DISTRIBUTION FREE MULTIPLE COMPARISONS FOR MONOTONICALLY ORDERED TREATMENT EFFECTS

A method of calculating simultaneous one-sided confidence intervals for all ordered pairwise differences of the treatment effects_j–_i, 1 i < j k, in a one-way model without any distributional assumptions is discussed. When it is known a priori that the treatment effects satisfy the simple ordering_1k, these simultaneous confidence intervals offer the experimenter a simple way of determining which treatment effects may be declared to be unequal, and is more powerful than the usual two-sided Steel-Dwass procedure. Some exact critical points required by the confidence intervals are presented for k= 3 and small sample sizes, and other methods of critical point determination such as asymptotic approximation and simulation are discussed.     

A circular-cone test for testing homogeneity against a simple tree order

Several methods exist for the problem of testing the equality of several treatments against the one-sided alternative that the treatments are better than the control. These methods include Dunnett's test, Bartholomew's likelihood-ratio test, the Abelson-Tukey-Schaafsma-Smid optimal-contrast test, and the multiple-contrast test of Mukerjee, Robertson, and Wright. A new test is proposed based on an approximation of the likelihood-ratio test of Bartholomew. This test involves using a circular cone in place of the alternative-hypothesis cone. The circular-cone test has excellent power characteristics similar to those of Bartholomew's test. Moreover, it has the advantages of being simpler to compute and may be used with unequal sample sizes.     

Exact Simultaneous Confidence Bands for Quadratic and Cubic Polynomial Regression with Applications in Dose Response Study

Exact simultaneous confidence bands (SCBs) for a polynomial regression model are available only in some special situations. In this paper, simultaneous confidence levels for both hyperbolic and constant width bands for a polynomial function over a given interval are expressed as multidimensional integrals. The dimension of these integrals is equal to the degree of the polynomial. Hence the values can be calculated quickly and accurately via numerical quadrature provided that the degree of the polynomial is small (e.g. 2 or 3). This allows the construction of exact SCBs for quadratic and cubic regression functions over any given interval and for any given design matrix. Quadratic and cubic regressions are frequently used to characterise dose response relationships in addition to many other applications. Comparison between the hyperbolic and constant width bands under both the average width and minimum volume confidence set criteria shows that the constant width band can be much less efficient than the hyperbolic band. For hyperbolic bands, comparison between the exact critical constant and conservative or approximate critical constants indicates that the exact critical constant can be substantially smaller than the conservative or approximate critical constants. Numerical examples from a dose response study are used to illustrate the methods.     

A test of simultaneous homogeneity against ordered alternatives in multifactorial design

A test of simultaneous homogeneity of main effects of several factors against an alternative hypothesis with simple order restrictions in main effects of more than one factor in a multifactorial design is considered. This can be regarded as an extension of Shorack's (1967) work where the alternative hypo t hqaLs involves simple order restriction in main effects of one faetor. We derive the likelihood ratio test statistic, E^-2 , and its null hypothesis distribution which involves the convolutions of PIrobabilities P(l,k) used in the statistical inference under order restriction , The powers of the E^-2 test and the usual F test are compar-ed by simulation.     

One-sided range test for testing against an ordered alternative under heteroscedasticity

In a one-way fixed effects analysis of variance model, when normal variances are unknown and possibly unequal, a one-sided range test for testing the null hypothesis H ₀ : μ ₁ = … = μ_k against an ordered alternative H_a : μ ₁ ≤ … ≤ μ_k by a single-stage and a two-stage procedure, respectively, is proposed. The critical values under H ₀ and the power under a specific alternative are calculated. Relation between the one-stage and the two-stage test procedures is discussed. A numerical example to illustrate these procedures is given.     

Tables of percentage points of the k-variate normal distribution for large values of k

This paper gives tabulations of the upper α percentage points of the maximum absolute value of the k-variate normal distribution with common correlation ρ for values of k as high as 500. The tables are useful for performing multiple comparison procedures in experiments with large numbers of treatments.     

