have higher power than other combination tests against many alternatives. Furthermore, we…
The $γ$-FDP and $k$-FWER multiple testing error metrics, which are tail probabilities of the respective error statistics, have become popular recently as less-stringent alternatives to the FDR and FWER. We propose general and flexible stepup and stepdown procedures for testing multiple hypotheses about sequential (or streaming) data that simultaneously control both the type I and II versions of $γ$-FDP, or $k$-FWER. The error control holds regardless of the dependence between data streams, which may be of arbitrary size and shape. All that is needed is a test statistic for each data stream tha…
We study the problem of identifying the set of \emph{active} variables, termed in the literature as \emph{variable selection} or \emph{multiple hypothesis testing}, depending on the pursued criteria. For a general \emph{robust setting} of non-normal, possibly dependent observations and a generalized notion of \emph{active set}, we propose a procedure that is used simultaneously for the both tasks, variable selection and multiple testing. The procedure is based on the \emph{risk hull minimization} method, but can also be obtained as a result of an empirical Bayes approach or a penalization stra…
In applications such as clinical safety analysis, the data of the experiments usually consists of frequency counts. In the analysis of such data, researchers often face the problem of multiple testing based on discrete test statistics, aimed at controlling family-wise error rate (FWER). Most existing FWER controlling procedures are developed for continuous data, which are often conservative when analyzing discrete data. By using minimal attainable $p$-values, several FWER controlling procedures have been specifically developed for discrete data in the literature. In this paper, by utilizing kn…
In many large scale multiple testing applications, the hypotheses often have a known graphical structure, such as gene ontology in gene expression data. Exploiting this graphical structure in multiple testing procedures can improve power as well as aid in interpretation. However, incorporating the structure into large scale testing procedures and proving that an error rate, such as the false discovery rate (FDR), is controlled can be challenging. In this paper, we introduce a new general approach for large scale multiple testing, which can aid in developing new procedures under various setting…
This paper is concerned with false discovery rate (FDR) control in large-scale multiple testing problems. We first propose a new data-driven testing procedure for controlling the FDR in large-scale t-tests for one-sample mean problem. The proposed procedure achieves exact FDR control in finite sample settings when the populations are symmetric no matter the number of tests or sample sizes. Comparing with the existing bootstrap method for FDR control, the proposed procedure is computationally efficient. We show that the proposed method can control the FDR asymptotically for asymmetric populatio…
Two recently introduced model based bias corrected estimators for proportion of true null hypotheses ($π_0$) under multiple hypotheses testing scenario have been restructured for exponentially distributed random observations available for each of the common hypotheses. Based on stochastic ordering, a new motivation behind formulation of some related estimators for $π_0$ is given. The reduction of bias for the model based estimators are theoretically justified and algorithms for computing the estimators are also presented. The estimators are also used to formulate a popular adaptive multiple te…
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 investigate asymptotically optimal multiple testing procedures for streams of sequential data in the context of prior information on the number of false null hypotheses ("signals"). We show that the "gap" and "gap-intersection" procedures, recently proposed and shown by Song and Fellouris (2017, Electron. J. Statist.) to be asymptotically optimal for controlling type 1 and 2 familywise error rates (FWEs), are also asymptotically optimal for controlling FDR/FNR when their critical values are appropriately adjusted. Generalizing this result, we show that these procedures, again with appropria…
We apply multiple testing procedures to the validation of estimated default probabilities in credit rating systems. The goal is to identify rating classes for which the probability of default is estimated inaccurately, while still maintaining a predefined level of committing type I errors as measured by the familywise error rate (FWER) and the false discovery rate (FDR). For FWER, we also consider procedures that take possible discreteness of the data resp. test statistics into account. The performance of these methods is illustrated in a simulation setting and for empirical default data.
