Generalized Behrens-Fisher Problem Research Articles

A Solution to the Generalized Behrens-Fisher Problem

A Solution to the Generalized Behrens-Fisher Problem

Adaptive Nonparametric Tests for the Generalized Behrens-Fisher Problem

Some adaptive test procedures are developed for the generalized Behrens-Fisher problem. The one having a deterministic approach is based on calculating a measure of symmetry from each sample and using them as a basis for choosing between the modifiedWilcoxon-Mann-Whitney test (Fligner and Policello, 1981) and the modified Mood’s median test (Fligner and Rust, 1982). The other one is a probabilistic approach which also uses a combination of the modified Wilcoxon-Mann-Whitney test and the modified Mood’smedian test according to an evidence of asymmetry provided by the p-value from the triples test for symmetry given in Randles, Fligner, Policello, and Wolfe (1980). This probabilistic approach is further modified by using a suitable function of the p-value from the triples test. A simulation study reveals that the modified procedure performs reasonably well in terms of power and attainment of the nominal size.

A Test for Generalized Behrens-Fisher Problem with an Adaptive Interim Analysis

Abtsrcat The present paper considers a generalization of the Behrens-Fisher problem without assuming that the underlying distribution functions are continuous so that the proposed procedure is also applicable to count or ordered categorical data. The proposed test is based on a generalization of the Wilcoxon-Mann-Whitney statistic where the data correspond to sequentially observed responses from patients allocated at random to two treatments in two stages. At stage 1, patients are allocated completely at random between two treatments. At the end of stage 1, the accumulated responses corresponding to the treatments are compared. At stage 2, the patients are assigned to the treatments in such a way that the more promising treatment receives more patients. Simulation comparisons of the proposed procedure with some natural competitors show a considerably larger number of allocations to the better treatment and a smaller average sample number with a little loss in power.

Resampling methods for the nonparametric and generalized Behrens-Fisher problems

Defining a location parameter as a generalization of the median, a robust test is proposed for (a) the nonparametric Behrens-Fisher problem, where the underlying distributions may have different scales and could be skewed, and (b) the generalized Behrens-Fisher problem, where the distributions may even have different shapes. We propose to bootstrap a signed rank statistic based on differences of sample values and derive rigorous bootstrap central limit theorems for its probabilistic justification, allowing for the so-called m-out-of-n bootstrap. The location parameter of interest is the pseudo-median of the distribution of the difference between a control measurement and an observation from the treatment group. It reduces to (a) the shift in the two sample location model and (b) the difference between the centers of symmetry in the nonparametric Behrens-Fisher model, under the additional assumption that the distributions are symmetric. Due to its importance for applications, we also extend our results to an ANOVA design where each treatment is compared with the control group. Finally, we compare our test with competitors on the basis of theory as well as simulation studies. It turns out that our approach yields a substantial improvement for distributions close to the generalized extreme value type, which makes it attractive for applications in engineering as well as finance. Several heteroscedastic data sets from electrical engineering, astro physics, energy research, analytical chemistry and psychology are used to illustrate our solution.

Partial Sequential Tests for Generalized Behrens-Fisher Problem

The present article provides two partially sequential nonparametric tests (see Wolfe, 1977a) for testing the identity of two location parameters without making any assumptions on the shapes of the distributions of the underlying populations. The proposed tests use 0–1 scores while defining the stopping variable, and are based on the partial sequential placement statistics introduced by Orban and Wolfe (1980). Some asymptotic results related to the proposed procedures are studied. Competitive tests based on fixed sample sizes are also found. Further various exact and asymptotic performance measures of the tests are investigated. Recommended by Edit Gombay

Exact tests based on the Baumgartner-Weiß-Schindler statistic—A survey

It is the purpose of this paper to review recently-proposed exact tests based on the Baumgartner-Weis-Schindler statistic and its modification. Except for the generalized Behrens-Fisher problem, these tests are broadly applicable, and they can be used to compare two groups irrespective of whether or not ties occur. In addition, a nonparametric trend test and a trend test for binomial proportions are possible. These exact tests are preferable to commonly-applied tests, such as the Wilcoxon rank sum test, in terms of both type I error rate and power.

Degrees of freedom and the likelihood ratio test for the generalized Behrens-Fisher problem

Several methods for testing the difference between two group means of k independent populations are compared. Simulation shows that the likelihood ratio test with the Bartlett correction factor and the t test with appropriate degrees of freedom perform better, particularly when the sample size is small. However, the latter is very good for all configurations.

Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem

It is shown that permutation tests based on studentized statistics are asymptotically exact of size α also under certain extended non-i.i.d. null hypotheses. To demonstrate the principle the results are applied to the generalized two-sample Behrens-Fisher problem for testing equality of the means under general non-parametric heterogeneous error distributions. Within this setting we propose a permutation version of the Welch test which is an extension of Pitman's two-sample permutation test. These results are special cases of a conditional central limit theorem for studentized permutation statistics which also applies to asymptotic power functions.

