Multivariate two‐sample permutation tests for trials with multiple time‐to‐event outcomes

Clinical trials involving multiple time‐to‐event outcomes are increasingly common. In this paper, permutation tests for testing for group differences in multivariate time‐to‐event data are proposed. Unlike other two‐sample tests for multivariate survival data, the proposed tests attain the nominal type I error rate. A simulation study shows that the proposed tests outperform their competitors when the degree of censored observations is sufficiently high. When the degree of censoring is low, it is seen that naive tests such as Hotelling's T² outperform tests tailored to survival data. Computational and practical aspects of the proposed tests are discussed, and their use is illustrated by analyses of three publicly available datasets. Implementations of the proposed tests are available in an accompanying R package.     

Some solutions to the multivariate Behrens–Fisher problem for dissimilarity‐based analyses

The essence of the generalised multivariate Behrens–Fisher problem (BFP) is how to test the null hypothesis of equality of mean vectors for two or more populations when their dispersion matrices differ. Solutions to the BFP usually assume variables are multivariate normal and do not handle high‐dimensional data. In ecology, species' count data are often high‐dimensional, non‐normal and heterogeneous. Also, interest lies in analysing compositional dissimilarities among whole communities in non‐Euclidean (semi‐metric or non‐metric) multivariate space. Hence, dissimilarity‐based tests by permutation (e.g., PERMANOVA, ANOSIM) are used to detect differences among groups of multivariate samples. Such tests are not robust, however, to heterogeneity of dispersions in the space of the chosen dissimilarity measure, most conspicuously for unbalanced designs. Here, we propose a modification to the PERMANOVA test statistic, coupled with either permutation or bootstrap resampling methods, as a solution to the BFP for dissimilarity‐based tests. Empirical simulations demonstrate that the type I error remains close to nominal significance levels under classical scenarios known to cause problems for the un‐modified test. Furthermore, the permutation approach is found to be more powerful than the (more conservative) bootstrap for detecting changes in community structure for real ecological datasets. The utility of the approach is shown through analysis of 809 species of benthic soft‐sediment invertebrates from 101 sites in five areas spanning 1960 km along the Norwegian continental shelf, based on the Jaccard dissimilarity measure.     

Multivariate analysis of variance for functional data

Functional data are being observed frequently in many scientific fields, and therefore most of the standard statistical methods are being adapted for functional data. The multivariate analysis of variance problem for functional data is considered. It seems to be of practical interest similarly as the one-way analysis of variance for such data. For the MANOVA problem for multivariate functional data, we propose permutation tests based on a basis function representation and tests based on random projections. Their performance is examined in comprehensive simulation studies, which provide an idea of the size control and power of the tests and identify differences between them. The simulation experiments are based on artificial data and real labeled multivariate time series data found in the literature. The results suggest that the studied testing procedures can detect small differences between vectors of curves even with small sample sizes. Illustrative real data examples of the use of the proposed testing procedures in practice are also presented.     

Lévy‐based Modelling in Brain Imaging

Abstract. A substantive problem in neuroscience is the lack of valid statistical methods for non‐Gaussian random fields. In the present study, we develop a flexible, yet tractable model for a random field based on kernel smoothing of a so‐called Lévy basis. The resulting field may be Gaussian, but there are many other possibilities, including random fields based on Gamma, inverse Gaussian and normal inverse Gaussian (NIG) Lévy bases. It is easy to estimate the parameters of the model and accordingly to assess by simulation the quantiles of test statistics commonly used in neuroscience. We give a concrete example of magnetic resonance imaging scans that are non‐Gaussian. For these data, simulations under the fitted models show that traditional methods based on Gaussian random field theory may leave small, but significant changes in signal level undetected, while these changes are detectable under a non‐Gaussian Lévy model.     

