Wilcoxon's signed‐rank statistic: what null hypothesis and why it matters

In statistical literature, the term ‘signed‐rank test’ (or ‘Wilcoxon signed‐rank test’) has been used to refer to two distinct tests: a test for symmetry of distribution and a test for the median of a symmetric distribution, sharing a common test statistic. To avoid potential ambiguity, we propose to refer to those two tests by different names, as ‘test for symmetry based on signed‐rank statistic’ and ‘test for median based on signed‐rank statistic’, respectively. The utility of such terminological differentiation should become evident through our discussion of how those tests connect and contrast with sign test and one‐sample t‐test. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

Non‐Parametric Change‐Point Tests for Long‐Range Dependent Data

Abstract. We propose a non‐parametric change‐point test for long‐range dependent data, which is based on the Wilcoxon two‐sample test. We derive the asymptotic distribution of the test statistic under the null hypothesis that no change occurred. In a simulation study, we compare the power of our test with the power of a test which is based on differences of means. The results of the simulation study show that in the case of Gaussian data, our test has only slightly smaller power minus.3pt than the ‘difference‐of‐means’ test. For heavy‐tailed data, our test outperforms the ‘difference‐of‐means’ test.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

Iso‐chi‐squared testing of 2 × k: ordered tables

The Pearson chi‐squared statistic for testing the equality of two multinomial populations when the categories are nominal is much less appropriate for ordinal categories. Test statistics typically used in this context are based on scorings of the ordinal levels, but the results of these tests are highly dependent on the choice of scores. The authors propose a test which naturally modifies the Pearson chi‐squared statistic to incorporate the ordinal information. The proposed test statistic does not depend on the scores and under the null hypothesis of equality of populations, it is asymptotically equivalent to the likelihood ratio test against the alternative of two‐sided likelihood ratio ordering.     

Simultaneous confidence interval for assessing non‐inferiority with assay sensitivity in a three‐arm trial with binary endpoints

A three‐arm trial including an experimental treatment, an active reference treatment and a placebo is often used to assess the non‐inferiority (NI) with assay sensitivity of an experimental treatment. Various hypothesis‐test‐based approaches via a fraction or pre‐specified margin have been proposed to assess the NI with assay sensitivity in a three‐arm trial. There is little work done on confidence interval in a three‐arm trial. This paper develops a hybrid approach to construct simultaneous confidence interval for assessing NI and assay sensitivity in a three‐arm trial. For comparison, we present normal‐approximation‐based and bootstrap‐resampling‐based simultaneous confidence intervals. Simulation studies evidence that the hybrid approach with the Wilson score statistic performs better than other approaches in terms of empirical coverage probability and mesial‐non‐coverage probability. An example is used to illustrate the proposed approaches.     

Comparing Conditional Quantile Curves

Abstract. We consider the problem of testing the equality of J quantile curves from independent samples. A test statistic based on an L²‐distance between non‐crossing non‐parametric estimates of the quantile curves from the individual samples is proposed. Asymptotic normality of this statistic is established under the null hypothesis, local and fixed alternatives, and the finite sample properties of a bootstrap‐based version of this test statistic are investigated by means of a simulation study.     

An affine‐invariant multivariate sign test for cluster correlated data

The author presents a multivariate location model for cluster correlated observations. He proposes an affine‐invariant multivariate sign statistic for testing the value of the location parameter. His statistic is an adaptation of that proposed by Randles (2000). The author shows, under very mild conditions, that his test statistic is asymptotically distributed as a chi‐squared random variable under the null hypothesis. In particular, the test can be used for skewed populations. In the context of a general multivariate normal model, the author obtains values of his test's Pitman asymptotic efficiency relative to another test based on the overall average. He shows that there is an improvement in the relative performance of the new test as soon as intra‐cluster correlation is present Even in the univariate case, the new test can be very competitive for Gaussian data. Furthermore, the statistic is easy to compute, even for large dimensional data. The author shows through simulations that his test performs well compared to the average‐based test. He illustrates its use with real data.     

