Fourth exact moment results for mrbp tests with two or three treatments

MRBP tests were proposed by Mielke and Iyer (1982) to analyze multivariate data for the randomized block design, based on permutation procedures. They obtained the first three exact moments of the MRBP test statistic to approximate its permutation distribution. Tracy and Khan (1991) derived its fourth exact moment, to obtain a better approximating distribution, when there are four or more treatments. In this paper we obtain the fourth exact moment when the number of treatments is less than four.     

Approximations to permutation distributions in the completely randomized design

The permutation distribution of a statistic T equivalent to the usual F ratio for the completely randomized design is considered. A correction to the second moment of T derived by Robinson (1983) is presented and the third and fourth moments are educed. Inadequacies in the conventional permutation distribution approximations are demonstrated.     

PERMUTATION TESTS AND OTHER TEST STATISTICS FOR ILL-BEHAVED DATA: EXPERIENCE OF THE NINDS t-PA STROKE TRIAL

Permutation tests are often used to analyze data since they may not require one to make assumptions regarding the form of the distribution to have a random and independent sample selection. We initially considered a permutation test to assess the treatment effect on computed tomography lesion volume in the National Institute of Neurological Disorders and Stroke (NINDS) t-PA Stroke Trial, which has highly skewed data. However, we encountered difficulties in summarizing the permutation test results on the lesion volume. In this paper, we discuss some aspects of permutation tests and illustrate our findings. This experience with the NINDS t-PA Stroke Trial data emphasizes that permutation tests are useful for clinical trials and can be used to validate assumptions of an observed test statistic. The permutation test places fewer restrictions on the underlying distribution but is not always distribution-free or an exact test, especially for ill-behaved data. Quasi-likelihood estimation using the generalized estimating equation (GEE) approach on transformed data seems to be a good choice for analyzing CT lesion data, based on both its corresponding permutation test and its clinical interpretation.     

Simulated Power Curves for some Location Tests

In this paper we compare the power properties of some location tests. The most widely used such test is Student's t. Recently bootstrap-based tests have received much attention in the literature. A bootstrap version of the t-test will be included in our comparison. Finally, the nonparametric tests based on the idea of permuting the signs will be represented in our comparison. Again, we will initially concentrate on a version of that test based on the mean. The permutation tests predate the bootstrap by about fourty years. Theoretical results of Pitman (1937) and Bickel & Freedman (1981) show that these three methods are asymptotically equivalent if the underlying distribution is symmetric and has finite second moment. In the modern literature, the use of the nonparametric techniques is advocated on the grounds that the size of the test would be either exact, or more nearly exact. In this paper we report on a simulation study that compares the power curves and we show that it is not necessary to use resampling tests with a statistic based on the mean of the sample.     

A Permutation Test for Planar Regression

Oja (1987) presents some distribution-free tests applicable in the presence of covariates when treatment values are randomly assigned. The formulas and calculations are cumbersome, however, and implementation of the tests relies on using a x² approximation to the exact null distribution. In this paper a re-formulation of his test statistic is given which has the advantages of ease of calculation, explicit formulas for permutation moments, and allowing a Beta distribution to be fitted to the exact null distribution.     

Extended moment results for improving inferences based on mre'p

The MRPP test statistic studied by Mielke and others is the weighted average distance between pairs of observations within a group. They defined 12 symmetric functions to obtainits first three moments. We define 23 additional symmetric functions to obtain the fourth moment. This can beuseful instudying further approximations to its sampling distribution. We also study the special case when the distance function is the Euclidean distance between ranks of observations     

Exact moments of the symmetric cubic assignment statistic

Hubert (1987Assignment Methods in Combinatorial Data Analysis) presented a class of permutation, or random assignment, techniques for assessing correspondence between general k-dimensional proximity measures on a set of “objects.” A major problem in higher-order assignment models is the prohibitive level of computation that is required. We present the first three exact moments of a test statistic for the symmetric cubic assignment model. Efficient computational formulas for the first three moments have been derived, thereby permitting approximation of the permutation distribution using well-known methods.     