Significance levels and confidence intervals for permutation tests

A computational algorithm is given which calculates exact significance levels of a wide class of permutation tests in the one and two sample problems. This class includes the permutation test based on the means, locally most powerful permutation tests and linear rank tests. When a shift model is assumed confidence intervals can also be obtained. Approximate methods, based on asymptotic expansions, are also presented.     

Testing nonparametric statistical functionals with applications to rank tests

The present paper discusses how nonparametric tests can be deduced from statistical functionals. Efficient and asymptotically most powerful maximin tests are derived. Their power function is calculated under implicit alternatives given by the functional for one – and two – sample testing problems. It is shown that the asymptotic power function does not depend on the special implicit direction of the alternatives but only on quantities of the functional. The present approach offers a nonparametric principle how to construct common rank tests as the Wilcoxon test, the log rank test, and the median test from special two-sample functionals. In addition it is shown that studentized permutation tests yield asymptotically valid tests for certain extended null hypotheses given by functionals which are strictly larger than the common i.i.d. null hypothesis. As example tests concerning the von Mises functional and the Wilcoxon two-sample test are treated.     

QuickMMCTest: quick multiple Monte Carlo testing

Multiple hypothesis testing is widely used to evaluate scientific studies involving statistical tests. However, for many of these tests, p values are not available and are thus often approximated using Monte Carlo tests such as permutation tests or bootstrap tests. This article presents a simple algorithm based on Thompson Sampling to test multiple hypotheses. It works with arbitrary multiple testing procedures, in particular with step-up and step-down procedures. Its main feature is to sequentially allocate Monte Carlo effort, generating more Monte Carlo samples for tests whose decisions are so far less certain. A simulation study demonstrates that for a low computational effort, the new approach yields a higher power and a higher degree of reproducibility of its results than previously suggested methods.     

Large sample results for permutation tests of association

This paper deals with the asymptotics of permutation tests based on a certain rather general class of measures of association for R by C contingency tables, given marginal totals. This class includes the classical chi-square test, the T _b and γ indices of Goodman and Kruskall (1954) and the popular Rand (1971) index. The asymptotic distribution of this class of permutation tests for association is a weighted sum of non-central (gen-erally speaking) chi-squares. The formulae for the asymptotic moments of such tests are also given. If non-centrality holds under the null hypothe-sis of independence, the distribution in question converges to the normal distribution. The efficacies for such measures of association are obtained. Several applications are analysed in detail, including the above mentioned indices. Approximations to the permutation distribution are also discussed.     

An exact distribution-free test comparing two multivariate distributions based on adjacency

Summary.  A new test is proposed comparing two multivariate distributions by using distances between observations. Unlike earlier tests using interpoint distances, the new test statistic has a known exact distribution and is exactly distribution free. The interpoint distances are used to construct an optimal non-bipartite matching, i.e. a matching of the observations into disjoint pairs to minimize the total distance within pairs. The cross-match statistic is the number of pairs containing one observation from the first distribution and one from the second. Distributions that are very different will exhibit few cross-matches. When comparing two discrete distributions with finite support, the test is consistent against all alternatives. The test is applied to a study of brain activation measured by functional magnetic resonance imaging during two linguistic tasks, comparing brains that are impaired by arteriovenous abnormalities with normal controls. A second exact distribution-free test is also discussed: it ranks the pairs and sums the ranks of the cross-matched pairs.     

Log‐rank permutation tests for trend: saddlepoint p‐values and survival rate confidence intervals

Suppose p + 1 experimental groups correspond to increasing dose levels of a treatment and all groups are subject to right censoring. In such instances, permutation tests for trend can be performed based on statistics derived from the weighted log‐rank class. This article uses saddlepoint methods to determine the mid‐P‐values for such permutation tests for any test statistic in the weighted log‐rank class. Permutation simulations are replaced by analytical saddlepoint computations which provide extremely accurate mid‐P‐values that are exact for most practical purposes and almost always more accurate than normal approximations. The speed of mid‐P‐value computation allows for the inversion of such tests to determine confidence intervals for the percentage increase in mean (or median) survival time per unit increase in dosage. The Canadian Journal of Statistics 37: 5‐16; 2009 © 2009 Statistical Society of Canada     