Strengthening the understanding of the relationship between survival designs and test statistics

In the planning of randomized survival trials, the role of follow‐up time of trial participants introduces a level of complexity not encountered in non‐survival trials. Of the two commonly used survival designs, one design fixes the follow‐up time whereas the other allows it to vary. When the follow‐up time is fixed the number of events varies. Conversely, when the number of events is fixed, the follow‐up time varies. These two designs influence test statistics in ways that have not been fully explored resulting in a misunderstanding of the design–test statistic relationship. We use examples from the literature to strengthen the understanding of this relationship. Group sequential trials are briefly discussed. When the number of events is fixed, we demonstrate why a two‐sample risk difference test statistic reduces to a one‐sample test statistic which is nearly equal to the risk ratio test statistic. Some aspects of fixed event designs that need further consideration are also discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

The score test for the two‐sample occupancy model

The score test statistic from the observed information is easy to compute numerically. Its large sample distribution under the null hypothesis is well known and is equivalent to that of the score test based on the expected information, the likelihood‐ratio test and the Wald test. However, several authors have noted that under the alternative hypothesis this no longer holds and in particular the score statistic from the observed information can take negative values. We extend the anthology on the score test to a problem of interest in ecology when studying species occurrence. This is the comparison of two zero‐inflated binomial random variables from two independent samples under imperfect detection. An analysis of eigenvalues associated with the score test in this setting assists in understanding why using the observed information matrix in the score test can be problematic. We demonstrate through a combination of simulations and theoretical analysis that the power of the score test calculated under the observed information decreases as the populations being compared become more dissimilar. In particular, the score test based on the observed information is inconsistent. Finally, we propose a modified rule that rejects the null hypothesis when the score statistic is computed using the observed information is negative or is larger than the usual chi‐square cut‐off. In simulations in our setting this has power that is comparable to the Wald and likelihood ratio tests and consistency is largely restored. Our new test is easy to use and inference is possible. Supplementary material for this article is available online as per journal instructions.     

Two‐Sample Bootstrap Hypothesis Tests for Three‐Dimensional Labelled Landmark Data

Abstract. We investigate resampling methodologies for testing the null hypothesis that two samples of labelled landmark data in three dimensions come from populations with a common mean reflection shape or mean reflection size‐and‐shape. The investigation includes comparisons between (i) two different test statistics that are functions of the projection onto tangent space of the data, namely the James statistic and an empirical likelihood statistic; (ii) bootstrap and permutation procedures; and (iii) three methods for resampling under the null hypothesis, namely translating in tangent space, resampling using weights determined by empirical likelihood and using a novel method to transform the original sample entirely within refection shape space. We present results of extensive numerical simulations, on which basis we recommend a bootstrap test procedure that we expect will work well in practise. We demonstrate the procedure using a data set of human faces, to test whether humans in different age groups have a common mean face shape.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Testing parametric models in linear‐directional regression

This paper presents a goodness‐of‐fit test for parametric regression models with scalar response and directional predictor, that is, a vector on a sphere of arbitrary dimension. The testing procedure is based on the weighted squared distance between a smooth and a parametric regression estimator, where the smooth regression estimator is obtained by a projected local approach. Asymptotic behaviour of the test statistic under the null hypothesis and local alternatives is provided, jointly with a consistent bootstrap algorithm for application in practice. A simulation study illustrates the performance of the test in finite samples. The procedure is applied to test a linear model in text mining.     

Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses

Abstract. A right‐censored version of a U ‐statistic with a kernel of degree m 1 is introduced by the principle of a mean preserving reweighting scheme which is also applicable when the dependence between failure times and the censoring variable is explainable through observable covariates. Its asymptotic normality and an expression of its standard error are obtained through a martingale argument. We study the performances of our U ‐statistic by simulation and compare them with theoretical results. A doubly robust version of this reweighted U ‐statistic is also introduced to gain efficiency under correct models while preserving consistency in the face of model mis‐specifications. Using a Kendall's kernel, we obtain a test statistic for testing homogeneity of failure times for multiple failure causes in a multiple decrement model. The performance of the proposed test is studied through simulations. Its usefulness is also illustrated by applying it to a real data set on graft‐versus‐host‐disease.     

Asymptotic properties of likelihood ratio test statistics in affected‐sib‐pair analysis

In an affected‐sib‐pair genetic linkage analysis, identical by descent data for affected sib pairs are routinely collected at a large number of markers along chromosomes. Under very general genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBD data should thus utilize this information to improve efficiency. At the same time, this constraint renders the usual regularity conditions for likelihood‐based statistical methods unsatisfied. In this paper, the authors study the asymptotic properties of the likelihood ratio test (LRT) under the possible triangle constraint. They derive the limiting distribution of the LRT statistic based on data from a single locus. They investigate the precision of the asymptotic distribution and the power of the test by simulation. They also study the test based on the supremum of the LRT statistics over the markers distributed throughout a chromosome. Instead of deriving a limiting distribution for this test, they use a mixture of chi‐squared distributions to approximate its true distribution. Their simulation results show that this approach has desirable simplicity and satisfactory precision.     

