Testing for No Effect by Cosine Series Methods

The properties of three lack-of-fit tests that are related to non-parametric cosine regression analysis are examined in the context of testing for a constant mean function. Analytic power comparisons of these tests vs a most powerful test are made using intermediate asymptotic relative efficiency. In particular, a data-driven test is produced which is asymptotically as efficient as the most powerful test over a class of alternatives. A small scale simulation experiment is conducted to ascertain the extent that the large sample comparisons are applicable to finite samples.     

Two permutation tests of equality of variances

The F-ratio test for equality of dispersion in two samples is by no means robust, while non-parametric tests either assume a common median, or are not very powerful. Two new permutation tests are presented, which do not suffer from either of these problems. Algorithms for Monte Carlo calculation of P values and confidence intervals are given, and the performance of the tests are studied and compared using Monte Carlo simulations for a range of distributional types. The methods used to speed up Monte Carlo calculations, e.g. stratification, are of wider applicability.     

The analysis of multicentre clinical trials when there is heterogeneity between centres

Two-treatment multicentre clinical trials are very common in practice. In cases where a non-parametric analysis is appropriate, a rank-sum test for grouped data called the van Elteren test can be applied. As an alternative approach, one may apply a combination test such as Fisher's combination test or the inverse normal combination test (also called Liptak's method) in order to combine centre-specific P-values. If there are no ties and no differences between centres with regard to the groups’ sample sizes, the inverse normal combination test using centre-specific Wilcoxon rank-sum tests is equivalent to the van Elteren test. In this paper, the van Elteren test is compared with Fisher's combination test based on Wilcoxon rank-sum tests. Data from two multicentre trials as well as simulated data indicate that Fisher's combination of P-values is more powerful than the van Elteren test in realistic scenarios, i.e. when there are large differences between the centres’ P-values, some quantitative interaction between treatment and centre, and/or heterogeneity in variability. The combination approach opens the possibility of using statistics other than the rank sum, and it is also a suitable method for more complicated designs, e.g. when covariates such as age or gender are included in the analysis.     

Power comparison of non-parametric tests: Small-sample properties from Monte Carlo experiments

SUMMARY Non-parametric tests that deal with two samples include scores tests (such as the Wilcoxon rank sum test, normal scores test, logistic scores test, Cauchy scores test, etc.) and Fisher's randomization test. Because the non-parametric tests generally require a large amount of computational work, there are few studies on small-sample properties, although asymptotic properties with regard to various aspects were studied in the past. In this paper, the non-parametric tests are compared with the t -test through Monte Carlo experiments. Also, we consider testing structural changes as an application in economics.     

On improving the min test for the analysis of combination drug trials

In the analysis of clinical trials of combination therapies, the min test is often used to demonstrate a combination therapy's superiority to its components. Although uniformly most powerful within a class of monotone tests, this test is excessively conservative with low power at certain alternatives. This paperdemonstrates that more powerful tests may be found outside of this class. Some such alternative tests are suggested and compared with the min tests on the basis of their actual significance levels and powers. The proposed tests are observed to be less conservative and uniformly more powerful than the min test.     

Maximum Test versus Adaptive Tests for the Two-Sample Location Problem

For the non-parametric two-sample location problem, adaptive tests based on a selector statistic are compared with a maximum and a sum test, respectively. When the class of all continuous distributions is not restricted, the sum test is not a robust test, i.e. it does not have a relatively high power across the different possible distributions. However, according to our simulation results, the adaptive tests as well as the maximum test are robust. For a small sample size, the maximum test is preferable, whereas for a large sample size the comparison between the adaptive tests and the maximum test does not show a clear winner. Consequently, one may argue in favour of the maximum test since it is a useful test for all sample sizes. Furthermore, it does not need a selector and the specification of which test is to be performed for which values of the selector. When the family of possible distributions is restricted, the maximin efficiency robust test may be a further robust alternative. However, for the family of t distributions this test is not as powerful as the corresponding maximum test.     