A Statistical Framework for Hypothesis Testing in Real Data Comparison Studies

In computational sciences, including computational statistics, machine learning, and bioinformatics, it is often claimed in articles presenting new supervised learning methods that the new method performs better than existing methods on real data, for instance in terms of error rate. However, these claims are often not based on proper statistical tests and, even if such tests are performed, the tested hypothesis is not clearly defined and poor attention is devoted to the Type I and Type II errors. In the present article, we aim to fill this gap by providing a proper statistical framework for hypothesis tests that compare the performances of supervised learning methods based on several real datasets with unknown underlying distributions. After giving a statistical interpretation of ad hoc tests commonly performed by computational researchers, we devote special attention to power issues and outline a simple method of determining the number of datasets to be included in a comparison study to reach an adequate power. These methods are illustrated through three comparison studies from the literature and an exemplary benchmarking study using gene expression microarray data. All our results can be reproduced using R codes and datasets available from the companion website http://www.ibe.med.uni-muenchen.de/organisation/mitarbeiter/020_professuren/boulesteix/compstud2013.     

MMCTest—A Safe Algorithm for Implementing Multiple Monte Carlo Tests

Consider testing multiple hypotheses using tests that can only be evaluated by simulation, such as permutation tests or bootstrap tests. This article introduces MMCTest , a sequential algorithm that gives, with arbitrarily high probability, the same classification as a specific multiple testing procedure applied to ideal p‐values. The method can be used with a class of multiple testing procedures that include the Benjamini and Hochberg false discovery rate procedure and the Bonferroni correction controlling the familywise error rate. One of the key features of the algorithm is that it stops sampling for all the hypotheses that can already be decided as being rejected or non‐rejected. MMCTest can be interrupted at any stage and then returns three sets of hypotheses: the rejected, the non‐rejected and the undecided hypotheses. A simulation study motivated by actual biological data shows that MMCTest is usable in practice and that, despite the additional guarantee, it can be computationally more efficient than other methods.     

DATA‐DRIVEN SMOOTH TESTS AND A DIAGNOSTIC TOOL FOR LACK‐OF‐FIT FOR CIRCULAR DATA

Two contributions to the statistical analysis of circular data are given. First we construct data‐driven smooth goodness‐of‐fit tests for the circular von Mises assumption. As a second method, we propose a new graphical diagnostic tool for the detection of lack‐of‐fit for circular distributions. We illustrate our methods on two real datasets.     

A comparison of recent nonparametric methods for testing effects in two-by-two factorial designs

The two-way two-levels crossed factorial design is a commonly used design by practitioners at the exploratory phase of industrial experiments. The F-test in the usual linear model for analysis of variance (ANOVA) is a key instrument to assess the impact of each factor and of their interactions on the response variable. However, if assumptions such as normal distribution and homoscedasticity of errors are violated, the conventional wisdom is to resort to nonparametric tests. Nonparametric methods, rank-based as well as permutation, have been a subject of recent investigations to make them effective in testing the hypotheses of interest and to improve their performance in small sample situations. In this study, we assess the performances of some nonparametric methods and, more importantly, we compare their powers. Specifically, we examine three permutation methods (Constrained Synchronized Permutations, Unconstrained Synchronized Permutations and Wald-Type Permutation Test), a rank-based method (Aligned Rank Transform) and a parametric method (ANOVA-Type Test). In the simulations, we generate datasets with different configurations of distribution of errors, variance, factor's effect and number of replicates. The objective is to elicit practical advice and guides to practitioners regarding the sensitivity of the tests in the various configurations, the conditions under which some tests cannot be used, the tradeoff between power and type I error, and the bias of the power on one main factor analysis due to the presence of effect of the other factor. A dataset from an industrial engineering experiment for thermoformed packaging production is used to illustrate the application of the various methods of analysis, taking into account the power of the test suggested by the objective of the experiment.     

