In this paper, we present novel methodologies that incorporate auxiliary variables for multiple hypotheses testing related to the main point of interest while effectively controlling the false discovery rate. When dealing with multiple tests concerning the primary variable of interest, researchers can use auxiliary variables to set preconditions for the significance of primary variables, thereby enhancing test efficacy. Depending on the auxiliary variable's role, we propose two approaches: one terminates testing of the primary variable if it does not meet predefined conditions, and the other adjusts the evaluation criteria based on the auxiliary variable. Employing the copula method, we elucidate the dependence between the auxiliary and primary variables by deriving their joint distribution from individual marginal distributions.Our numerical studies, compared with existing methods, demonstrate that the proposed methodologies effectively control the FDR and yield greater statistical power than previous approaches solely based on the primary variable. As an illustrative example, we apply our methods to the Set4$Δ$ mutant dataset. Our findings highlight the distinctions between our methodologies and traditional approaches, emphasising the potential advantages of our methods in introducing the auxiliary variable for selecting more genes.
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…
Simultaneously finding multiple influential variables and controlling the false discovery rate (FDR) for linear regression models is a fundamental problem. We here propose the Gaussian Mirror (GM) method, which creates for each predictor variable a pair of mirror variables by adding and subtracting a randomly generated Gaussian perturbation, and proceeds with a certain regression method, such as the ordinary least-square or the Lasso (the mirror variables can also be created after selection). The mirror variables naturally lead to test statistics effective for controlling the FDR. Under a mild…
Investigating the problem of setting control limits in the case of parameter uncertainty is more accessible when monitoring the variance because only one parameter has to be estimated. Simply ignoring the induced uncertainty frequently leads to control charts with poor false alarm performances. Adjusting the unconditional in-control (IC) average run length (ARL) makes the situation even worse. Guaranteeing a minimum conditional IC ARL with some given probability is another very popular approach to solving these difficulties. However, it is very conservative as well as more complex and more dif…
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…
Recent tools for interactive data exploration significantly increase the chance that users make false discoveries. The crux is that these tools implicitly allow the user to test a large body of different hypotheses with just a few clicks thus incurring in the issue commonly known in statistics as the multiple hypothesis testing error. In this paper, we propose solutions to integrate multiple hypothesis testing control into interactive data exploration tools. A key insight is that existing methods for controlling the false discovery rate (such as FDR) are not directly applicable for interactive…
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…
Several classical methods exist for controlling the false discovery exceedance (FDX) for large scale multiple testing problems, among them the Lehmann-Romano procedure ([LR] below) and the Guo-Romano procedure ([GR] below). While these two procedures are the most prominent, they were originally designed for homogeneous test statistics, that is, when the null distribution functions of the $p$-values $F_i$, $1\leq i\leq m$, are all equal. In many applications, however, the data are heterogeneous which leads to heterogeneous null distribution functions. Ignoring this heterogeneity usually induces…
New York University（纽约大学）；Nomura Asset Management Co.（野村资产管理公司）；Temple University（天普大学）
We consider controlling the false discovery rate for testing many time series with an unknown cross-sectional correlation structure. Given a large number of hypotheses, false and missing discoveries can plague an analysis. While many procedures have been proposed to control false discovery, most of them either assume independent hypotheses or lack statistical power. A problem of particular interest is in financial asset pricing, where the goal is to determine which ``factors" lead to excess returns out of a large number of potential factors. Our contribution is two-fold. First, we show the con…
Variable selection has been widely used in data analysis for the past decades, and it becomes increasingly important in the Big Data era as there are usually hundreds of variables available in a dataset. To enhance interpretability of a model, identifying potentially relevant features is often a step before fitting all the features into a regression model. A good variable selection method should effectively control the fraction of false discoveries and ensure large enough power of its selection set. In a lot of contemporary data applications, a great portion of features are coded as binary var…
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…
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…
Controlling the joint local false discovery rate is more powerful than meta-analysis methods in joint analysis of summary statistics from multiple genome-wide association studies
The Hong Kong University of Science and Technology（香港科技大学）
The analysis of environmental mixtures is of growing importance in environmental epidemiology, and one of the key goals in such analyses is to identify exposures and their interactions that are associated with adverse health outcomes. Typical approaches utilize flexible regression models combined with variable selection to identify important exposures and estimate a potentially nonlinear relationship with the outcome of interest. Despite this surge in interest, no approaches to date can identify exposures and interactions while controlling any form of error rates with respect to exposure selec…
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…
Bayesian networks can represent directed gene regulations and therefore are favored over co-expression networks. However, hardly any Bayesian network study concerns the false discovery control (FDC) of network edges, leading to low accuracies due to systematic biases from inconsistent false discovery levels in the same study. We design four empirical tests to examine the FDC of Bayesian networks from three p-value based lasso regression variable selections --- two existing and one we originate. Our method, lassopv, computes p-values for the critical regularization strength at which a predictor…
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）
Scientists often must simultaneously localize and discover signals. For instance, in genetic fine-mapping, high correlations between nearby genetic variants make it hard to identify the exact locations of causal variants. So the statistical task is to output as many disjoint regions containing a signal as possible, each as small as possible, while controlling false positives. Similar problems arise in any application where signals cannot be perfectly localized, such as locating stars in astronomical surveys and changepoint detection in sequential data. Common Bayesian approaches to these probl…
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…
Extracting associations that recur across multiple studies while controlling the false discovery rate is a fundamental challenge. Here, we consider an extension of Efron's single-study two-groups model to allow joint analysis of multiple studies. We assume that given a set of p-values obtained from each study, the researcher is interested in associations that recur in at least $k>1$ studies. We propose new algorithms that differ in how the study dependencies are modeled. We compared our new methods and others using various simulated scenarios. The top performing algorithm, SCREEN (Scalable Clu…
In many domains, data measurements can naturally be associated with the leaves of a tree, expressing the relationships among these measurements. For example, companies belong to industries, which in turn belong to ever coarser divisions such as sectors; microbes are commonly arranged in a taxonomic hierarchy from species to kingdoms; street blocks belong to neighborhoods, which in turn belong to larger-scale regions. The problem of tree-based aggregation that we consider in this paper asks which of these tree-defined subgroups of leaves should really be treated as a single entity and which of …
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…
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…
Phase III randomized clinical trials play a monumentally critical role in the evaluation of new medical products. Because of the intrinsic nature of uncertainty embedded in our capability in assessing the efficacy of a medical product, interpretation of trial results relies on statistical principles to control the error of false positives below desirable level. The well-established statistical hypothesis testing procedure suffers from two major limitations, namely, the lack of flexibility in the thresholds to claim success and the lack of capability of controlling the total number of false pos…
Large-scale multiple testing is a fundamental problem in high dimensional statistical inference. It is increasingly common that various types of auxiliary information, reflecting the structural relationship among the hypotheses, are available. Exploiting such auxiliary information can boost statistical power. To this end, we propose a framework based on a two-group mixture model with varying probabilities of being null for different hypotheses a priori, where a shape-constrained relationship is imposed between the auxiliary information and the prior probabilities of being null. An optimal reje…
The popularity of penalized regression in high-dimensional data analysis has led to a demand for new inferential tools for these models. False discovery rate control is widely used in high-dimensional hypothesis testing, but has only recently been considered in the context of penalized regression. Almost all of this work, however, has focused on lasso-penalized linear regression. In this paper, we derive a general method for controlling the marginal false discovery rate that can be applied to any penalized likelihood-based model, such as logistic regression and Cox regression. Our approach is …
