Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian covariance matrices

Power studies of tests of equality of covariance matrices of two p-variate complex normal populations σ₁ = σ₂ against two-sided alternatives have been made based on the following five criteria: (1) Roy's largest root, (2) Hotelling's trace, (4) Wilks' criterion and (5) Roy's largest and smallest roots. Some theorems on transformations and Jacobians in the two-sample complex Gaussian case have been proved in order to obtain a general theorem for establishing the local unbiasedness conditions connecting the two critical values for tests (1)–(5). Extensive unbiased power tabulations have been made for p=2, for various values of n₁, n₂, λ₁ and λ₂ where n₁ is the df of the SP matrix from the ith sample and λ₁ is the ith latent root of σ₁σ^-1₂ (i=1, 2). Equal tail areas approach has also been used further to compute powers of tests (1)–(4) for p=2 for studying the bias and facilitating comparisons with powers in the unbiased case. The inferences have been found similar to those in the real case. (Chu and Pillai, Ann. Inst. Statist. Math. 31.     

Combination tests for the equality of the covariance matrices of two dependent bivariate normals

To test the equality of the covariance matrices of two dependent bivariate normals, we derive five combination tests using the Simes method. We compare the performance of these tests using simulation to each other and to the competing tests. In particular, simulations show that one of the combination tests has the best performance in terms of controlling the type I error rate even for small samples with similar power compared to other tests. We also apply the recommended test to real data from a crossover bioavailability study.     

Generalized orthogonal contrast tests for homogeneity of ordered means

Several procedures have been proposed for testing equality of ordered means. The best-known of these is the likelihood-ratio test introduced by Bartholomew, which possesses generally superior power characteristics to those of its competitors. Difficulties in implementing this test have led to the development of alternative approaches, such as tests based on single and multiple contrasts. Some recent approaches have utilized approximations to the polyhedral cone defining the restricted parameter space, including those of Akkerboom (circular cone) and Mudholkar & McDermott (orthant). This article proposes a class of tests based on an improved orthant approximation to the polyhedral cone. These tests may be viewed as generalizations of the orthogonal contrast test proposed by Mukerjee, Robertson & Wright. Studies of the power functions of several competing tests indicate that the generalized orthogonal contrast tests are effective alternatives to the likelihood-ratio test, especially when the latter is difficult to implement.     

Distances between normal populations when covariance matrices are unequal

The definition of distance between two populations of equal covariance matrices is extended to two and more than two populations with unequal covariance matrices and Rao’s U test for testing the conditional contribution of a subset of variables to the distance is extended to this situation, even when sample sizes are not necessarily the same.     

Linear rank tests for homogeneity against ordered alternatives in anova and anocova

A unified method of constructing rank tests for homogeneity against ordered alternatives in unbalanced analysis of variance and analysis of covariance is considered. The relationship between these tests with some of the existing methods are studied. The normal theory likelihood ratio tests are also derived and the asymptotic relative efficiency comparisons, in Pitman sense, of the rank tests with respect to the likelihood ratio tests are carried out.     

Improved tests for homogeneity of variances

Equality of variances is one of the key assumptions of analysis of variances (ANOVA). There are several testing procedures available to validate this assumption, but it is rare to find a test procedure which controls the type I error rate while providing high statistical power. In this article, we introduce a bootstrap test based on the ratio of mean absolute deviances (RMD). We also propose a two-stage testing procedure where we first quantify the skewness of the distributions and then choose an appropriate test for homogeneity of variances. The performances of these test procedures are studied via a simulation study.     

A note on the power of Fisher's least significant difference procedure

Fisher's least significant difference (LSD) procedure is a two-step testing procedure for pairwise comparisons of several treatment groups. In the first step of the procedure, a global test is performed for the null hypothesis that the expected means of all treatment groups under study are equal. If this global null hypothesis can be rejected at the pre-specified level of significance, then in the second step of the procedure, one is permitted in principle to perform all pairwise comparisons at the same level of significance (although in practice, not all of them may be of primary interest). Fisher's LSD procedure is known to preserve the experimentwise type I error rate at the nominal level of significance, if (and only if) the number of treatment groups is three. The procedure may therefore be applied to phase III clinical trials comparing two doses of an active treatment against placebo in the confirmatory sense (while in this case, no confirmatory comparison has to be performed between the two active treatment groups). The power properties of this approach are examined in the present paper. It is shown that the power of the first step global test--and therefore the power of the overall procedure--may be relevantly lower than the power of the pairwise comparison between the more-favourable active dose group and placebo. Achieving a certain overall power for this comparison with Fisher's LSD procedure--irrespective of the effect size at the less-favourable dose group--may require slightly larger treatment groups than sizing the study with respect to the simple Bonferroni alpha adjustment. Therefore if Fisher's LSD procedure is used to avoid an alpha adjustment for phase III clinical trials, the potential loss of power due to the first-step global test should be considered at the planning stage.     