One‐Way anova for Functional Data via Globalizing the Pointwise F‐test

In this paper, we propose and study a new global test, namely, GPF test, for the one‐way anova problem for functional data, obtained via globalizing the usual pointwise F‐test. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be root‐n consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the one‐way anova for functional data. Via some simulation studies, it is found that in terms of size‐controlling and power, the GPF test is comparable with two existing tests adopted for the one‐way anova problem for functional data. A real data example illustrates the GPF test.     

Enhancements of Non‐parametric Generalized Likelihood Ratio Test: Bias Correction and Dimension Reduction

Non‐parametric generalized likelihood ratio test is a popular method of model checking for regressions. However, there are two issues that may be the barriers for its powerfulness: existing bias term and curse of dimensionality. The purpose of this paper is thus twofold: a bias reduction is suggested and a dimension reduction‐based adaptive‐to‐model enhancement is recommended to promote the power performance. The proposed test statistic still possesses the Wilks phenomenon and behaves like a test with only one covariate. Thus, it converges to its limit at a much faster rate and is much more sensitive to alternative models than the classical non‐parametric generalized likelihood ratio test. As a by‐product, we also prove that the bias‐corrected test is more efficient than the one without bias reduction in the sense that its asymptotic variance is smaller. Simulation studies and a real data analysis are conducted to evaluate of proposed tests.     

An improved test of equality of mean directions for the Langevin‐von Mises‐Fisher distribution

A multi‐sample test for equality of mean directions is developed for populations having Langevin‐von Mises‐Fisher distributions with a common unknown concentration. The proposed test statistic is a monotone transformation of the likelihood ratio. The high‐concentration asymptotic null distribution of the test statistic is derived. In contrast to previously suggested high‐concentration tests, the high‐concentration asymptotic approximation to the null distribution of the proposed test statistic is also valid for large sample sizes with any fixed nonzero concentration parameter. Simulations of size and power show that the proposed test outperforms competing tests. An example with three‐dimensional data from an anthropological study illustrates the practical application of the testing procedure.     

Testing Semiparametric Hypotheses in Locally Stationary Processes

In this paper, we investigate the problem of testing semiparametric hypotheses in locally stationary processes. The proposed method is based on an empirical version of the L₂‐distance between the true time varying spectral density and its best approximation under the null hypothesis. As this approach only requires estimation of integrals of the time varying spectral density and its square, we do not have to choose a smoothing bandwidth for the local estimation of the spectral density – in contrast to most other procedures discussed in the literature. Asymptotic normality of the test statistic is derived both under the null hypothesis and the alternative. We also propose a bootstrap procedure to obtain critical values in the case of small sample sizes. Additionally, we investigate the finite sample properties of the new method and compare it with the currently available procedures by means of a simulation study. Finally, we illustrate the performance of the new test in two data examples, one regarding log returns of the S&P 500 and the other a well‐known series of weekly egg prices.     

Sample size calculation for the one‐sample log‐rank test

In this paper, an exact variance of the one‐sample log‐rank test statistic is derived under the alternative hypothesis, and a sample size formula is proposed based on the derived exact variance. Simulation results showed that the proposed sample size formula provides adequate power to design a study to compare the survival of a single sample with that of a standard population. Copyright © 2014 John Wiley & Sons, Ltd.     

Two‐Sample Test Against One‐Sided Alternatives

Abstract. This paper proposes, implements and investigates a new non‐parametric two‐sample test for detecting stochastic dominance. We pose the question of detecting the stochastic dominance in a non‐standard way. This is motivated by existing evidence showing that standard formulations and pertaining procedures may lead to serious errors in inference. The procedure that we introduce matches testing and model selection. More precisely, we reparametrize the testing problem in terms of Fourier coefficients of well‐known comparison densities. Next, the estimated Fourier coefficients are used to form a kind of signed smooth rank statistic. In such a setting, the number of Fourier coefficients incorporated into the statistic is a smoothing parameter. We determine this parameter via some flexible selection rule. We establish the asymptotic properties of the new test under null and alternative hypotheses. The finite sample performance of the new solution is demonstrated through Monte Carlo studies and an application to a set of survival times.