CONFOUNDER ADJUSTMENT IN MULTIPLE HYPOTHESIS TESTING.

Multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously. One multiple hypothesis testing error measure is the false discovery rate (FDR), which is loosely defined to be the expected proportion of false positives among all significant hypotheses. The FDR is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests. In this work, we introduce a modified version of the FDR called the “positive false discovery rate” (pFDR). We discuss the advantages and disadvantages of the pFDR and investigate its statistical properties. When assuming the test statistics follow a mixture distribution, we show that the pFDR can be written as a Bayesian posterior probability and can be connected to classification theory. These properties remain asymptotically true under fairly general conditions, even under certain forms of dependence. Also, a new quantity called the “q-value” is introduced and investigated, which is a natural “Bayesian posterior p-value,” or rather the pFDR analogue of the p-value. 1. Introduction. When testing a single hypothesis, one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists. In statistical terms, we maximize the power conditional on the Type I error rate being at or below some level. The field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously. One must define an appropriate compound error measure according to the rate of false positives one is willing to encounter. Then a procedure is developed that allows one to control the error rate at a desired level, while maintaining the power of each test as much as possible. The most commonly controlled quantity when testing multiple hypotheses is the family wise error rate (FWER), which is the probability of yielding one or more false positives out of all hypotheses tested. The most familiar example of this is the Bonferroni method. If there are m hypothesis tests, each test is controlled so that the probability of a false positive is less than or equal to α/m for some chosen value of α. It then follows that the overall FWER is less than or equal to α .M any

Large-scale retrospective analysis suggests medications and dietary supplements, such as rosuvastatin, ezetimibe, ubiquinone, risedronate, vitamin D, and magnesium, are associated with a lower rate of severe COVID-19 disease.

In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistent homology. The computational efficiency of our procedure is based on the observation that one can empirically simulate a null distribution that is universal across many hypothesis testing applications involving persistence homology. Our observation suggests that one can simulate the null distribution efficiently based on a small number of summaries of the collected data and use this null in the same way that p-value tables were used in classical statistics. To illustrate the efficiency and utility of the null distribution we provide procedures for rejecting acyclicity with both control of the Family-Wise Error Rate (FWER) and the False Discovery Rate (FDR). We will argue that the empirical null we propose is very general conditional on a few summaries of the data based on simulations and limit theorems for persistent homology for point processes.

Large-scale multiparameter screening has become increasingly feasible and straightforward to perform thanks to developments in technologies such as high-content microscopy and high-throughput flow cytometry. The automated toolkits for analyzing similarities and differences between large numbers of tested conditions have not kept pace with these technological developments. Thus, effective analysis of multiparameter screening datasets becomes a bottleneck and a limiting factor in unbiased interpretation of results. Here we introduce compaRe, a toolkit for large-scale multiparameter data analysis, which integrates quality control, data bias correction, and data visualization methods with a mass-aware gridding algorithm-based similarity analysis providing a much faster and more robust analyses than existing methods. Using mass and flow cytometry data from acute myeloid leukemia and myelodysplastic syndrome patients, we show that compaRe can reveal interpatient heterogeneity and recognizable phenotypic profiles. By applying compaRe to high-throughput flow cytometry drug response data in AML models, we robustly identified multiple types of both deep and subtle phenotypic response patterns, highlighting how this analysis could be used for therapeutic discoveries. In conclusion, compaRe is a toolkit that uniquely allows for automated, rapid, and precise comparisons of large-scale multiparameter datasets, including high-throughput screens.

False Discovery Rate (FDR) and the Bayes risk are two different statistical measures, which can be used to evaluate and compare multiple testing procedures. Recent results show that under sparsity FDR controlling procedures, like the popular Benjamini-Hochberg (BH) procedure, perform also very well in terms of the Bayes risk. In particular asymptotic Bayes optimality under sparsity (ABOS) of BH was shown previously for location and scale models based on log-concave densities. This article extends previous work to a substantially larger set of distributions of effect sizes under the alternative, where the alternative distribution of true signals does not change with the number of tests $m$, while the sample size $n$ slowly increases. ABOS of BH and the corresponding step-down procedure based on FDR levels proportional to $n^{-1/2}$ are proved. A simulation study shows that these asymptotic results are relevant already for relatively small values of $m$ and $n$. Apart from showing asymptotic optimality of BH, our results on the optimal FDR level provide a natural extension of the well known results on the significance levels of Bayesian tests.

