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Given independent samples from P and Q, two-sample permutation tests allow one to construct exact level tests when the null hypothesis is P = Q. On the other hand, when comparing or testing particular parameters θ of P and Q, such as their means or medians, permutation tests need not be level α, or even approximately level α in large samples. Under very weak assumptions for comparing estimators, we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability α in finite samples when the underlying distributions are identical. The ideas are broadly applicable and special attention is given to the k-sample problem of comparing general parameters, whereby a permutation test is constructed which is exact level α under the hypothesis of identical distributions, but has asymptotic rejection probability α under the more general null hypothesis of equality of parameters. A Monte Carlo simulation study is performed as well. A quite general theory is possible based on a coupling construction, as well as a key contiguity argument for the multinomial and multivariate hypergeometric distributions.

The most popular ways to test for independence of two ordinal random variables are by means of Kendall's tau and Spearman's rho. However, such tests are not consistent, only having power for alternatives with \"monotonie\" association. In this paper, we introduce a natural extension of Kendall's tau, called τ*, which is non-negative and zero if and only if independence holds, thus leading to a consistent independence test. Furthermore, normalization gives a rank correlation which can be used as a measure of dependence, taking values between zero and one. A comparison with alternative measures of dependence for ordinal random variables is given, and it is shown that, in a well-defined sense, τ* is the simplest, similarly to Kendall's tau being the simplest of ordinal measures of monotone association. Simulation studies show our test compares well with the alternatives in terms of average p-values.

In biomedical research, weighted logrank tests are frequently applied to compare two samples of randomly right censored survival times. We address the question how to combine a number of weighted logrank statistics to achieve good power of the corresponding survival test for a whole linear space or cone of alternatives, which are given by hazard rates. This leads to a new class of semiparametric projection tests that are motivated by likelihood ratio tests for an asymptotic model. We show that these tests can be carried out as permutation tests and discuss their asymptotic properties. A simulation study together with the analysis of a classical data set illustrates the advantages.

The search for species associations is one of the classical problems of community ecology. This article proposes to use Kendall's coefficient of concordance (W) to identify groups of significantly associated species in field survey data. An overall test of independence of all species is first carried out. If the null hypothesis is rejected, one looks for groups of correlated species and, within each group, tests the contribution of each species to the overall statistic, using a permutation test. A field survey of oribatid mites in the peat blanket surrounding a bog lake is presented as an example. In the permutation framework, an a posteriori test of the contribution of each \"judge\" (species) to the overall W concordance statistic is possible; this is not the case in the classical testing framework. A simulation study showed that when the number of judges is small, which is the case in most real-life applications of Kendall's test of concordance, the classical X2 test is overly conservative, whereas the permutation test has correct Type I error; power of the permutation test is thus also higher. The interpretation and usefulness of the a posteriori tests are discussed in the framework of environmental studies. They can help identify groups of concordant species that can be used as indices of the quality of environment, in particular in cases of pollution or contamination of the environment.

Segmented line regression models, which are composed of continuous linear phases, have been applied to describe changes in rate trend patterns. In this article, we propose a procedure to compare two segmented line regression functions, specifically to test (i) whether the two segmented line regression functions are identical or (ii) whether the two mean functions are parallel allowing different intercepts. A general form of the test statistic is described and then the permutation procedure is proposed to estimate the p‐value of the test. The permutation test is compared to an approximate F‐test in terms of the p‐value estimation and the performance of the permutation test is studied via simulations. The tests are applied to compare female lung cancer mortality rates between two registry areas and also to compare female breast cancer mortality rates between two states.

In observational microarray studies, issues of confounding invariably arise. One approach to account for measured confounders is to include them as covariates in a multivariate linear model. For this model, however, the application of permutation‐based multiple testing procedures is problematic because exchangeability of responses, in general, does not hold. Nevertheless, it is possible to achieve rotatability of transformed responses at the cost of a distributional assumption. We argue that rotation‐based multiple testing, by allowing for adjustments for confounding, represents an important extension of permutation‐based multiple testing procedures. The proposed methodology is illustrated with a microarray observational study on breast cancer tumors. Software to perform the procedure described in this article is available in the flip R package.

Random graph null models have found widespread application in diverse research communities analyzing network datasets, including social, information, and economic networks, as well as food webs, protein-protein interactions, and neuronal networks. The most popular random graph null models, called configuration models, are defined as uniform distributions over a space of graphs with a fixed degree sequence. Commonly, properties of an empirical network are compared to properties of an ensemble of graphs from a configuration model in order to quantify whether empirical network properties are meaningful or whether they are instead a common consequence of the particular degree sequence. In this work we study the subtle but important decisions underlying the specification of a configuration model, and we investigate the role these choices play in graph sampling procedures and a suite of applications. We place particular emphasis on the importance of specifying the appropriate graph labeling—stub-labeled or vertex-labeled—under which to consider a null model, a choice that closely connects the study of random graphs to the study of random contingency tables. We show that the choice of graph labeling is inconsequential for studies of simple graphs, but can have a significant impact on analyses of multigraphs or graphs with self-loops. The importance of these choices is demonstrated through a series of three in-depth vignettes, analyzing three different network datasets under many different configuration models and observing substantial differences in study conclusions under different models. We argue that in each case, only one of the possible configuration models is appropriate. While our work focuses on undirected static networks, it aims to guide the study of directed networks, dynamic networks, and all other network contexts that are suitably studied through the lens of random graph null models.

Segmented line regression has been used in many applications, and the problem of estimating the number of change-points in segmented line regression has been discussed in Kim et al. (2000). This paper studies asymptotic properties of the number of change-points selected by the permutation procedure of Kim et al. (2000). This procedure is based on a sequential application of likelihood ratio type tests, and controls the over-fitting probability by its design. In this paper we show that, under some conditions, the number of change-points selected by the permutation procedure is consistent. Via simulations, the permutation procedure is compared with such information-based criterior as the Bayesian Information Criterion (BIC), the Akaike Information Criterion (AIC), and Generalized Cross Validation (GCV).

In this article, we use a Hilbert-Schmidt Independence Criterion to propose a new method for estimating directions in single-index models. This approach enjoys a model free property and requires no link function to be smoothed or estimated. Further, we propose a permutation test to check whether the estimated single-index is sufficient. The sampling distribution of our estimator is established. Finite sample performance of proposed estimates is examined through simulation studies and compared with two well-established methods: the refined Minimum Average Variance Estimation method (rMAVE, Xia et al. (2002)) and the Estimating Function Method (EFM, Cui, Härdle, and Zhu (2011)). A New Zealand Horse Mussels data set is analyzed via our approach to demonstrate the efficacy of our proposed approach.

Minimization as an alternative to randomization is gaining popularity for small clinical trials. In response to critics’ questions about the proper analysis of such a trial, proponents have argued that a rerandomization approach, akin to a permutation test with conventional randomization, can be used. However, they add that this computationally intensive approach is not necessary because its results are very similar to those of a t‐test or test of proportions unless the sample size is very small. We show that minimization applied with unequal allocation causes problems that challenge this conventional wisdom.