Investigation of the robust rank-order test for non-normal populations with unequal variances: The case of reaction time.

Abstract This article examines the performance of the robust rank-order (Fligner-Policello) test of treatment effects for populations with unequal variances. Both symmetric (normal) and skewed (ex-Gaussian) distributions are examined. The ex-Gaussian distribution is particularly relevant because it is representative of skewed distributions found in experimental psychology, most notably reaction time data. The results indicate that when testing the hypothesis of equal medians for the symmetric distribution, the Fligner-Policello test is conservative, whereas it performs inconsistently for the skewed population. In light of the findings, we discuss the various options presently available to researchers confronted with unequal variances and skewed data, including a discussion of whether testing for treatment effects for populations with unequal variances is of practical relevance. Experimental psychologists are often interested in testing hypotheses about treatment effects in their experimental designs. The commonly used significance tests using t or F statistics involve the assumptions that the populations are normally distributed and the population variances are equal. The violation of the latter assumption is often referred to as the Behrens-Fisher problem (Scheffe, 1970) in honour of two pioneering statisticians who drew attention to the violation of this assumption and provided an early solution. In this article, we consider the simultaneous violation of both normality and equal variances, and we evaluate a nonparametric statistical method that is designed for use when neither of these assumptions is met. Consider a common scenario in experimental research: A researcher is interested in the reaction times (recorded in milliseconds) in a perceptual judgement task. The design consists of two independent groups, wherein group A is a control group and group B is an experimental group. As expected with reaction time data, the distribution of scores in each group is skewed (mean minus median equals 92 milliseconds); however, the variance in the experimental group is 2.25 times larger than that in the control group. The researcher is interested in testing the equality of the two sample means or, alternatively, medians. Given that the samples are skewed, it is generally recommended that a nonparametric inference procedure, usually the Wilcoxon-Mann-Whitney test, be used (Harwell & Serlin, 1989). However, the Wilcoxon-Mann-Whitney test is inappropriate for cases with unequal variances (see, e.g., Harwell, Rubinstein, Hayes, & Olds, 1992; Zimmerman & Zumbo, 1993a; 1993b). In fact, it is noteworthy that the Wilcoxon-Mann-Whitney test behaves very much like the Student t test: When the larger variance is associated with the larger sample size, there is a depression of the Type I error rate; when the larger variance is associated with the smaller sample size, there is an elevation of the Type I error rate (Zimmerman & Zumbo, 1993a; 1993b). Siegel and Castellan (1988), however, recommend that in the case of non-normality and unequal variances, an alternative nonparametric inference procedure is the Fligner-Policello (1981) test, also known as the robust rank-order test. It should be noted that the Behrens-Fisher problem was originally formulated only for the case of normal population distributions (Scheffe, 1970). The scenario described in our example, non-normality and unequal variances, is therefore referred to in the statistical literature as the generalized Behrens-Fisher problem. Although solutions to the Behrens-Fisher problem are discussed in some textbooks on experimental design (e.g., Winer, Brown, & Michels, 1991), to our knowledge, the only such discussion of solutions to the generalized Behrens-Fisher problem is Siegel and Castellan's (1988; pp. 137-144) presentation of the robust rank-order test (Fligner & Policello, 1981). Therefore, the purposes of the present article are: (a) to present the Fligner-Policello test in more detail, noting some important points overlooked by Siegel and Castellan in their presentation, and (b) to examine the performance of the Fligner-Policello test when applied to a non-normal distribution representative of those found in experimental psychology, especially in reaction time data. …

A modification of Mood's median test for the generalized Behrens-Fisher problem

SUMMARY A modification of Mood's median test is described which allows one to test for differences between two medians without making any assumptions on the shapes of the underlying populations. The proposed procedure is exactly distribution-free when the populations have the same shape and is asymptotically distribution-free when they do not. It is the retention of the exact distribution-free property when the populations have the same shape which distinguishes the proposed procedure from previous work.

Use of a median test for a generalized Behrens-Fisher problem

A non-parametric test is proposed for testing differences of location in two independent samples without assuming equal scale parameters of the respective populations. The test is based on simultaneous application of two modified median tests, each using its sample median of the single samples.

Use of the Wilcoxon Statistic for a Generalized Behrens-Fisher Problem

Heretofore, the ordinary Wilcoxon statistic for the two-sample problem [9], [5] has been used only to test the null hypothesis that the two parent populations are identical. This paper presents a technique for utilizing the Wilcoxon statistic to test a broader type of null hypothesis, like that encountered in the Behrens-Fisher problem: we show that the usual Wilcoxon test, with $(m + n + 1)/12mn$ replaced by $1/\lbrack 4 \min(m, n)\rbrack$, may be used to test the null hypothesis of the equality of the medians of two symmetrical (continuous) distributions which are of the same form but which have different (unknown) scale parameters; more generally, the test still works for testing the equality of the medians of any two symmetrical distributions.