Multivariate generalizations of jonckheere's test for ordered alternatives

In many dose-response studies, each of several independent groups of animals is treated with a different dose of a substance. Many response variables are then measured on each animal. The distributions of the response variables may be nonnormal, and Jonckheere's (1954) test for ordered alternatives in the one-way layout is sometimes used to test whether the level of a single variable increases with increasing dose. In some applications, however, it is important to consider a set of response variables simultaneously. For instance, an increase in each of certain enzymes in the blood serum may suggest liver damage. To test whether these enzyme levels increase with increasing dose, it may be preferable to consider these enzymes as a group, rather than individually.

I propose two multivariate generalizations of Jonckheere's univariate test. Each multivariate test statistic is a function of coordinate-wise Jonckheere statistics—one a sum, the other a quadratic form. The sum statistic can be used to test the alternative hypothesis that each variable is stochastically increasing with increasing dose. The quadratic form statistic is designed for the more general alternative hypothesis that each variable is stochastically ordered with increasing dose.

For each of these two alternatives, I also propose a multivariate generalization of a normal theory test described by Puri (1965). I examine the asymptotic distributions of the four test statistics under the null hypothesis and under translation alternatives and compare each distribution-free test to the corresponding normal theory test in terms of asymptotic relative efficiency.

The multivariate Jonckheere tests are illustrated using does-response data from a subchronic toxicology study carried out by the National Toxicology Program. Four groups of ten male rats each were treated with increasing doses of vinylidene flouride, and the serum enzymes SDH, SGOT, and SGPT were measured. A comparison of univariate Jonckheere tests on each variable, bivariate tests on SDH and SGOT, and multivariate tests on all three variables gives insight into the behavior of the various procedures.     

Rank-based analysis of repeated measures block designs

A rank-based inference is developed for repeated measures balanced incomplete block and randomized complete block designs using a suitable dispersion function. Asymptotic distributions of rank estimators are developed after establishing approximate linearity of the gradient vector of the dispersion function. Unlike available nonparametric procedures for those designs, estimation and testing are tied together. Three different test statistics are developed for testing the linear hypotheses. Friedman's (1937) statistic and Durbin's (1951) statistic are particular cases of one of the three proposed statistics. An estimate of a scale parameter which appears in the ARE expression as well as as in the variences and covariances of the rank estimators is discussed.     

Comparison of unweighted and weighted rank based tests for an ordered alternative in randomized complete block designs

In randomized complete block designs, a monotonic relationship among treatment groups may already be established from prior information, e.g., a study with different dose levels of a drug. The test statistic developed by Page and another from Jonckheere and Terpstra are two unweighted rank based tests used to detect ordered alternatives when the assumptions in the traditional two-way analysis of variance are not satisfied. We consider a new weighted rank based test by utilizing a weight for each subject based on the sample variance in computing the new test statistic. The new weighted rank based test is compared with the two commonly used unweighted tests with regard to power under various conditions. The weighted test is generally more powerful than the two unweighted tests when the number of treatment groups is small to moderate.     

Distributional Studies and the Computer: An Analysis of Durbin's Rank Test

A study of the distribution of a statistic involves two major steps: (a) working out its asymptotic, large n, distribution, and (b) making the connection between the asymptotic results and the distribution of the statistic for the sample sizes used in practice. This crucial second step is not included in many studies. In this article, the second step is applied to Durbin's (1951) well-known rank test of treatment effects in balanced incomplete block designs (BIB's). We found that asymptotic, χ², distributions do not provide adequate approximations in most BIB's. Consequently, we feel that several of Durbin's recommendations should be altered.     

A comparative study on detection of influential observations in linear regression

A large number of statistics are used in the literature to detect outliers and influential observations in the linear regression model. In this paper comparison studies have been made for determining a statistic which performs better than the other. This includes: (i) a detailed simulation study, and (ii) analyses of several data sets studied by different authors. Different choices of the design matrix of regression model are considered. Design A studies the performance of the various statistics for detecting the scale shift type outliers, and designs B and C provide information on the performance of the statistics for identifying the influential observations. We have used cutoff points using the exact distributions and Bonferroni's inequality for each statistic. The results show that the studentized residual which is used for detection of mean shift outliers is appropriate for detection of scale shift outliers also, and the Welsch's statistic and the Cook's distance are appropriate for detection of influential observations.     

A note on a parametric extension of rank tests to the analysis of multifactor experiments within blocks

A general rank test procedure based on an underlying multinomial distribution is suggested for randomized block experiments with multifactor treatment combinations within each block. The Wald statistic for the multinomial is used to test hypotheses about the within–block rankings. This statistic is shown to be related to the one–sample Hotellingt's T² statistic, suggesting a method for computing the test statistic using the standard statistical computer packages.     

Testing equality of several exponential distributions

Roy's union-intersection principle is used to develop a test procedure to test the equality of scale parameters of several exponential distributions. Upper five and one percent values of the test statistic for two and three exponential distributions are tabulated and an illustrative simulated example is qiven.     

General orthogonal designs for parameter estimation of ANOVA models under weighted sum-to-zero constraints

ABSTRACT

Traditional studies on optimal designs for ANOVA parameter estimation are based on the framework of equal probabilities of appearance for each factor's levels. However, this premise does not hold in a variety of experimental problems, and it is of theoretical and practical interest to investigate optimal designs for parameters with unequal appearing odds. In this paper, we propose a general orthogonal design via matrix image, in which all columns’ matrix images are orthogonal with each other. Our main results show that such designs have A- and E-optimalities on the estimation of ANOVA parameters which have unequal appearing odds. In addition, we develop two simple methods to construct the proposed designs. The optimality of the design is also validated by a simulation study.     

