Limitations of the rank transform statistic in tests for interactions

Conover and Iman (1976), Iman (1974), and Iman and Conover (1976) have found the rank transform test to be robust and powerful when testing for interaction in experimental designs.The current study shows that, insofar as tests for interactions

are concerned, the rank transform test is robust and powerful in some circumstances but is dramatically nonrobust and manifests power significantly below that of the usual F test in some cases. Therefore, this procedure should be used only withcaution when employed in designs suchas those examined here.     

Comparing mean ranks for repeated measures data

Rank tests are considered that compare t treatments in repeated measures designs. A statistic is given that contains as special cases several that have been proposed for this problem, including one that corresponds to the randomized block ANOVA statistic applied to the rank transformed data. Another statistic is proposed, having a null distribution holding under more general conditions, that is the rank transform of the Hotelling statistic for repeated measures. A statistic of this type is also given for data that are ordered categorical rather than fully rankedo Unlike the Friedman statistic, the statistics discussed in this article utilize a single ranking of the entire sample. Power calculations for an underlying normal distribution indicate that the rank transformed ANOVA test can be substantially more powerful than the Friedman test.     

Approximate and exact distributions of rank tests for balanced incomplete block designs

Judges rank k out of t objects according to m replic ations of abasic balanced incomplete block design with bblocks. In Alvo and Cabilio(1991),it is shown that the Durbin test, which is the usual test in this situation, can be written in terms of Spearman correlations between the blocks, and using a Kendall correlation, they generated a new statistic for this situation.This Kendall tau based statistic has a richer support than the Durbin statistic, and is at least as efficient.In the present paper,exact and simulation based tables are generated for both statistics, and various approximations to these null distributions are considered and compared.     

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.     

On some alternative procedures using ranks for the analysis of experimental designs

The analysis of data from eseperisental designs is often hampered by the lack of more than one procedure available for the analysis, especially when that procedure is based on assumptions which do not apply in the situation at hand. In this paper tvo classes of alternative procedures are discussed and compared, One is the aligned ranks procedure which first standardises the data by subtracting an appropriate estimate of location, then replaces the data with ranks t and finally uses an appropriate test statistic which has asymptotically a chi-square distribution The second procedure is the rank transform which first replaces all of the data with the ranks, and then employs the usual parametric methods, but computed on the ranks instead of the data Some Monte Carlo simulations for a test of interaction in a two way layout with replication enable the robustness and pover of these tvo methods to be compared with the usual analysis of variancs.     

Estimated confidence under ancillary statistic everywhere-valid constraint

Consider the problem of estimating the coverage function of an usual confidence interval for a randomly chosen linear combination of the elements of the mean vector of a p-dimensional normal distribution. The usual constant coverage probability estimator is shown to be admissible under the ancillary statistic everywhere-valid constraint. Note that this estimator is not admissible under the usual sense if p⩾5. Since the criterion of admissibility under the ancillary statistic everywhere-valid constraint is a reasonable one, that the constant coverage probability estimator has been commonly accepted is justified.     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

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.     

On adaptive estimation

Point estimates that are weighted averages of other estimates are considered. They are adaptive because the weights are also functions of the sample observations.In particular, the weights are functions of new measures of peakedness and skewness. Five adaptive estimators are compared (in a Monte Carlo study using the swindle) to some of the usual estimators, including those robust ones of Huber and Tukey. In addition, the swindle constant is considered in some detail. All of the adaptive estimators do extremely well with an adaptive biweight statistic being the best one in this study. Suggestions are made about future directions in this area.     

Linear Signed Rank Test for Model Selection

In this article, we consider a linear signed rank test for non-nested distributions in the context of the model selection. Introducing a new test, we show that, it is asymptotically more efficient than the Vuong test and the test statistic based on B statistic introduced by Clarke. However, here, we let the magnitude of the data give a better performance to the test statistic. We have shown that this test is an unbiased one. The results of simulations show that the rank test has the greater statistical power than the Vuong test where the underline distributions is symmetric.     

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.     

An approximated distribution of the gini's rank associatio coefficient

An approximate distribution is proposed for the Gini's rank association coefficient g which is, like Kendall's and Spearman's rank correlation coefficient, a statistic to test independence between two random variables. The purposed distribution can be simply transformed into a Student's T distribution; so, hypothesis testing is made much easier.     

Using categorical markers as auxiliary variables in log‐rank tests and hazard ratio estimation

Markers, which are prognostic longitudinal variables, can be used to replace some of the information lost due to right censoring. They may also be used to remove or reduce bias due to informative censoring. In this paper, the authors propose novel methods for using markers to increase the efficiency of log‐rank tests and hazard ratio estimation, as well as parametric estimation. They propose a «plug‐in» methodology that consists of writing the test statistic or estimate of interest as a functional of Kaplan–Meier estimators. The latter are then replaced by an efficient estimator of the survival curve that incorporates information from markers. Using simulations, the authors show that the resulting estimators and tests can be up to 30% more efficient than the usual procedures, provided that the marker is highly prognostic and that the frequency of censoring is high.     

