Permutation testing in linear models, where the number of nuisance coefficients is smaller than the sample size, is a well-studied topic. The common approach of such tests is to permute residuals after regressing on the nuisance covariates. Permutation-based tests are valuable in particular because they can be highly robust to violations of the standard linear model, such as non-normality and heteroscedasticity. Moreover, in some cases they can be combined with existing, powerful permutation-based multiple testing methods. Here, we propose permutation tests for models where the number of nuisance coefficients exceeds the sample size. The performance of the novel tests is investigated with simulations. In a wide range of simulation scenarios our proposed permutation methods provided appropriate type I error rate control, unlike some competing tests, while having good power.
Neural networks are powerful predictive models, but they provide little insight into the nature of relationships between predictors and outcomes. Although numerous methods have been proposed to quantify the relative contributions of input features, statistical inference and hypothesis testing of feature associations remain largely unexplored. We propose a permutation-based approach to testing that uses the partial derivatives of the network output with respect to specific inputs to assess both the significance of input features and whether significant features are linearly associated with the …
Permutation methods are commonly used to test significance of regressors of interest in general linear models (GLMs) for functional (image) data sets, in particular for neuroimaging applications as they rely on mild assumptions. Permutation inference for GLMs typically consists of three parts: choosing a relevant test statistic, computing pointwise permutation tests and applying a multiple testing correction. We propose new multiple testing methods as an alternative to the commonly used maximum value of test statistics across the image. The new methods improve power and robustness against inho…
In this article, we derive and compare methods to derive \textit{p}-values and sets of confidence intervals with strong control of the family-wise error rates and coverage for estimates of treatment effects in cluster randomised trials with multiple outcomes. There are few methods for \textit{p}-value corrections and deriving confidence intervals, limiting their application in this setting. We discuss the methods of Bonferroni, Holm, and Romano \& Wolf (2005) and adapt them to cluster randomised trial inference using permutation-based methods with different test statistics. We develop a novel …
Motivated by the prevalence of high dimensional low sample size datasets in modern statistical applications, we propose a general nonparametric framework, Direction-Projection-Permutation (DiProPerm), for testing high dimensional hypotheses. The method is aimed at rigorous testing of whether lower dimensional visual differences are statistically significant. Theoretical analysis under the non-classical asymptotic regime of dimension going to infinity for fixed sample size reveals that certain natural variations of DiProPerm can have very different behaviors. An empirical power study both confi…
When permutation methods are used in practice, often a limited number of random permutations are used to decrease the computational burden. However, most theoretical literature assumes that the whole permutation group is used, and methods based on random permutations tend to be seen as approximate. There exists a very limited amount of literature on exact testing with random permutations and only recently a thorough proof of exactness was given. In this paper we provide an alternative proof, viewing the test as a "conditional Monte Carlo test" as it has been called in the literature. We also p…
Confidence intervals are central to statistical inference as a tool to evaluate the type I error risk at a given significance level. We devise a method to construct confidence intervals using a single run of a permutation test. This methodology is extended to a multivariate setting, where we are able to handle multiple testing under arbitrary dependence. We demonstrate the method on a weather data set and in a simulation example.
We present a general approach to constructing permutation tests that are both exact for the null hypothesis of equality of distributions and asymptotically correct for testing equality of parameters of distributions while allowing the distributions themselves to differ. These robust permutation tests transform a given test statistic by a consistent estimator of its limiting distribution function before enumerating its permutation distribution. This transformation, known as prepivoting, aligns the unconditional limiting distribution for the test statistic with the probability limit of its permu…
Test statistics are often strongly dependent in large-scale multiple testing applications. Most corrections for multiplicity are unduly conservative for correlated test statistics, resulting in a loss of power to detect true positives. We show that the Westfall--Young permutation method has asymptotically optimal power for a broad class of testing problems with a block-dependence and sparsity structure among the tests, when the number of tests tends to infinity.