Two-stage procedures for the difference of two multinormal means with covariance matrices different only by unknown scalar multipliers

We consider the problem of constructing a fixed-size confidence region for the difference of means of two multivariate normal populations It is assumed that the variance-covariance matrices of two populations are different only by unknown scalar multipliers Two-stage procedures are presented to derive such a confidence region We also discuss the asymptotic efficiency of the procedure.     

A two-stage procedure for estimating a linear function of K multinormal mean vectors when covariance matrices are unknown

We consider the problem of constructing a fixed-size confidence region for a linear function of mean vectors of k multinormal populations, where all covariance matrices are completely unknown. A two-stage procedure is proposed to construct such a confidence region. It is shown that the proposed two-stage procedure is consistent and its asymptotic property for the expected sample size is also given. A Monte Carlo simulation study is given for an illustration.     

A note on the distribution of the resistance of rank tests

Measurement of the effect of extreme observations on test statistics has been discussed by many authors. Test resistance to rejection (acceptance) is one of the most appealing methods. Using the distribution of the test statistic, the exact and asymptotic distributions of the test resistance to rejection (acceptance) are introduced. The usage of the distribution is emphasized in the case of the sign test and Spearman’s rho.     

A note on asymptotic expansions for the power of perturbed tests

For simplicity or tractability reasons one sometimes uses modified test statistics, which differ from the original ones up to O_p(a_n) terms with a_n→0. In this note, some technical conditions are provided under which a corresponding expansion for the powers of such perturbed tests holds. The necessity of some of these conditions is discussed and illustrated by examples. An application to invariant testing multivariate normality is presented.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).     

Robustness of the two-sample t-test umdeh violations of the homogeneity of farjance assumptioi

When the two-sample t-test has equal sample slies, it is widely considered to be a robust procedure (with respect to the significaoce level) under violatioa of the assuaptioo of equal variances. This paper is coa-earned with a quantification of the amount of robustness which this procedure has under such violations, The approach is through the concept of "religion of robustness" and the resluts show an extremely strong degree of robustness for the equal an extremely strong degree of robustness for the equal sample size t-test, probably more so than most statistyicians realise. This extremely high level of robustness, however, reduces quickly as the sample sizes begin to vary from equality. The regions of robustnes obtained show that while most users would likely be satisfied with the degree of robustness inherent when the two sample sizes each vary by 10% from equality, most would wish to be much more cautions when the variation is 20%. The study covers sample sizes n₁ -= n ² = 5(5)30(10)50 plus 10% and 20% variations thereof for the two-tailed test and nominal significance levels of 0.01 and 0.05.     

A class of distribution-free tests for two-way layout

In this article a class of distribution-free tests for the hypothesis of no row (treatment) effect in a two-way layout design, with several observations per cell, is proposed. The tests are based on U-statistics, constructed by considering minima of all possible subsamples of same size from each cell.The proposed class of tests is compared with the parametric test, Mack and Skillings test and Yate's test for two-way layout, in terms of Pitman ARE sense. It is seen that for the case of equal number of observations per cell, the proposed tests have better efficiency for exponential and uniform error distributions.     

A family of IDMRL tests with unknown turning point

In this paper we propose a family of tests for exponentiality against the IDMRL alternative. Here we assume that the turning point or the proportion before the turning point is unknown. We derive the asymptotic null distributions of the test statistics and obtain their asymptotic critical values based on Durbin's approximation method. A simulation study is conducted to evaluate the proposed tests.     

Goodness of fit and homogeneity tests on the basis of N-distances

In this paper we propose an application of N-distance theory [Klebanov, L.B., 2005. N-distances and their applications. Karolinum, Prague] for testing simple hypotheses of goodness of fit and homogeneity. The asymptotic null distribution of test statistics is established and coincides with the distribution of infinite quadratic form of independent standard normal random variables. A construction of multivariate free-of-distribution homogeneity test is considered. The power of proposed criteria is compared with classical tests using Monte-Carlo simulations.     

Hierarchic likelihood ratio tests for equality of covariance matrices

Powers and sizes are simulated for hierarchic components of an adjusted likelihood ratio test for equality of covariance matrices.     

Pearson-type goodness-of-fit tests for regression

A procedure for testing the goodness of fit of linear regression models is introduced. For a given partition of the real line into cells, the proposed test is a quadratic form based on the vector of observed minus expected frequencies of the residuals obtained by maximum-likelihood estimation of the regression parameters. The quadratic form is of the same computational difficulty as the traditional Pearson-type tests with uncensored data. A statistic based on only one cell is particularly easy to apply and is used for testing the normality assumption in a real data set from astronomy. A simulation study examines the finite-sample properties of the proposed tests.