The 'improved' brown and forsythe test for mean equality: some things can't be fixed

Mehrotra (1997) presented an ‘;improved’ Brown and Forsythe (1974) statistic which is designed to provide a valid test of mean equality in independent groups designs when variances are heterogeneous. In particular, the usual Brown and Fosythe procedure was modified by using a Satterthwaite approximation for numerator degrees of freedom instead of the usual value of number of groups minus one. Mehrotra then, through Monte Carlo methods, demonstrated that the ‘improved’ method resulted in a robust test of significance in cases where the usual Brown and Forsythe method did not. Accordingly, this ‘improved’ procedure was recommended. We show that under conditions likely to be encountered in applied settings, that is, conditions involving heterogeneous variances as well as nonnormal data, the ‘improved’ Brown and Forsythe procedure results in depressed or inflated rates of Type I error in unbalanced designs. Previous findings indicate, however, that one can obtain a robust test by adopting a heteroscedastic statistic with the robust estimators, rather than the usual least squares estimators, and further improvement can be expected when critical significance values are obtained through bootstrapping methods.     

Asymptotic relative efficiency of wald tests in measurement error models

In this paper, asymptotic relative efficiency (ARE) of Wald tests for the Tweedie class of models with log-linear mean, is considered when the aux¬iliary variable is measured with error. Wald test statistics based on the naive maximum likelihood estimator and on a consistent estimator which is obtained by using Nakarnura's (1990) corrected score function approach are defined. As shown analytically, the Wald statistics based on the naive and corrected score function estimators are asymptotically equivalents in terms of ARE. On the other hand, the asymptotic relative efficiency of the naive and corrected Wald statistic with respect to the Wald statistic based on the true covariate equals to the square of the correlation between the unobserved and the observed co-variate. A small scale numerical Monte Carlo study and an example illustrate the small sample size situation.     

Correction methods for ties in rank correlations

Equal values are common when rank methods are applied to rounded data or data consisting solely of small integers. A popular technique for resolving ties in rank correlation is the mid-rank method: the mean of the rankings remains unaltered, but the variance is reduced and modified according to the number and location of ties. Although other methods for breaking ties were proposed in the literature as early as 1939, no such procedure has gained such wide acceptance as mid-ranks. This research analyses various techniques for assigning ranks to tied values, with two objectives: (1) to enable the computation of rank correlation coefficients, such as those of Spearman, Kendall and Gini, by using the usual definition applied in the absence of ties, and (2) to determine whether it really makes a difference which of the various techniques is selected and, if so, which technique is most appropriate for a given application.     

Approximations of the critical region of the fbietkan statistic

The Friedman (1937) test for the randomized complete block design is used to test the hypothesis of no treatment effect among k treatments with b blocks. Difficulty in determination of the size of the critical region for this hypothesis is com¬pounded by the facts that (1) the most recent extension of exact tables for the distribution of the test statistic by Odeh (1977) go up only to the case with k6 and b6, and (2) the usual chi-square approximation is grossly inaccurate for most commonly used combinations of (k,b). The purpose of this paper 2 is to compare two new approximations with the usual x² and F large sample approximations. This work represents an extension to the two-way layout of work done earlier by the authors for the one-way Kruskal-Wallis test statistic.     

Bootstrap Bartlett correction in inflated beta regression

The inflated beta regression model aims to enable the modeling of responses in the intervals (0, 1], [0, 1), or [0, 1]. In this model, hypothesis testing is often performed based on the likelihood ratio statistic. The critical values are obtained from asymptotic approximations, which may lead to distortions of size in small samples. In this sense, this article proposes the bootstrap Bartlett correction to the statistic of likelihood ratio in the inflated beta regression model. The proposed adjustment only requires a simple Monte Carlo simulation. Through extensive Monte Carlo simulations the finite sample performance (size and power) of the proposed corrected test is compared to the usual likelihood ratio test and the Skovgaard adjustment already proposed in the literature. The numerical results evidence that inference based on the proposed correction is much more reliable than that based on the usual likelihood ratio statistics and the Skovgaard adjustment. At the end of the work, an application to real data is also presented.     

The limiting distribution of a test for multivariate structure

We define a chi-squared statistic for p-dimensional data as follows. First, we transform the data to remove the correlations between the p variables. Then, we discretize each variable into groups of equal size and compute the cell counts in the resulting p-way contingency table. Our statistic is just the usual chi-squared statistic for testing independence in a contingency table. Because the cells have been chosen in a data-dependent manner, this statistic does not have the usual limiting distribution. We derive the limiting joint distribution of the cell counts and the limiting distribution of the chi-squared statistic when the data is sampled from a multivariate normal distribution. The chi-squared statistic is useful in detecting hidden structure in raw data or residuals. It can also be used as a test for multivariate normality.     

GOODNESS OF FIT STATISTICS BASED ON THE SPACINGS OF COMPLETE OR CENSORED SAMPLES

A goodness-of-fit statistic Z is defined in terms of the spacings generated by the order statistics of a complete or a censored sample from a distribution of the type (l/)f((x-μ)/), μ and unknown. The distribution of Z is studied, mostly through Monte Carlo methods. The power properties of Z for testing Exponential, Uniform, Normal, Gamma and Logistic distributions are discussed; Z is shown to be more powerful than the Smith & Bain (1976) correlation statistic, except for testing Uniform, Normal and Logistic (symmetric distributions) against symmetric alternatives. The statistic Z is generalized to test the goodness-of-fit from κ 2 independent complete or censored samples.