The Power of Jonckheere's Test

This paper discusses the power of Jonckheere's test under the ordered alternative hypothesis. It is shown that the power of the test is bounded significantly away from one under certain shift alternatives and sample sizes.     

Testing the Equality of Two Exponential Distributions

This paper proposes an overlapping-based test statistic for testing the equality of two exponential distributions with different scale and location parameters. The test statistic is defined as the maximum likelihood estimate of the Weitzman's overlapping coefficient, which estimates the agreement of two densities. The proposed test statistic is derived in closed form. Simulated critical points are generated for the proposed test statistic for various sample sizes and significance levels via Monte Carlo Simulations. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing Log likelihood ratio test.     

Incomplete block designs for asymmetric parallel line assays

Abstract

A method of construction of A-optimal binary block designs for asymmetrical parallel line assays, i.e., the assays in which the number doses for standard and test preparation are unequal has been considered. The method is illustrated with examples. Two cases of this method have been considered. In the first case, designs obtained are of equal replications of the doses. In the second case, designs with unequal replications are obtained.     

P-values for the multi-sample kolmogorov-smirnov test using the expanded bonferroni appoximation

Birnbaum and Hall (1960) introduced a natural statistic for a k-sample generlization of the Kolomogorov-Smirnov test. Using an expansion of Bonferroni's Inequality, this paper determines approximate p-values for the Birnbaum and Hall statistic up to ten samples. This approximation is found to be very accurate under most circumstances. The statistic is also generalized to unequal sample sizes. An example of its use is presented.     

A consistent correlation-type goodness-of-fit test;with application to the two-parameter weibull distribution

In this article the properties of a general univariate JiT-sample rank tests for complete block designs are investigated. Especially, the asymptotic distribution of the test .statistic under H0 and under contiguous alternatives is derived. Some asymptotic relative'PITMAN efficiencies are computed.

AMSX 1980 subject classifications: Primary 62G10; secondary 62K10     

Hypothesis testing on the common location parameter of several shifted exponential distributions: A note

This note deals with hypothesis testing on the common location parameter of several shifted exponential distributions with unknown and possibly unequal scale parameters. No exact test is available for the above mentioned problem; and one does not have the luxury of applying the asymptotic Chi-square test for the likelihood ratio test statistic since the distributions do not satisfy the usual regularity conditions. Therefore, we have proposed a few approximate tests based on the parametric bootstrap method which appear to work well even for small samples in terms of attaining the level. Powers of the proposed tests have been provided along with a recommendation of their usage.     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

E- and R-optimality of block designs for treatment-control comparisons

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple linear constraints. Based on this characterization, we obtain a class of E-optimal exact designs for unequal block sizes. In the studied model, we provide a statistical interpretation for wide classes of E-optimal designs. Moreover, we show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.     

Inequality for equireplicated n-ary block designs with unequal block sizes

It is shown that certain inequalities known for binary, equireplicated, equiblock-sized block designs remain valid for equireplicated n-ary block designs with unequal block sizes. The approach used here is based on the spectral expansion of the C-matrix of the block design. The main theorems include some useful and combinatorially interesting results.     

Data driven chi-square test for uniformity with unequal cells

The power of Pearson's chi-square test for uniformity depends heavily on the choice of the partition of the unit interval involved in the form of the test statistic. We propose a selection rule which chooses a proper partition based on the data. This selection rule leads usually to essentially unequal cells well suited to the observed distribution. We investigate the corresponding data driven chi-square test and present a Monte Carlo simulation study. The conclusion is that this test achieves a high and very stable power for a large class of alternatives, and is much more stable than any other test we compare to.     

Testtesting for the equality of scale parameters of k(> 2) exponential populations based on complete and type ii censored samples

This paper is concerned with testing the equality of scale parameters of K(> 2) two-parameter exponential distributions in presence of unspecified location parameters based on complete and type II censored samples. We develop a marginal likelihood ratio statistic, a quadratic statistic (Qu) (Nelson, 1982) based on maximum marginal likelihood estimates of the scale parameters under the null and the alternative hypotheses, a C(a) statistic (CPL) (Neyman, 1959) based on the profile likelihood estimate of the scale parameter under the null hypothesis and an extremal scale parameter ratio statistic (ESP) (McCool, 1979). We show that the marginal likelihood ratio statistic is equivalent to the modified Bartlett test statistic. We use Bartlett's small sample correction to the marginal likelihood ratio statistic and call it the modified marginal likelihood ratio statistic (MLB). We then compare the four statistics, MLBi Qut CPL and ESP in terms of size and power by using Monte Carlo simulation experiments. For the variety of sample sizes and censoring combinations and nominal levels considered the statistic MLB holds nominal level most accurately and based on empirically calculated critical values, this statistic performs best or as good as others in most situations. Two examples are given.     

The generalized Cucconi test statistic for the two-sample problem

When testing hypotheses in two-sample problem, the Lepage test statistic is often used to jointly test the location and scale parameters, and this test statistic has been discussed by many authors over the years. Since two-sample nonparametric testing plays an important role in biometry, the Cucconi test statistic is generalized to the location, scale, and location–scale parameters in two-sample problem. The limiting distribution of the suggested test statistic is derived under the hypotheses. Deriving the exact critical value of the test statistic is difficult when the sample sizes are increased. A gamma approximation is used to evaluate the upper tail probability for the proposed test statistic given finite sample sizes. The asymptotic efficiencies of the proposed test statistic are determined for various distributions. The consistency of the original Cucconi test statistic is shown on the specific cases. Finally, the original Cucconi statistic is discussed in the theory of ties.