Competition-based control of the false discovery proportion
Dong Luo, Arya Ebadi, Yilun He, Kristen Emery, William Stafford Noble, Uri Keich
备注：
This revision focuses only on FDP-SD described in the original submission. A later submission will further develop the procedures for simultaneous bounds on the FDP
Recently, Barber and Candès laid the theoretical foundation for a general framework for false discovery rate (FDR) control based on the notion of "knockoffs." A closely related FDR control methodology has long been employed in the analysis of mass spectrometry data, referred to there as "target-decoy competition" (TDC). However, any approach that aims to control the FDR, which is defined as the expected value of the false discovery proportion (FDP), suffers from a problem. Specifically, even when successfully controlling the FDR at level $α$, the FDP in the list of discoveries can significantly exceed $α$. We offer FDP-SD, a new procedure that rigorously controls the FDP in the competition (knockoff / TDC) setup by guaranteeing that the FDP is bounded by $α$ at any desired confidence level. Compared with the just-published general framework of Katsevich and Ramdas, FDP-SD generally delivers more power and often substantially so in simulated as well as real data.
Multiple hypothesis testing has been widely applied to problems dealing with high-dimensional data, e.g., selecting significant variables and controlling the selection error rate. The most prevailing measure of error rate used in the multiple hypothesis testing is the false discovery rate (FDR). In recent years, local false discovery rate (fdr) has drawn much attention, due to its advantage of accessing the confidence of individual hypothesis. However, most methods estimate fdr through p-values or statistics with known null distributions, which are sometimes not available or reliable. Adopting…
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 …
We consider the class of all multiple testing methods controlling tail probabilities of the false discovery proportion, either for one random set or simultaneously for many such sets. This class encompasses methods controlling familywise error rate, generalized familywise error rate, false discovery exceedance, joint error rate, simultaneous control of all false discovery proportions, and others, as well as seemingly unrelated methods such as gene set testing in genomics and cluster inference methods in neuroimaging. We show that all such methods are either equivalent to a closed testing metho…
We consider multiple testing means of many dependent Normal random variables that do not necessarily follow a joint Normal distribution. Under weak dependence, we show the uniform consistency of proportion estimators that are constructed as solutions to Lebesgue-Stieltjes equations for the setting of a point, bounded and one-sided null, respectively, and characterize via the index of weak dependence the sparsest proportion these estimators can consistently estimate. On the other hand, under a principal correlation structure and employing a suitable definition of p-value for composite null hypo…
We propose a ranking and selection procedure to prioritize relevant predictors and control false discovery proportion (FDP) of variable selection. Our procedure utilizes a new ranking method built upon the de-sparsified Lasso estimator. We show that the new ranking method achieves the optimal order of minimum non-zero effects in ranking relevant predictors ahead of irrelevant ones. Adopting the new ranking method, we develop a variable selection procedure to asymptotically control FDP at a user-specified level. We show that our procedure can consistently estimate the FDP of variable selection …
The proportion of false null hypotheses is a very important quantity in statistical modelling and inference based on the two-component mixture model and its extensions, and in control and estimation of the false discovery rate and false non-discovery rate. Most existing estimators of this proportion threshold p-values, deconvolve the mixture model under constraints on its components, or depend heavily on the location-shift property of distributions. Hence, they usually are not consistent, applicable to non-location-shift distributions, or applicable to discrete statistics or p-values. To elimi…
Large-scale multiple testing with highly correlated test statistics arises frequently in many scientific research. Incorporating correlation information in estimating false discovery proportion has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan, Han & Gu (2012) provided a consistent estimate of False Discovery Proportion (FDP) under arbitrary dependence structure. However, the covariance matrix is often unknown in many applications and such dependence information has to be estimated before estimating FDP (Efron, 2010). The estimation…
When multiple hypotheses are tested, interest is often in ensuring that the proportion of false discoveries (FDP) is small with high confidence. In this paper, confidence upper bounds for the FDP are constructed, which are simultaneous over all rejection cut-offs. In particular this allows the user to select a set of hypotheses post hoc such that the FDP lies below some constant with high confidence. Our method uses permutations to account for the dependence structure in the data. So far only Meinshausen provided an exact, permutation-based and computationally feasible method for simultaneous …