Customizing Generalizations of the Kolmogorov-Smirnov Goodness-of-fit Test

This paper presents a method of customizing goodness-of-fit tests that transforms the empirical distribution function in such a way as to create tests for certain alternatives. Using the @ , g transform described in Blom(1958), one can create non-parametric tests for an assortment of alternative distributions. As examples, three new ( f , g )-corrected Kolmogorov-Smirnov tests for goodness-of-fit are discussed. One of these tests is powerful for testing whether or not the data come from an alternative that is heavier in the tails. Another test identifies whether or not the data come from an alternative which is heavier in the middle of the distribution. The last test identifies if the data come from an alternative in which the first or third quartile is far from the corresponding quartile of the hypothesized distribution. The behavior of the three new tests is investigated through a power study.     

Combining the t test and Wilcoxon's rank-sum test

In the two-sample location-shift problem, Student's t test or Wilcoxon's rank-sum test are commonly applied. The latter test can be more powerful for non-normal data. Here, we propose to combine the two tests within a maximum test. We show that the constructed maximum test controls the type I error rate and has good power characteristics for a variety of distributions; its power is close to that of the more powerful of the two tests. Thus, irrespective of the distribution, the maximum test stabilizes the power. To carry out the maximum test is a more powerful strategy than selecting one of the single tests. The proposed test is applied to data of a clinical trial.     

Non-parametric Analysis of Covariance – The Case of Inhomogeneous and Heteroscedastic Noise

Abstract.  The purpose of this paper was to propose a procedure for testing the equality of several regression curves f _i in non-parametric regression models when the noise is inhomogeneous and heteroscedastic, i.e. when the variances depend on the regressor and may vary between groups. The presented approach is very natural because it transfers the maximum likelihood statistic from a heteroscedastic one-way analysis of variance to the context of non-parametric regression. The maximum likelihood estimators will be replaced by kernel estimators of the regression functions f _i . It is shown that the asymptotic distribution of the obtained test-statistic is nuisance parameter free. Asymptotic efficiency is compared with a test of Dette & Neumeyer [Annals of Statistics (2001) Vol. 29, 1361–1400] and it is shown that the new test is asymptotically uniformly more powerful. For practical purposes, a bootstrap variant is suggested. In a simulation study, level and power of this test will be briefly investigated and compared with other procedures. In summary, our theoretical findings are supported by this study. Finally, a crop yield experiment is reanalysed.     

A simple approach for testing constant failure rate against different ageing classes for discrete data

In this paper, we develop simple non-parametric test based on U-statistics for testing constant failure rate against IFR, IFRA, DMRL, NBU and NBUE alternatives. The asymptotic properties of the test statistics are studied. In particular, the test statistics are shown to be asymptotically normal and consistent against the relevant alternatives. Some numerical results are presented to demonstrate the performance of the proposed tests.     

Two non-parametric, analysis-of-means-type tests for homogeneity of variances

After a brief review of the literature, two non-parametric tests for homogeneity of variances are presented. The first test is based on the analysis of means for ranks, which is a non-parametric version of the analysis of means (ANOM) that uses ranks as input for an ANOM test. The second test uses inverse normal scores of the ranks of scale transformations of the observations as input to the ANOM. Both homogeneity of variances tests can be presented in a graphical form, which makes it easy for practitioners to assess the practical and the statistical significance. A Monte Carlo study is used to show that these tests have power comparable with that of well-known robust tests for homogeneity of variances.     