Hypotheses testing for generalized order statistics with simple order restrictions on model parameters under the alternative

In applications of generalized order statistics as, for instance, reliability analysis of engineering systems, prior knowledge about the order of the underlying model parameters is often available and may therefore be incorporated in inferential procedures. Taking this information into account, we establish the likelihood ratio test, Rao's score test, and Wald's test for test problems arising from the question of appropriate model selection for ordered data, where simple order restrictions are put on the parameters under the alternative hypothesis. For simple and composite null hypothesis, explicit representations of the corresponding test statistics are obtained along with some properties and their asymptotic distributions. A simulation study is carried out to compare the order restricted tests in terms of their power. In the set-up considered, the adapted tests significantly improve the power of the associated omnibus versions for small sample sizes, especially when testing a composite null hypothesis.     

Order restricted statistical inference for scale parameters based on sequential order statistics

In the situation of a multi-sample experiment consisting of differently equipped sequential k-out-of-n systems, scale parameters of underlying distributions from a general location-scale family of distributions are estimated under an order restriction. In each sample, the case of missing the smallest observations is included. Moreover, based on a profile likelihood a homogeneity test against an ordered alternative is proposed and analyzed. This work extends an approach of Bhattacharya [2007. Testing for ordered failure rates under general progressive censoring. J. Statist. Plann. Inference 137, 1775–1786] in the progressive Type-II censoring framework.     

A test for the equality of k medians against the simple tree alternative under right censorship

We consider a test for the equality of k population medians, θ_i i=1,2,….,k, when it is believed a priori, that θ _i: The observations are subject to right censorhip. The distributions of the censoring variables for each population are assumed to be equal. This test is compared with the general k-sample test proposed by Breslow     

A comparison of some nonparametric tests for the simple tree alternative

This paper develops a new test statistic for testing hypotheses of the form H_O M₀=M₁=…=M_k versus H_a: M₀≦M₁,M₂,…,M_k] where at least one inequality is strict. M₀ is the median of the control population and M₁ is the median of the population receiving trearment i, i=1,2,…,k. The population distributions are assumed to be unknown but to differ only in their location parameters if at all. A simulation study is done comparing the new test statistic with the Chacko and the Kruskal-Wallis when the underlying population distributions are either normal, uniform, exponential, or Cauchy. Sample sizes of five, eight, ten, and twenty were considered. The new test statistic did better than the Chacko and the Kruskal-Wallis when the medians of the populations receiving the treatments were approximately the same     

A tow-stage procedure for testing homogeneity of means against ordered alternatives in analysis of variance with unequal variances

In this paper we present a two-stage sampling procedure for testing the equality of normal means against ordered alternatives in one-way analysis of variance with unequal unknown variances. A table of approximated percentiles needed for implementation is provided. Some Monte Carlo results for estimating the power of the proposed test statistic are presented.     

A test for zero point triserial correlation coefficient and its relationship with tests for homogeneity of means with ordered alternatives

The point triserial correlation coefficient is defined and, under appropriate order restrictions, an exact test that this correlation coefficient equals zero is developed. The power function of that test is derived and partially tabulated. The general problem of testing for homogeneity of means under ordered alternatives is discussed. The available procedures for performing such tests are considered, are seen to provide alternative approaches to the test developed herein, and are compared with that test. An exact test for the equality of dependent point triserial correlation coefficients is described through application of a procedure suggested by Wolfe ‘1976’     

A modified chi-square test for testing equality of two multinomial populations against an order restricted alternative

A modified chi-square test for testing the equality of two multinomial populations against an order restricted alternative in one sample and two sample cases is constructed. The relation between the concepts of dependence by cM-square and stochastic ordering is established, The asymptotic distribution of the test statistic is the chi-bar-square type discussed by Robertson, Wright and Dykstra (1988). Simulations are used to compare the power of this test with the power of the likelihood ratio test of stochastic ordering of the two multinomial populations.     

A simple algorithm for the adaptation of scores and power behavior of the corresponding rank test

The purpose of this paper is twofold:On one hand we want to give a very simple algorithm for evaluating a special rank estimator of the type given in Behnen, Neuhaus, and Ruymgaart (1983) for the approximate optimal choice of the scores-generating function of a two-sample linear rank test for the general testing problem H_o:F=G versus H₁:F ≤ G, F ≠ G, in order to demonstrate that the corresponding adaptive rank statistic is simple enough for practical applications. On the other hand we prove the asymptotic normality of the adaptive rank statistic under H (leading to approximate critical values) and demonstrate the adaptive behavior of the corresponding rank test by a Monte Carlo power simulation for sample sizes as low as m=10, n=10.