ABSTRACTIn many scientific fields, it is interesting and important to determine whether an observed data stream comes from a prespecified model or not, particularly when the number of data streams is of large scale, where multiple hypotheses testing is necessary. In this article, we consider large-scale model checking under certain dependence among different data streams observed at the same time. We propose a false discovery rate (FDR) control procedure to check those unusual data streams. Specifically, we derive an approximation of false discovery and construct a point estimate of FDR. Theoretical results show that, under some mild assumptions, our proposed estimate of FDR is simultaneously conservatively consistent with the true FDR, and hence it is an asymptotically strong control procedure. Simulation comparisons with some competing procedures show that our proposed FDR procedure behaves better in general settings. Application of our proposed FDR procedure is illustrated by the StarPlus fMRI data.

Micropulse transscleral cyclophotocoagulation (MP-TSCPC) lowers intraocular pressure (IOP), but the mechanisms underlying its IOP-reducing effect remain incompletely understood. Although early postoperative flare elevation has been reported in association with IOP reduction, the relationship during the mid-term postoperative period has not been well characterized. In this study, the mid-term period was defined a priori as 3–18 months after surgery. This retrospective exploratory study included 25 eyes of 25 patients who underwent initial MP-TSCPC at Ishida Eye Clinic between March 2019 and May 2022 and had available postoperative flare measurements. Aqueous flare was assessed using a laser flare meter, and IOP was measured primarily by Goldmann applanation tonometry. Flare increase ratio (%) and IOP reduction ratio (%) were calculated relative to baseline. Associations were evaluated using Spearman’s rank correlation coefficient. To account for multiple comparisons across postoperative time points, p-values were adjusted using the Benjamini–Hochberg procedure, primarily considering a false discovery rate (FDR) threshold of 0.05; findings at FDR < 0.10 were interpreted as exploratory. The mean age was 80.4 ± 10.3 years. The mean preoperative IOP was 17.6 ± 6.9 mmHg, and the mean preoperative flare value was 13.9 ± 7.7 photon counts/ms. Postoperative IOP showed significant reductions at 1 and 12 months, and flare values were significantly elevated at multiple time points through 24 months. Significant positive correlations between flare increase ratio and IOP reduction ratio were observed at 3 months (ρ = 0.54, p = 0.026), 18 months (ρ = 0.964, p = 0.0027), and at the final visit (ρ = 0.41, p = 0.022). After adjustment for multiple comparisons, only the 18-month correlation remained significant at FDR < 0.05 (q = 0.022), whereas the correlations at 3 months (q = 0.069) and at the final visit (q = 0.088) were significant at FDR < 0.10 and were therefore considered exploratory. In this small exploratory cohort, sustained postoperative aqueous flare elevation was associated with greater mid-term IOP reduction after MP-TSCPC. However, given the limited sample size and observational design, these findings should be interpreted cautiously, and causal mechanisms cannot be established. Approval Number: R2025063 (University of Toyama Ethics Committee). Registration date: July 10, 2025.

Much effort has been made to improve the famous step up procedure of Benjamini and Hochberg given by linear critical values $\frac{i\alpha}{n}$. It is pointed out by Gavrilov, Benjamini and Sarkar that step down multiple testing procedures based on the critical values $\beta_{i}=\frac{i\alpha}{n+1-i(1-\alpha)}$ still control the false discovery rate (FDR) at the upper bound $\alpha$ under basic independence assumptions. Since that result is no longer true for step up procedures and for step down procedures, if the p-values are dependent, a big discussion about the corresponding FDR starts in the literature. The present paper establishes finite sample formulas and bounds for the FDR and the expected number of false rejections for multiple testing procedures using critical values $\beta_{i}$ under martingale and reverse martingale dependence models. It is pointed out that martingale methods are natural tools for the treatment of local FDR estimators which are closely connected to the present coefficients $\beta_{i}$. The martingale approach also yields new results and further inside for the special basic independence model.