This paper addresses the following general scenario: A scientist wishes to perform a battery of experiments, each generating a sequential stream of data, to investigate some phenomenon. The scientist would like to control the overall error rate in order to draw statistically-valid conclusions from each experiment, while being as efficient as possible. The between-stream data may differ in distribution and dimension but also may be highly correlated, even duplicated exactly in some cases. Treating each experiment as a hypothesis test and adopting the familywise error rate (FWER) metric, we give…
Penn State University（宾夕法尼亚州立大学）；Snap Inc.（Snap公司）
Controlling false discovery rate (FDR) while leveraging the side information of multiple hypothesis testing is an emerging research topic in modern data science. Existing methods rely on the test-level covariates while ignoring possible hierarchy among the covariates. This strategy may not be optimal for complex large-scale problems, where hierarchical information often exists among those test-level covariates. We propose NeurT-FDR which boosts statistical power and controls FDR for multiple hypothesis testing while leveraging the hierarchy among test-level covariates. Our method parametrizes …
We present a unifying approach to multiple testing procedures for sequential (or streaming) data by giving sufficient conditions for a sequential multiple testing procedure to control the familywise error rate (FWER), extending to the sequential domain the work of Goeman and Solari (2010) who accomplished this for fixed sample size procedures. Together we call these conditions the "rejection principle for sequential tests," which we then apply to some existing sequential multiple testing procedures to give simplified understanding of their FWER control. Next the principle is applied to derive …
The False Discovery Rate (FDR) paradigm aims to attain certain control on Type I errors with relatively high power for multiple hypothesis testing. The Benjamini--Hochberg (BH) procedure is a well-known FDR controlling procedure. Under a random effects model, we show that, in general, unlike the FDR, the positive FDR (pFDR) of the BH procedure cannot be controlled at an arbitrarily low level due to the limited evidence provided by the observations to separate false and true nulls. This results in a criticality phenomenon, which is characterized by a transition of the procedure's power from bei…
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）
Controlling the false discovery rate (FDR) is a powerful approach to multiple testing. In many applications, the tested hypotheses have an inherent hierarchical structure. In this paper, we focus on the fixed sequence structure where the testing order of the hypotheses has been strictly specified in advance. We are motivated to study such a structure, since it is the most basic of hierarchical structures, yet it is often seen in real applications such as statistical process control and streaming data analysis. We first consider a conventional fixed sequence method that stops testing once an ac…
We consider the problem of identifying whether findings replicate from one study of high dimension to another, when the primary study guides the selection of hypotheses to be examined in the follow-up study as well as when there is no division of roles into the primary and the follow-up study. We show that existing meta-analysis methods are not appropriate for this problem, and suggest novel methods instead. We prove that our multiple testing procedures control for appropriate error-rates. The suggested FWER controlling procedure is valid for arbitrary dependence among the test statistics with…
In this paper, a noisy version of the stochastic block model (NSBM) is introduced and we investigate the three following statistical inferences in this model: estimation of the model parameters, clustering of the nodes and identification of the underlying graph. While the two first inferences are done by using a variational expectation-maximization (VEM) algorithm, the graph inference is done by controlling the false discovery rate (FDR), that is, the average proportion of errors among the edges declared significant, and by maximizing the true discovery rate (TDR), that is, the average proport…
We investigate the performance of a family of multiple comparison procedures for strong control of the False Discovery Rate ($\mathsf{FDR}$). The $\mathsf{FDR}$ is the expected False Discovery Proportion ($\mathsf{FDP}$), that is, the expected fraction of false rejections among all rejected hypotheses. A number of refinements to the original Benjamini-Hochberg procedure [1] have been proposed, to increase power by estimating the proportion of true null hypotheses, either implicitly, leading to one-stage adaptive procedures [4, 7] or explicitly, leading to two-stage adaptive (or plug-in) proced…
The PC algorithm allows investigators to estimate a complete partially directed acyclic graph (CPDAG) from a finite dataset, but few groups have investigated strategies for estimating and controlling the false discovery rate (FDR) of the edges in the CPDAG. In this paper, we introduce PC with p-values (PC-p), a fast algorithm which robustly computes edge-specific p-values and then estimates and controls the FDR across the edges. PC-p specifically uses the p-values returned by many conditional independence tests to upper bound the p-values of more complex edge-specific hypothesis tests. The alg…
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…
We establish a generic theoretical tool to construct probabilistic bounds for algorithms where the output is a subset of objects from an initial pool of candidates (or more generally, a probability distribution on said pool). This general device, dubbed "Occam's hammer'', acts as a meta layer when a probabilistic bound is already known on the objects of the pool taken individually, and aims at controlling the proportion of the objects in the set output not satisfying their individual bound. In this regard, it can be seen as a non-trivial generalization of the "union bound with a prior'' ("Occa…