A similarity analysis of curves

The authors propose a method for comparing two samples of curves. The notion of similarity between two curves is the basis of three statistics they suggest for testing the null hypothesis of no difference between the two groups. They exploit standard tools from functional data analysis to preprocess the observed curves and use the permutation distribution under the null hypothesis to obtain p‐values for their tests. They explore the operating characteristics of these tests through simulations and as an application, compare the ganglioside distribution in brain tissue between old and young rats.     

Multivariate multi-sample tests for location based on data depth

A notion of data depth is used to measure centrality or outlyingness of a data point in a given data cloud. In the context of data depth, the point (or points) having maximum depth is called as deepest point (or points). In the present work, we propose three multi-sample tests for testing equality of location parameters of multivariate populations by using the deepest point (or points). These tests can be considered as extensions of two-sample tests based on the deepest point (or points). The proposed tests are implemented through the idea of Fisher's permutation test. Performance of earlier tests is studied by simulation. Illustration with two real datasets is also provided.     

Hypothesis testing, power and sample size determination for between group comparisons in fMRI experiments

Modern methods for imaging the human brain, such as functional magnetic resonance imaging (fMRI) present a range of challenging statistical problems. In this paper, we first develop a large sample based test for between group comparisons and use it to determine the necessary sample size in order to obtain a target power via simulation under various alternatives for a given pre-specified significance level. Both testing and sample size calculations are particularly critical for neuroscientists who use these new techniques, since each subject is expensive to image.     

IDENTIFYING TREATMENT EFFECTS IN MULTI-CHANNEL MEASUREMENTS IN ELECTROENCEPHALOGRAPHIC STUDIES: MULTIVARIATE PERMUTATION TESTS AND MULTIPLE COMPARISONS

A versatile procedure is described comprising an application of statistical techniques to the analysis of the large, multi‐dimensional data arrays produced by electroencephalographic (EEG) measurements of human brain function. Previous analytical methods have been unable to identify objectively the precise times at which statistically significant experimental effects occur, owing to the large number of variables (electrodes) and small number of subjects, or have been restricted to two‐treatment experimental designs. Many time‐points are sampled in each experimental trial, making adjustment for multiple comparisons mandatory. Given the typically large number of comparisons and the clear dependence structure among time‐points, simple Bonferroni‐type adjustments are far too conservative. A three‐step approach is proposed: (i) summing univariate statistics across variables; (ii) using permutation tests for treatment effects at each time‐point; and (iii) adjusting for multiple comparisons using permutation distributions to control family‐wise error across the whole set of time‐points. Our approach provides an exact test of the individual hypotheses while asymptotically controlling family‐wise error in the strong sense, and can provide tests of interaction and main effects in factorial designs. An application to two experimental data sets from EEG studies is described, but the approach has application to the analysis of spatio‐temporal multivariate data gathered in many other contexts.     

A fast algorithm for constructing efficient event-related functional magnetic resonance imaging designs

We propose a novel, efficient approach for obtaining high-quality experimental designs for event-related functional magnetic resonance imaging (ER-fMRI), a popular brain mapping technique. Our proposed approach combines a greedy hill-climbing algorithm and a cyclic permutation method. When searching for optimal ER-fMRI designs, the proposed approach focuses only on a promising restricted class of designs with equal frequency of occurrence across stimulus types. The computational time is significantly reduced. We demonstrate that our proposed approach is very efficient compared with a recently proposed genetic algorithm approach. We also apply our approach in obtaining designs that are robust against misspecification of error correlations.     

Exact tests for matched pairs:studies with covariates

In prospective or retrospective studies with matched pairs one often wishes to control for covariates other than those used in the matching process.Large sample procedures assuming a logistic model are available for this problem.The present paper presents some exact permutation tests which are uniformly most powerful unbiased within a large class of tests.