Hypothesis tests

Multidimensional statistics, especially in the case where the sample size and the number of estimated parameters (or hypotheses tested) are comparable, is a natural area of application of the theory of random matrices (see V. L. Girko, Statistical Analysis of Observations of Increasing Dimension (Theory and Decision Library B). Springer, New York, 1995.). In this paper we consider procedures for multiple hypothesis testing that may be used in the case, which is even further from the classical statistics: the number of hypotheses may be much larger than the sample size.

False discovery rate (FDR) controlling procedures provide important statistical guarantees for replicability in signal identification based on multiple hypotheses testing. In many fields of study, FDR controling 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 biomarkers of multiple types of cancers. These studies provide us with opportunities to identify signals by considering information from different sources (with potential heterogeneity) jointly. This paper is about how to provide FDR control guarantees for the tests of union null hypotheses of conditional independence. We present a knockoff-based variable selection method (Simultaneous knockoffs) to identify mutual signals from multiple independent datasets, providing exact FDR control guarantees under finite sample settings. This method can work with very general model settings and test statistics. We demonstrate the performance of this method with extensive numerical studies and two real-dataexamples.

Holm's (1979) step-down multiple-testing procedure (MTP) is appealing for its flexibility, transparency, and general validity, but the derivation of corresponding simultaneous confidence regions has remained an unsolved problem. This article provides such confidence regions. In fact, simultanenous confidence regions are provided for any MTP in the class of short-cut consonant closed-testing procedures based on marginal p -values and weighted Bonferroni tests for intersection hypotheses considered by Hommel, Bretz and Maurer (2007). In addition to Holm's MTP, this class includes the fixed-sequence MTP, recently proposed gatekeeping MTPs, and the fallback MTP. The simultaneous confidence regions are generally valid if underlying marginal p -values and corresponding marginal confidence regions (assumed to be available) are valid. The marginal confidence regions and estimated quantities are not assumed to be of any particular kinds/dimensions. Compared to the rejections made by the MTP for the family of null hypotheses H under consideration, the proposed confidence regions provide extra free information. In particular, with Holm's MTP, such extra information is provided: for all nonrejected H s, in case not all H s are rejected; or for certain (possibly all) H s, in case all H s are rejected. In case not all H s are rejected, no extra information is provided for rejected H s. This drawback seems however difficult to overcome. Illustrations concerning clinical studies are given.

When a large number of hypotheses are investigated the false discovery rate (FDR) is commonly applied in gene expression analysis or gene association studies. Conventional single-stage designs may lack power due to low sample sizes for the individual hypotheses. We propose two-stage designs where the first stage is used to screen the 'promising' hypotheses which are further investigated at the second stage with an increased sample size. A multiple test procedure based on sequential individual P-values is proposed to control the FDR for the case of independent normal distributions with known variance. The power of optimal two-stage designs is impressively larger than the power of the corresponding single-stage design with equal costs. Extensions to the case of unknown variances and correlated test statistics are investigated by simulations. Moreover, it is shown that the simple multiple test procedure using first stage data for screening purposes and deriving the test decisions only from second stage data is a very powerful option.

Identifying influential variables in high-dimensional matrix-valued data while controlling the false discovery rate (FDR) is a critical challenge in modern data science. We propose a novel, model-free procedure specifically designed for simultaneous row and column selection in matrix predictor regression. Our approach utilizes folding selection subspaces (FSS) to formulate structured hypotheses and employs data splitting to construct mirror statistics from FSS estimators. This design bypasses restrictive model specification and the need for p-value computation. Key theoretical contributions include establishing the asymptotic distribution of FSS estimators and proving the mirror statistic is asymptotically symmetric with respect to zero under the null hypothesis. Using the symmetry, we develop a multiple hypothesis testing procedure with data-driven thresholds that provably controls the FDR for row and column at the desired level asymptotically. The framework is further extended to control element-wise FDR under specific structural assumptions. Extensive simulations and a real data analysis demonstrate the superior performance of the proposed method over existing approaches across various settings.

Multiple hypothesis testing in proteomics: A strategy for experimental work.