1,721,717 research outputs found
Nonparametric methods in factorial designs
No full textIn this paper, we summarize some recent developments in the analysis of nonparametric models where the classical models of ANOVA are generalized in such a way that not only the assumption of normality is relaxed but also the structure of the designs is introduced in a broader framework and also the concept of treatment effects is redefined. The continuity of the distribution functions is not assumed so that not only data from continuous distributions but also data with ties are included in this general setup. In designs with independent observations as well as in repeated measures designs, the hypotheses are formulated by means of the distribution functions. The main results are given in a unified form. Some applications to special designs are considered, where in simple designs, some well known statistics (such as the Kruskal-Wallis statistic and the chi (2)-statistic for dichotomous data) come out as special cases. The general framework presented here enables the nonparametric analysis of data with continuous distribution functions as well as arbitrary discrete data such as count data, ordered categorical and dichotomous data
Nonparametric analysis of ordered categorical data in designs with longitudinal observations and small sample sizes
No full textFor designs with longitudinal observations of ordered categorical data, a nonparametric model is considered where treatment effects and interactions are defined by means of the marginal distributions. These treatment effects are estimated consistently by ranking methods. The hypotheses in this nonparametric setup are formulated by means of the distribution functions. The asymptotic distribution of the estimators for the nonparametric effects are given under the hypotheses. For small samples, a rather accurate approximation is suggested. A clinical trial with ordered categorical data is used to motivate the ideas and to explain the procedures which are extensions of the Wilcoxon-Mann-Whitney test to factorial designs with longitudinal observations. The application of the procedures requires only some trivial regularity assumptions
Nonparametric tests in the unbalanced multivariate one-way design
No full textA nonparametric model for the multivariate one-way design is discussed which entails continuous as well as discontinuous distributions and, therefore, allows for ordinal data. Nonparametric hypotheses are formulated by the normalized version of the marginal distribution functions as well as the common distribution functions. The differences between the distribution functions are described by means of the so-called relative treatment effects, for which unbiased and consistent estimators are derived. The asymptotic distribution of the vector of the effect estimators is derived and under the marignal hypothesis a consistent estimator for the asymptotic covariance matrix is given. Nonparametric Versions of the Wald-type statistic, the ANOVA-type statistic and the Lawley-Hotelling statistic are considered and compared by means of a simulation study. Finally, these tests are applied to a psychiatric clinical trial
Nonparametric methods in multivariate factorial designs
No full textA nonparametric approach to the analysis of multivariate data is presented that is based on seperate rankings for different variables and extends the results of Akritas ct al. (1997. J. Amer. Statist. Assoc. 89, 336-343.) to multivariate designs. Factorial designs are considered including degenerate distributions as well as discontinuous distributions. The asymptotic normality of a linear combination of dependent linear rank score statistics is shown under rather weak assumptions. Wald-type and ANOVA-type statistics are derived and their properties are compared in simulation studies. The theory is applied to a completely randomised two-way layout. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: 62G05, 62G10, 62G20, 62H10, 62H12, 62H15
The nonparametric Behrens-Fisher problem: Asymptotic theory and a small-sample approximation
No full textA generalization of the Behrens-Fisher problem for two samples is examined in a nonparametric model. It is not assumed that the underlying distribution functions are continuous so that data with arbitrary ties can be handled. A rank rest is considered where the asymptotic variance is estimated consistently by using the ranks over all observations as well as the ranks within each sample. The consistency of the estimator is derived in the appendix. For small samples (n(1), n(2) greater than or equal to 10), a simple approximation by a central t-distribution is suggested where the degrees of freedom are taken from the Satterthwaite-Smith-Welch approximation in the parametric Behrens-Fisher problem. It is demonstrated by means of a simulation study that the Wilcoxon-Mann-Whitney-test may be conservative or liberal depending on the ratio of the sample sizes and the variances of the underlying distribution functions. For the suggested approximation, however, it turns out that the nominal level is maintained rather accurately. The suggested nonparametric procedure is applied to a data set from a clinical trial. Moreover, a confidence interval for the nonparametric treatment effect is given
An exact paired rank test
No full textAn exact rank test for two dependent samples based on overall mid-ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data
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