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…
Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any SNPs are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a novel method based on principal factor approximation, which successfully subtracts the common dependence and weakens significantly the correl…
Post hoc false discovery proportion inference under a Hidden Markov Model
Université de Paris, CNRS（巴黎大学）；Université Paris-Saclay, CNRS, Inria Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学）；Institut de Mathématiques de Toulouse UMR 5219, Université de Toulouse, CNRS UPS（图卢兹数学研究所）；Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, Université de Paris, CNRS（索邦大学）
This paper addresses the challenge of efficiently capturing a high proportion of true signals for subsequent data analyses when sample sizes are relatively limited with respect to data dimension. We propose the signal missing rate as a new measure for false negative control to account for the variability of false negative proportion. Novel data-adaptive procedures are developed to control signal missing rate without incurring many unnecessary false positives under dependence. We justify the efficiency and adaptivity of the proposed methods via theory and simulation. The proposed methods are ap…
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）
Statistical dependence between hypotheses poses a significant challenge to the stability of large scale multiple hypotheses testing. Ignoring it often results in an unacceptably large spread in the false positive proportion even though the average value is acceptable [21, 39, 40, 49]. However, the statistical dependence structure of data is often unknown. Using a generic signalprocessing model, Bayesian multiple testing, and simulations, we demonstrate that the variance of the false positive proportion can be substantially reduced even under unknown short range dependence. We do this by modeli…
This paper presents a survey on some recent advances for the type I error rate control in multiple testing methodology. We consider the problem of controlling the $k$-family-wise error rate (kFWER, probability to make $k$ false discoveries or more) and the false discovery proportion (FDP, proportion of false discoveries among the discoveries). The FDP is controlled either via its expectation, which is the so-called false discovery rate (FDR), or via its upper-tail distribution function. We aim at deriving general and unified results together with concise and simple mathematical proofs. Further…
Large-scale multiple testing with correlated and heavy-tailed data arises in a wide range of research areas from genomics, medical imaging to finance. Conventional methods for estimating the false discovery proportion (FDP) often ignore the effect of heavy-tailedness and the dependence structure among test statistics, and thus may lead to inefficient or even inconsistent estimation. Also, the commonly imposed joint normality assumption is arguably too stringent for many applications. To address these challenges, in this paper we propose a Factor-Adjusted Robust Multiple Testing (FarmTest) proc…
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…
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…
There is a growing interest in cell-type-specific analysis from bulk samples with a mixture of different cell types. A critical first step in such analyses is the accurate estimation of cell-type proportions in a bulk sample. Although many methods have been proposed recently, quantifying the uncertainties associated with the estimated cell-type proportions has not been well studied. Lack of consideration of these uncertainties can lead to missed or false findings in downstream analyses. In this article, we introduce a flexible statistical deconvolution framework that allows a general and subje…
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 pro…
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…
When simultaneously testing multiple hypotheses, the usual approach in the context of confirmatory clinical trials is to control the familywise error rate (FWER), which bounds the probability of making at least one false rejection. In many trial settings, these hypotheses will additionally have a hierarchical structure that reflects the relative importance and links between different clinical objectives. The graphical approach of Bretz et al. (2009) is a flexible and easily communicable way of controlling the FWER while respecting complex trial objectives and multiple structured hypotheses. Ho…
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…
Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any genes are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a new methodology based on principal factor approximation, which successfully substracts the common dependence and weakens significantly the c…
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…
Cluster-level inference procedures are widely used for brain mapping. These methods compare the size of clusters obtained by thresholding brain maps to an upper bound under the global null hypothesis, computed using Random Field Theory or permutations. However, the guarantees obtained by this type of inference - i.e. at least one voxel is truly activated in the cluster - are not informative with regards to the strength of the signal therein. There is thus a need for methods to assess the amount of signal within clusters; yet such methods have to take into account that clusters are defined base…