A combined test for differences in scale based on the interquantile range

A class of tests due to Shoemaker (Commun Stat Simul Comput 28: 189–205, 1999) for differences in scale which is valid for a variety of both skewed and symmetric distributions when location is known or unknown is considered. The class is based on the interquantile range and requires that the population variances are finite. In this paper, we firstly propose a permutation version of it that does not require the condition of finite variances and is remarkably more powerful than the original one. Secondly we solve the question of what quantile choose by proposing a combined interquantile test based on our permutation version of Shoemaker tests. Shoemaker showed that the more extreme interquantile range tests are more powerful than the less extreme ones, unless the underlying distributions are very highly skewed. Since in practice you may not know if the underlying distributions are very highly skewed or not, the question arises. The combined interquantile test solves this question, is robust and more powerful than the stand alone tests. Thirdly we conducted a much more detailed simulation study than that of Shoemaker (1999) that compared his tests to the F and the squared rank tests showing that his tests are better. Since the F and the squared rank test are not good for differences in scale, his results suffer of such a drawback, and for this reason instead of considering the squared rank test we consider, following the suggestions of several authors, tests due to Brown–Forsythe (J Am Stat Assoc 69:364–367, 1974), Pan (J Stat Comput Simul 63:59–71, 1999), O’Brien (J Am Stat Assoc 74:877–880, 1979) and Conover et al. (Technometrics 23:351–361, 1981).     

A TEST OF HOMOGENEITY AGAINST SCALE ALTERNATIVES USING SUBSAMPLE EXTREMA1

In this paper a new class of non-parametric tests for testing homogeneity of several populations against scale alternatives is proposed. For this, independent samples of fixed sizes are drawn from each population and from these samples, all possible sub-samples of the same size are drawn and their maxima and minima are computed. Using these extreme the class of tests is obtained. Tests of this type have been offered for the two-sample slippage problem by Kochar (1978). Under certain conditions, this class of tests is shown to be consistent against ‘difference in scale’ alternatives. The test has been compared with Bhapkar's V-test (1961), Deshpande's D-test (1965), Sugiura's D_rs-test (1965) and with a classical test given by Lehmann (1959, pp. 273–275). It is shown that some members of this proposed class of tests are more efficient than the first three tests in the case of uniform, Laplace and normal distributions, when the number of populations compared is small.     

Regions of alternatives with high and low power for goodness-of-fit tests

In the present paper we find finite dimensional spaces W of alternatives with high power for a given class of tests and non-parametric alternatives. On the orthogonal complement of W the power function is flat. These methods can be used to reduce the dimension of interesting alternatives. We sketch a device how to calculate (approximately) an alternative with maximum power of a fixed test on a given ball of certain non-parametric alternatives.

The calculations are done within different asymptotic models specified by signal detection tests. Specific tests are Kolmogorov–Smirnov type tests, integral tests (like the Anderson and Darling test) and Rényi tests for hazard based models. The statistical meaning and interpretation of the spaces of alternatives with high power is discussed. These alternatives belong to least favorable directions of a class of statistical functionals which are linear combinations of quantile functions. For various cases their meaning is explained for parametric submodels, in particular for location alternatives.     

Testing for Serial Independence: Beyond the Portmanteau Approach

Portmanteau tests are typically used to test serial independence even if, by construction, they are generally powerful only in presence of pairwise dependence between lagged variables. In this article, we present a simple statistic defining a new serial independence test, which is able to detect more general forms of dependence. In particular, differently from the Portmanteau tests, the resulting test is powerful also under a dependent process characterized by pairwise independence. A diagram, based on p-values from the proposed test, is introduced to investigate serial dependence. Finally, the effectiveness of the proposal is evaluated in a simulation study and with an application on financial data. Both show that the new test, used in synergy with the existing ones, helps in the identification of the true data-generating process. Supplementary materials for this article are available online.     

A comparison of some conditional and unconditional exact tests for 2x2 contingency tables

In 1935, R.A. Fisher published his well-known “exact” test for 2x2 contingency tables. This test is based on the conditional distribution of a cell entry when the rows and columns marginal totals are held fixed. Tocher (1950) and Lehmann (1959) showed that Fisher s test, when supplemented by randomization, is uniformly most powerful among all the unbiased tests UMPU). However, since all the practical tests for 2x2 tables are nonrandomized - and therefore biased the UMPU test is not necessarily more powerful than other tests of the same or lower size. Inthis work, the two-sided Fisher exact test and the UMPU test are compared with six nonrandomized unconditional exact tests with respect to their power. In both the two-binomial and double dichotomy models, the UMPU test is often less powerful than some of the unconditional tests of the same (or even lower) size. Thus, the assertion that the Tocher-Lehmann modification of Fisher's conditional test is the optimal test for 2x2 tables is unjustified.     