A sequential approximation to a permutation test

A sequential method for approximating a general permutation test (SAPT) is proposed and evaluated. Permutations are randomly generated from some set G, and a sequential probability ratio test (SPRT) is used to determine whether an observed test statistic falls sufficiently far in the tail of the permutation distribution to warrant rejecting some hypothesis. An estimate and bounds on the power function of the SPRT are used to find bounds on the effective significance level of the SAPT. Guidelines are developed for choosing parameters in order to obtain a desired significance level and minimize the number of permutations needed to reach a decision. A theoretical estimate of the average number of permutations under the null hypothesis is given along with simulation results demonstrating the power and average number of permutations for various alternatives. The sequential approximation retains the generality of the permutation test,- while avoiding the computational complexities that arise in attempting to computer the full permutation distribution exactly     

A permutation test for multivariate data with grouped components

In this paper, we consider a nonparametric test procedure for multivariate data with grouped components under the two sample problem setting. For the construction of the test statistic, we use linear rank statistics which were derived by applying the likelihood ratio principle for each component. For the null distribution of the test statistic, we apply the permutation principle for small or moderate sample sizes and derive the limiting distribution for the large sample case. Also we illustrate our test procedure with an example and compare with other procedures through simulation study. Finally, we discuss some additional interesting features as concluding remarks.     

Exact distribution-free tests for equality of several linear models

Exact tests for the equality of several linear models are developed using permutation techniques. Two cases of the linear model, characterized by either stochastic or nonstochastic predictors, are considered: the linear regression model (LRM) and the general linear model (GLM). A general class of test statistics using the volume of simplexes as the basic unit of analysis is proposed for this problem. The resulting class of statistics is shown to be a natural generalization of the multi-response permutation procedure (MRPP) test statistics which have been shown to comprise many of the statistics used in both parametric and nonparametric analysis of the standard g—sample problem. In the LRM case, exact moments of all orders are derived for the permutation distribution of any test statistic in the general class. Moment-based approximation of significance levels is shown to be computationally feasible in the simple LRM.     

Small-sample comparisons for the Rukhin goodness-of-fit-statistics

Rukhin's statistic family for goodness-of-fit, under the null hypothesis, has asymptotic chi-squared distribution; however, for small samples the chi-squared approximation in some cases does not well agree with the exact distribution. In this paper we consider this approximation and other three to get appropriate test levels in comparison with the exact level. Moreover, exact power comparisons for several values of the parameter under specified alternatives provide that the classical Pearson's statistic, obtained as a particular case of Rukhin statistic, can be improved by choosing other statistics from the family. An explanation is proposed in terms of the effects of individual cell frequencies on the Rukhin statistic. This work was supported in part by the DGCYT grants No. PR156/97-7159 and PB96-0635     

A minimum variance adaptive technique for parameter estimation and hypothesis testing

This report presents numerical results of an approach for parameter estimation and hypothesis testing that does not rely on specific assumptions about the underlying distribution of errors in the measured data. This approach combines robust estimation procedures, the bootstrap method for estimation of parameter uncertainties, permutation techniques for hypothesis testing, and adaptive approaches to estimation in order to obtain the minimum variance estimator or test statistic (within a predefined class) for the data under consideration. The technique produces efficient estimators of central tendency and powerful test statistics, even for small sample sizes. (Portions of this work have been presented in preliminary form (Turkheimer et al., 1996)).     

Nearly exact tests in factorial experiments using the aligned rank transform

A procedure is studied that uses rank-transformed data to perform exact and estimated exact tests, which is an alternative to the commonly used F-ratio test procedure. First, a common parametric test statistic is computed using rank-transformed data, where two methods of ranking-ranks taken for the original observations and ranks taken after aligning the observations-are studied. Significance is then determined using either the exact permutation distribution of the statistic or an estimate of this distribution based on a random sample of all possible permutations. Simulation studies compare the performance of this method with the normal theory parametric F-test and the traditional rank transform procedure. Power and nominal type I error rates are compared under conditions when normal theory assumptions are satisfied, as well as when these assumptions are violated. The method is studied for a two-factor factorial arrangement of treatments in a completely randomized design and for a split-unit experiment. The power of the tests rivals the parametric F-test when normal theory assumptions are satisfied, and is usually superior when normal theory assumptions are not satisfied. Based on the evidence of this study, the exact aligned rank procedure appears to be the overall best choice for performing tests in a general factorial experiment.     