False discovery rates (FDR) are an essential component of statistical inference, representing the propensity for an observed result to be mistaken. FDR estimates should accompany observed results to help the user contextualize the relevance and potential impact of findings. This paper introduces a new user-friendly R package for computing FDRs and adjusting p-values for FDR control. These tools respect the critical difference between the adjusted p-value and the estimated FDR for a particular finding, which are sometimes numerically identical but are often confused in practice. Newly augmented…
This paper continues the line of research initiated in Liu et. al. (2016) on developing a novel framework for multiple testing of hypotheses grouped in a one-way classified form using hypothesis-specific local false discovery rates (Lfdr's). It is built on an extension of the standard two-class mixture model from single to multiple groups, defining hypothesis-specific Lfdr as a function of the conditional Lfdr for the hypothesis given that it is within an important group and the Lfdr for the group itself and involving a new parameter that measures grouping effect. This definition captures the …
Barber and Candes recently introduced a feature selection method called knockoff+ that controls the false discovery rate (FDR) among the selected features in the classical linear regression problem. Knockoff+ uses the competition between the original features and artificially created knockoff features to control the FDR [1]. We generalize Barber and Candes' knockoff construction to generate multiple knockoffs and use those in conjunction with a recently developed general framework for multiple competition-based FDR control [9]. We prove that using our initial multiple-knockoff construction the…
We propose a new empirical Bayes method for covariate-assisted multiple testing with false discovery rate (FDR) control, where we model the local false discovery rate for each hypothesis as a function of both its covariates and p-value. Our method refines the adaptive p-value thresholding (AdaPT) procedure by generalizing its masking scheme to reduce the bias and variance of its false discovery proportion estimator, improving the power when the rejection set is small or some null p-values concentrate near 1. We also introduce a Gaussian mixture model for the conditional distribution of the tes…
In many scientific fields, researchers are interested in discovering features with substantial effect on the response from a large number of features while controlling the proportion of false discoveries. By incorporating the knockoff procedure in the Bayesian framework, we develop the Bayesian knockoff filter (BKF) for selecting features that have important effect on the response. In contrast to the fixed knockoff variables in a frequentist procedure, we allow the knockoff variables to be continuously updated using the Markov chain Monte Carlo. Based on the posterior samples and the elaborate…
Selecting relevant features associated with a given response variable is an important issue in many scientific fields. Quantifying quality and uncertainty of a selection result via false discovery rate (FDR) control has been of recent interest. This paper introduces a way of using data-splitting strategies to asymptotically control the FDR while maintaining a high power. For each feature, the method constructs a test statistic by estimating two independent regression coefficients via data splitting. FDR control is achieved by taking advantage of the statistic's property that, for any null feat…
We develop a new class of distribution--free multiple testing rules for false discovery rate (FDR) control under general dependence. A key element in our proposal is a symmetrized data aggregation (SDA) approach to incorporating the dependence structure via sample splitting, data screening and information pooling. The proposed SDA filter first constructs a sequence of ranking statistics that fulfill global symmetry properties, and then chooses a data--driven threshold along the ranking to control the FDR. The SDA filter substantially outperforms the knockoff method in power under moderate to s…
Control charts are important tools to monitor quality of products. One of useful applications is to monitor the proportion of non-conforming products. However, in practical applications, measurement error is ubiquitous and may occur due to false records or misclassification, which makes the observed proportion different from the underlying true proportion. It is also well-known that ignoring measurement error effects provides biases, and is expected that the resulting control charts may incur wrong detection. In this paper, we study this important problem and propose a valid method to correct …
In the multiple testing problem with independent tests, the classical linear step-up procedure controls the false discovery rate (FDR) at level $π_0α$, where $π_0$ is the proportion of true null hypotheses and $α$ is the target FDR level. Adaptive procedures can improve power by incorporating estimates of $π_0$, which typically rely on a tuning parameter. Fixed adaptive procedures set their tuning parameters before seeing the data and can be shown to control the FDR in finite samples. We develop theoretical results for dynamic adaptive procedures whose tuning parameters are determined by the d…
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…