We study the problem of independence testing given independent and identically distributed pairs taking values in a $σ$-finite, separable measure space. Defining a natural measure of dependence $D(f)$ as the squared $L^2$-distance between a joint density $f$ and the product of its marginals, we first show that there is no valid test of independence that is uniformly consistent against alternatives of the form $\{f: D(f) \geq ρ^2 \}$. We therefore restrict attention to alternatives that impose additional Sobolev-type smoothness constraints, and define a permutation test based on a basis expansi…
In nonlinear dynamics, and to a lesser extent in other fields, a widely used measure of complexity is the Permutation Entropy. But there is still no known method to determine the accuracy of this measure. There has been little research on the statistical properties of this quantity that characterize time series. The literature describes some resampling methods of quantities used in nonlinear dynamics - as the largest Lyapunov exponent - but all of these seems to fail. In this contribution we propose a parametric bootstrap methodology using a symbolic representation of the time series in order …
We present a novel algorithm, Westfall-Young light, for detecting patterns, such as itemsets and subgraphs, which are statistically significantly enriched in one of two classes. Our method corrects rigorously for multiple hypothesis testing and correlations between patterns through the Westfall-Young permutation procedure, which empirically estimates the null distribution of pattern frequencies in each class via permutations. In our experiments, Westfall-Young light dramatically outperforms the current state-of-the-art approach in terms of both runtime and memory efficiency on popular real-wor…
A number of biomedical problems require performing many hypothesis tests, with an attendant need to apply stringent thresholds. Often the data take the form of a series of predictor vectors, each of which must be compared with a single response vector, perhaps with nuisance covariates. Parametric tests of association are often used, but can result in inaccurate type I error at the extreme thresholds, even for large sample sizes. Furthermore, standard two-sided testing can reduce power compared to the doubled $p$-value, due to asymmetry in the null distribution. Exact (permutation) testing is a…
Measures of local and global spatial association are key tools for exploratory spatial data analysis. Many such measures exist including Moran's $I$, Geary's $C$, and the Getis-Ord $G$ and $G^*$ statistics. A parametric approach to testing for significance relies on strong assumptions, which are often not met by real world data. Alternatively, the most popular nonparametric approach, the permutation test, imposes a large computational burden especially for massive graphical networks. Hence, we propose a computation-free approach to nonparametric permutation testing for local and global measure…
William S. Middleton Memorial VA Hospital（威廉·S·米德尔顿纪念退伍军人事务医院）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
Multiple hypothesis testing is a significant problem in nearly all neuroimaging studies. In order to correct for this phenomena, we require a reliable estimate of the Family-Wise Error Rate (FWER). The well known Bonferroni correction method, while simple to implement, is quite conservative, and can substantially under-power a study because it ignores dependencies between test statistics. Permutation testing, on the other hand, is an exact, non-parametric method of estimating the FWER for a given $α$-threshold, but for acceptably low thresholds the computational burden can be prohibitive. In t…
In this paper, we investigate local permutation tests for testing conditional independence between two random vectors $X$ and $Y$ given $Z$. The local permutation test determines the significance of a test statistic by locally shuffling samples which share similar values of the conditioning variables $Z$, and it forms a natural extension of the usual permutation approach for unconditional independence testing. Despite its simplicity and empirical support, the theoretical underpinnings of the local permutation test remain unclear. Motivated by this gap, this paper aims to establish theoretical …
In this study, we propose a two-stage procedure for hypothesis testing, where the first stage is conventional hypothesis testing and the second is an equivalence testing procedure using an introduced Empirical Equivalence Bound. In 2016, the American Statistical Association released a policy statement on P-values to clarify the proper use and interpretation in response to the criticism of reproducibility and replicability in scientific findings. A recent solution to improve reproducibility and transparency in statistical hypothesis testing is to integrate P-values (or confidence intervals) wit…
Modern application of A/B tests is challenging due to its large scale in various dimensions, which demands flexibility to deal with multiple testing sequentially. The state-of-the-art practice first reduces the observed data stream to always-valid p-values, and then chooses a cut-off as in conventional multiple testing schemes. Here we propose an alternative method called AMSET (adaptive multistage empirical Bayes test) by incorporating historical data in decision-making to achieve efficiency gains while retaining marginal false discovery rate (mFDR) control that is immune to peeking. We also …
In this paper, our interest is in the problem of simultaneous hypothesis testing when the test statistics corresponding to the individual hypotheses are possibly correlated. Specifically, we consider the case when the test statistics together have a multivariate normal distribution (with equal correlation between each pair) with an unknown mean vector and our goal is to decide which components of the mean vector are zero and which are non-zero. This problem was taken up earlier in Bogdan et al. (2011) for the case when the test statistics are independent normals. Asymptotic optimality in a Bay…