Empirical power comparisons of two mrpp rank tests

Simulated powers of the MRPP two-sample rank test statistic ?₁- are compared with the powers of the MRPP test statistic ?₂(the two-sided Wilcoxon-Mann-Whitney test) for large samples from several underlying populations. Powers are obtained using two approximate distributions of ?₁ involving three and four moments

respectively, The use of the fourth moment indicates that an approximation to the null distribution of ? based on four moments can perform better     

The Multi-Sample Block-Scalar Sphericity Test: Exact and Near-Exact Distributions for Its Likelihood Ratio Test Statistic

In this article the authors show how by adequately decomposing the null hypothesis of the multi-sample block-scalar sphericity test it is possible to obtain the likelihood ratio test statistic as well as a different look over its exact distribution. This enables the construction of well-performing near-exact approximations for the distribution of the test statistic, whose exact distribution is quite elaborate and non-manageable. The near-exact distributions obtained are manageable and perform much better than the available asymptotic distributions, even for small sample sizes, and they show a good asymptotic behavior for increasing sample sizes as well as for increasing number of variables and/or populations involved.     

On tests of linearity for dose response data: asymptotic, exact conditional and exact unconditional tests

The approximate chi-square statistic, X 2 Q , which is calculated as the difference between the usual chi-square statistic for heterogeneity and the Cochran-Armitage trend test statistic, has been widely applied to test the linearity assumption for dose-response data. This statistic can be shown to be asymptotically distributed as chi-square with K - 2 degrees of freedom. However, this asymptotic property could be quite questionable if the sample size is small, or if there is a high degree of sparseness or imbalance in the data. In this article, we consider how exact tests based on this X 2 Q statistic can be performed. Both the exact conditional and unconditional versions will be studied. Interesting findings include: (i) the exact conditional test is extremely sensitive to a small change in dosages, which may eventually produce a degenerate exact conditional distribution; and (ii) the exact unconditional test avoids the problem of degenerate distribution and is shown to be less sensitive to the change in dosages. A real example involving an animal carcinogenesis experiment as well as a fictitious data set will be used for illustration purposes.     

A NONPARAMETRIC TEST FOR THE EQUALITY OF DEPENDENT CORRELATION COEFFICIENTS UNDER NORMALITY

ABSTRACT

A nonparametric testing method for the equality of two correlation coefficients in trivariate normal distribution, namely, one of the variables are common, is discussed. Using a permutation test, we obtain asymptotically exact solutions. The performance of this test is compared with the likelihood ratio test and a method of using the limiting distribution of correlation coefficients.     

Near-exact distributions for the sphericity likelihood ratio test statistic

In this paper three near-exact distributions are developed for the sphericity test statistic. The exact probability density function of this statistic is usually represented through the use of the Meijer G function, which renders the computation of quantiles impossible even for a moderately large number of variables. The main purpose of this paper is to obtain near-exact distributions that lie closer to the exact distribution than the asymptotic distributions while, at the same time, correspond to density and cumulative distribution functions practical to use, allowing for an easy determination of quantiles. In addition to this, two asymptotic distributions that lie closer to the exact distribution than the existing ones were developed. Two measures are considered to evaluate the proximity between the exact and the asymptotic and near-exact distributions developed. As a reference we use the saddlepoint approximations developed by Butler et al. [1993. Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc., Ser. B 55, 171–183] as well as the asymptotic distribution proposed by Box.     

A Laguerre polynomial approximation for a goodness-of-fit test for exponential distribution based on progressively censored data

This paper proposes an approximation to the distribution of a goodness-of-fit statistic proposed recently by Balakrishnan et al. [Balakrishnan, N., Ng, H.K.T. and Kannan, N., 2002, A test of exponentiality based on spacings for progressively Type-II censored data. In: C. Huber-Carol et al. (Eds.), Goodness-of-Fit Tests and Model Validity (Boston: Birkhäuser), pp. 89–111.] for testing exponentiality based on progressively Type-II right censored data. The moments of this statistic can be easily calculated, but its distribution is not known in an explicit form. We first obtain the exact moments of the statistic using Basu's theorem and then the density approximants based on these exact moments of the statistic, expressed in terms of Laguerre polynomials, are proposed. A comparative study of the proposed approximation to the exact critical values, computed by Balakrishnan and Lin [Balakrishnan, N. and Lin, C.T., 2003, On the distribution of a test for exponentiality based on progressively Type-II right censored spacings. Journal of Statistical Computation and Simulation, 73 (4), 277–283.], is carried out. This reveals that the proposed approximation is very accurate.     

Permutation Test for Equality of Individual an Eigenvalue from a Covariance Matrix in High-Dimension

A test statistic for examining the equality of an individual eigenvalue in two populations of high dimension is proposed. The asymptotic distribution of the proposed statistic is derived and the validity of the permutation test is discussed. Simulations were used to investigate the power of the suggested statistic. The proposed statistic is applied to hand measurement data on white and aboriginal Australians to test the equality of the eigenvalues.