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…
False discovery rate (FDR) has been widely used as an error measure in large scale multiple testing problems, but most research in the area has been focused on procedures for controlling the FDR based on independent test statistics or the properties of such procedures for test statistics with certain types of stochastic dependence. Based on an approach proposed in Tang and Zhang (2005), we further develop in this paper empirical Bayes methods for controlling the FDR with dependent data. We implement our methodology in a time series model and report the results of a simulation study to demonstr…
In neuroimaging, hundreds to hundreds of thousands of tests are performed across a set of brain regions or all locations in an image. Recent studies have shown that the most common family-wise error (FWE) controlling procedures in imaging, which rely on classical mathematical inequalities or Gaussian random field theory, yield FWE rates that are far from the nominal level. Depending on the approach used, the FWER can be exceedingly small or grossly inflated. Given the widespread use of neuroimaging as a tool for understanding neurological and psychiatric disorders, it is imperative that reliab…
The Pennsylvania State University（宾夕法尼亚州立大学）；Weber State University（韦伯州立大学）；Snap Inc.（色拉布公司）；Duke University（杜克大学）
Controlling False Discovery Rate (FDR) while leveraging the side information of multiple hypothesis testing is an emerging research topic in modern data science. Existing methods rely on the test-level covariates while ignoring metrics about test-level covariates. This strategy may not be optimal for complex large-scale problems, where indirect relations often exist among test-level covariates and auxiliary metrics or covariates. We incorporate auxiliary covariates among test-level covariates in a deep Black-Box framework controlling FDR (named as NeurT-FDR) which boosts statistical power and …
Controlling false discovery rate (FDR) is crucial for variable selection, multiple testing, among other signal detection problems. In literature, there is certainly no shortage of FDR control strategies when selecting individual features, but the relevant works for structural change detection, such as profile analysis for piecewise constant coefficients and integration analysis with multiple data sources, are limited. In this paper, we propose a generalized knockoff procedure (GKnockoff) for FDR control under such problem settings. We prove that the GKnockoff possesses pairwise exchangeability…
Competition-based approach to controlling the false discovery rate (FDR) recently rose to prominence when, generalizing it to sequential hypothesis testing, Barber and Candès used it as part of their knockoff-filter. Control of the FDR implies that the, arguably more important, false discovery proportion is only controlled in an average sense. We present TDC-SB and TDC-UB that provide upper prediction bounds on the FDP in the list of discoveries generated when controlling the FDR using competition. Using simulated and real data we show that, overall, our new procedures offer significantly tigh…
False discovery rate (FDR) controlling procedures provide important statistical guarantees for the replicability in signal identification based on multiple hypotheses testing. In many fields of study, FDR controlling procedures are used in high-dimensional (HD) analyses to discover features that are truly associated with the outcome. In some recent applications, data on the same set of candidate features are independently collected in multiple different studies. For example, gene expression data are collected at different facilities and with different cohorts, to identify the genetic biomarker…
This article considers the problem of multiple hypothesis testing using $t$-tests. The observed data are assumed to be independently generated conditional on an underlying and unknown two-state hidden model. We propose an asymptotically valid data-driven procedure to find critical values for rejection regions controlling the $k$-familywise error rate ($k$-FWER), false discovery rate (FDR) and the tail probability of false discovery proportion (FDTP) by using one-sample and two-sample $t$-statistics. We only require a finite fourth moment plus some very general conditions on the mean and varian…
We present a novel method for controlling the $k$-familywise error rate ($k$-FWER) in the linear regression setting using the knockoffs framework first introduced by Barber and Candès. Our procedure, which we also refer to as knockoffs, can be applied with any design matrix with at least as many observations as variables, and does not require knowing the noise variance. Unlike other multiple testing procedures which act directly on $p$-values, knockoffs is specifically tailored to linear regression and implicitly accounts for the statistical relationships between hypothesis tests of different …
Multiple hypothesis testing, a situation when we wish to consider many hypotheses, is a core problem in statistical inference that arises in almost every scientific field. In this setting, controlling the false discovery rate (FDR), which is the expected proportion of type I error, is an important challenge for making meaningful inferences. In this paper, we consider the problem of controlling FDR in an online manner. Concretely, we consider an ordered, possibly infinite, sequence of hypotheses, arriving one at each timestep, and for each hypothesis we observe a p-value along with a set of fea…