The traditional approaches to false discovery rate (FDR) control in multiple hypothesis testing are usually based on the null distribution of a test statistic. However, all types of null distributions, including the theoretical, permutation-based and empirical ones, have some inherent drawbacks. For example, the theoretical null might fail because of improper assumptions on the sample distribution. Here, we propose a null distribution-free approach to FDR control for multiple hypothesis testing. This approach, named target-decoy procedure, simply builds on the ordering of tests by some statist…
The Gini correlation plays an important role in measuring dependence of random variables with heavy tailed distributions, whose properties are a mixture of Pearson's and Spearman's correlations. Due to the structure of this dependence measure, there are two Gini correlations between each pair of random variables, which are not equal in general. Both the Gini correlation and the equality of the two Gini correlations play important roles in Economics. In the literature, there are limited papers focusing on the inference of the Gini correlations and their equality testing. In this paper, we devel…
We propose a new approach for clustering DNA features using array CGH data from multiple tumor samples. We distinguish data-collapsing: joining contiguous DNA clones or probes with extremely similar data into regions, from clustering: joining contiguous, correlated regions based on a maximum likelihood principle. The model-based clustering algorithm accounts for the apparent spatial patterns in the data. We evaluate the randomness of the clustering result by a cluster stability score in combination with cross-validation. Moreover, we argue that the clustering really captures spatial genomic de…
Machine learning models are central to people's lives and impact society in ways as fundamental as determining how people access information. The gravity of these models imparts a responsibility to model developers to ensure that they are treating users in a fair and equitable manner. Before deploying a model into production, it is crucial to examine the extent to which its predictions demonstrate biases. This paper deals with the detection of bias exhibited by a machine learning model through statistical hypothesis testing. We propose a permutation testing methodology that performs a hypothes…
We formulate and analyze a novel hypothesis testing problem for inferring the edge structure of an infection graph. In our model, a disease spreads over a network via contagion or random infection, where the random variables governing the rates of contracting the disease from neighbors or random infection are independent exponential random variables with unknown rate parameters. A subset of nodes is also censored uniformly at random. Given the statuses of nodes in the network, the goal is to determine the underlying graph. We present a procedure based on permutation testing, and we derive suff…
Hypothesis tests based on linear models are widely accepted by organizations that regulate clinical trials. These tests are derived using strong assumptions about the data-generating process so that the resulting inference can be based on parametric distributions. Because these methods are well understood and robust, they are sometimes applied to data that depart from assumptions, such as ordinal integer scores. Permutation tests are a nonparametric alternative that require minimal assumptions which are often guaranteed by the randomization that was conducted. We compare analysis of covariance…
In genome-wide association (GWA) studies the goal is to detect associations between genetic markers and a given phenotype. The number of genetic markers can be large and effective methods for control of the overall error rate is a central topic when analyzing GWA data. The Bonferroni method is known to be conservative when the tests are dependent. Permutation methods give exact control of the overall error rate when the assumption of exchangeability is satisfied, but are computationally intensive for large datasets. For regression models the exchangeability assumption is in general not satisfi…
The Hilbert--Schmidt Independence Criterion (HSIC) is a popular measure of the dependency between two random variables. The statistic dHSIC is an extension of HSIC that can be used to test joint independence of $d$ random variables. Such hypothesis testing for (joint) independence is often done using a permutation test, which compares the observed data with randomly permuted datasets. The main contribution of this work is proving that the power of such independence tests converges to 1 as the sample size converges to infinity. This answers a question that was asked in (Pfister, 2018) Additiona…
We propose a permutation-based method for testing a large collection of hypotheses simultaneously. Our method provides lower bounds for the number of true discoveries in any selected subset of hypotheses. These bounds are simultaneously valid with high confidence. The methodology is particularly useful in functional Magnetic Resonance Imaging cluster analysis, where it provides a confidence statement on the percentage of truly activated voxels within clusters of voxels, avoiding the well-known spatial specificity paradox. We offer a user-friendly tool to estimate the percentage of true discove…
A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations
CNRS, Univ. Grenoble-Alpes, GIPSA-lab, France e1；MaLGa Center - DIBRIS, Università di Genova, Genoa, Italy e2,e4；CNRS, Laboratoire d'Informatique Gaspard Monge, Champs-sur-Marne, France e3；CBMM - Massachusets Institute of Technology, Cambridge, MA, USA
We consider a permutation method for testing whether observations given in their natural pairing exhibit an unusual level of similarity in situations where any two observations may be similar at some unknown baseline level. Under a null hypotheses where there is no distinguished pairing of the observations, a normal approximation with explicit bounds and rates is presented for determining approximate critical test levels.