The approximation of an unbiased test with a small level for the bioequivalence problem

In the bioequivalence problem. Brown. Hwang and Munk (1997) constructed an unbiased level a test and other tests which are uniformly more powerful than the two one-sided tests procedures when a iscomparatively larger. In this paper, for a small level, an unbiased test is shown to be approxirnately constructeQ lor tnis prooiem oy using tneir Metnog. ine numerical construction is also given.     

An analysis of pulsation periods of long period variable stars

Summary.  We report the results of a period change analysis of time series observations for 378 pulsating variable stars. The null hypothesis of no trend in expected periods is tested for each of the stars. The tests are non-parametric in that potential trends are estimated by local linear smoothers. Our testing methodology has some novel features. First, the null distribution of a test statistic is defined to be the distribution that results in repeated sampling from a population of stars. This distribution is estimated by means of a bootstrap algorithm that resamples from the collection of 378 stars. Bootstrapping in this way obviates the problem that the conditional sampling distribution of a statistic, given a particular star, may depend on unknown parameters of that star. Another novel feature of our test statistics is that one-sided cross-validation is used to choose the smoothing parameters of the local linear estimators on which they are based. It is shown that doing so results in tests that are tremendously more powerful than analogous tests that are based on the usual version of cross-validation. The positive false discovery rate method of Storey is used to account for the fact that we simultaneously test 378 hypotheses. We ultimately find that 56 of the 378 stars have changes in mean pulsation period that are significant when controlling the positive false discovery rate at the 5% level.     

The glm log-rank test:general linear modeling of log-rank scores as a method of analysis for survival data

The use of general linear modeling (GLM) procedures based on log-rank scores is proposed for the analysis of survival data and compared to standard survival analysis procedures. For the comparison of two groups, this approach performed similarly to the traditional log-rank test. In the case of more complicated designs - without ties in the survival times - the approach was only marginally less powerful than tests from proportional hazards models, and clearly less powerful than a likelihood ratio test for a fully parametric model; however, with ties in the survival time, the approach proved more powerful than tests from Cox's semi-parametric proportional hazards procedure. The method appears to provide a reasonably powerful alternative for the analysis of survival data, is easily used in complicated study designs, avoids (semi-)parametric assumptions, and is quite computationally easy and inexpensive to employ.     

Linear and nonlinear causality tests in an LSTAR model: wavelet decomposition in a nonlinear environment

In this paper, we use simulated data to investigate the power of different causality tests in a two-dimensional vector autoregressive (VAR) model. The data are presented in a nonlinear environment that is modelled using a logistic smooth transition autoregressive function. We use both linear and nonlinear causality tests to investigate the unidirection causality relationship and compare the power of these tests. The linear test is the commonly used Granger causality F test. The nonlinear test is a non-parametric test based on Baek and Brock [A general test for non-linear Granger causality: Bivariate model. Tech. Rep., Iowa State University and University of Wisconsin, Madison, WI, 1992] and Hiemstra and Jones [Testing for linear and non-linear Granger causality in the stock price–volume relation, J. Finance 49(5) (1994), pp. 1639–1664]. When implementing the nonlinear test, we use separately the original data, the linear VAR filtered residuals, and the wavelet decomposed series based on wavelet multiresolution analysis. The VAR filtered residuals and the wavelet decomposition series are used to extract the nonlinear structure of the original data. The simulation results show that the non-parametric test based on the wavelet decomposition series (which is a model-free approach) has the highest power to explore the causality relationship in nonlinear models.