We propose a flexible and identifiable version of the two-groups model, motivated by hierarchical Bayes considerations, that features an empirical null and a semiparametric mixture model for the non-null cases. We use a computationally efficient predictive recursion marginal likelihood procedure to estimate the model parameters, even the nonparametric mixing distribution. This leads to a nonparametric empirical Bayes testing procedure, which we call PRtest, based on thresholding the estimated local false discovery rates. Simulations and real-data examples demonstrate that, compared to existing…
We investigate several distribution free dependence detection procedures, mainly based on bootstrap principles and their approximation properties. Thanks to this study, we introduce a new distribution free Unitary Events (UE) method, named Permutation UE, which consists in a multiple testing procedure based on permutation and delayed coincidence count. Each involved single test of this procedure achieves the prescribed level, so that the corresponding multiple testing procedure controls the False Discovery Rate (FDR), and this with as few assumptions as possible on the underneath distribution.…
In nonparametric independence testing, we observe i.i.d.\ data $\{(X_i,Y_i)\}_{i=1}^n$, where $X \in \mathcal{X}, Y \in \mathcal{Y}$ lie in any general spaces, and we wish to test the null that $X$ is independent of $Y$. Modern test statistics such as the kernel Hilbert-Schmidt Independence Criterion (HSIC) and Distance Covariance (dCov) have intractable null distributions due to the degeneracy of the underlying U-statistics. Thus, in practice, one often resorts to using permutation testing, which provides a nonasymptotic guarantee at the expense of recalculating the quadratic-time statistics …
University of Pennsylvania（宾夕法尼亚大学）；Temple University（天普大学）；University of Michigan（密歇根大学）
Large-scale modern data often involves estimation and testing for high-dimensional unknown parameters. It is desirable to identify the sparse signals, ``the needles in the haystack'', with accuracy and false discovery control. However, the unprecedented complexity and heterogeneity in modern data structure require new machine learning tools to effectively exploit commonalities and to robustly adjust for both sparsity and heterogeneity. In addition, estimates for high-dimensional parameters often lack uncertainty quantification. In this paper, we propose a novel Spike-and-Nonparametric mixture …
In this paper, we generalize the metric-based permutation test for the equality of covariance operators proposed by Pigoli et al. (2014) to the case of multiple samples of functional data. To this end, the non-parametric combination methodology of Pesarin and Salmaso (2010) is used to combine all the pairwise comparisons between samples into a global test. Different combining functions and permutation strategies are reviewed and analyzed in detail. The resulting test allows to make inference on the equality of the covariance operators of multiple groups and, if there is evidence to reject the …
University of Washington（华盛顿大学）；Google Inc（谷歌公司）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Fred Hutchinson Cancer Research Center（弗雷德·哈钦森癌症研究中心）
This paper investigates the theoretical and empirical performance of Fisher-Pitman-type permutation tests for assessing the equality of unknown Poisson mixture distributions. Building on nonparametric maximum likelihood estimators (NPMLEs) of the mixing distribution, these tests are theoretically shown to be able to adapt to complicated unspecified structures of count data and also consistent against their corresponding ANOVA-type alternatives; the latter is a result in parallel to classic claims made by Robinson (Robinson, 1973). The studied methods are then applied to a single-cell RNA-seq d…
This paper explores a connection between empirical Bayes posterior distributions and false discovery rate (FDR) control. In the Gaussian sequence model, this work shows that empirical Bayes-calibrated spike and slab posterior distributions allow a correct FDR control under sparsity. Doing so, it offers a frequentist theoretical validation of empirical Bayes methods in the context of multiple testing. Our theoretical results are illustrated with numerical experiments.
We propose a new adaptive empirical Bayes framework, the Bag-Of-Null-Statistics (BONuS) procedure, for multiple testing where each hypothesis testing problem is itself multivariate or nonparametric. BONuS is an adaptive and interactive knockoff-type method that helps improve the testing power while controlling the false discovery rate (FDR), and is closely connected to the "counting knockoffs" procedure analyzed in Weinstein et al. (2017). Contrary to procedures that start with a $p$-value for each hypothesis, our method analyzes the entire data set to adaptively estimate an optimal $p$-value …
The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest are defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-diverge…
Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This paper proposes permutation tests for equality of parameters that are estimated at root-$n$ or slower rates. Our general framework applies to both parametric and nonparametric models, with two samples or one sample split into two subsamples. Our tests have correct size asymptotically while preserving exact size in finite samples when distributions are equal. They have no loss in local asymp…
