## Details

Original language | English |
---|---|

Pages (from-to) | 1895-1905 |

Number of pages | 11 |

Journal | Computational Statistics and Data Analysis |

Volume | 54 |

Issue number | 8 |

Publication status | Published - 6 Mar 2010 |

## Abstract

The several sample case of the so-called nonparametric Behrens-Fisher problem in repeated measures designs is considered. That is, even under the null hypothesis, the marginal distribution functions in the different groups may have different shapes, and are not assumed to be equal. Moreover, the continuity of the marginal distribution functions is not required so that data with ties and, particularly, ordered categorical data are covered by this model. A multiple relative treatment effect is defined which can be estimated by using the mid-ranks of the observations within pairwise samples. The asymptotic distribution of this estimator is derived, along with a consistent estimator of its asymptotic covariance matrix. In addition, a multiple contrast test and related simultaneous confidence intervals for the relative marginal effects are derived and compared to rank-based Wald-type and ANOVA-type statistics. Simulations show that the ANOVA-type statistic and the multiple contrast test appear to maintain the pre-assigned level of the test quite accurately (even for rather small sample sizes) while the Wald-type statistic leads, as expected, to somewhat liberal decisions. Regarding the power, none of the statistics is uniformly superior. A real data set illustrates the application.

## Keywords

- Behrens-Fisher problem, Nonparametric hypothesis, Ordered categorical data, Rank test, Repeated measures design, Ties

## ASJC Scopus subject areas

- Mathematics(all)
**Statistics and Probability**- Mathematics(all)
**Computational Mathematics**- Computer Science(all)
**Computational Theory and Mathematics**- Mathematics(all)
**Applied Mathematics**

## Cite this

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**Testing and estimation of purely nonparametric effects in repeated measures designs.**/ Konietschke, F.; Bathke, A. C.; Hothorn, L. A. et al.

In: Computational Statistics and Data Analysis, Vol. 54, No. 8, 06.03.2010, p. 1895-1905.

Research output: Contribution to journal › Article › Research › peer review

*Computational Statistics and Data Analysis*, vol. 54, no. 8, pp. 1895-1905. https://doi.org/10.1016/j.csda.2010.02.019

*Computational Statistics and Data Analysis*,

*54*(8), 1895-1905. https://doi.org/10.1016/j.csda.2010.02.019

}

TY - JOUR

T1 - Testing and estimation of purely nonparametric effects in repeated measures designs

AU - Konietschke, F.

AU - Bathke, A. C.

AU - Hothorn, L. A.

AU - Brunner, E.

PY - 2010/3/6

Y1 - 2010/3/6

N2 - The several sample case of the so-called nonparametric Behrens-Fisher problem in repeated measures designs is considered. That is, even under the null hypothesis, the marginal distribution functions in the different groups may have different shapes, and are not assumed to be equal. Moreover, the continuity of the marginal distribution functions is not required so that data with ties and, particularly, ordered categorical data are covered by this model. A multiple relative treatment effect is defined which can be estimated by using the mid-ranks of the observations within pairwise samples. The asymptotic distribution of this estimator is derived, along with a consistent estimator of its asymptotic covariance matrix. In addition, a multiple contrast test and related simultaneous confidence intervals for the relative marginal effects are derived and compared to rank-based Wald-type and ANOVA-type statistics. Simulations show that the ANOVA-type statistic and the multiple contrast test appear to maintain the pre-assigned level of the test quite accurately (even for rather small sample sizes) while the Wald-type statistic leads, as expected, to somewhat liberal decisions. Regarding the power, none of the statistics is uniformly superior. A real data set illustrates the application.

AB - The several sample case of the so-called nonparametric Behrens-Fisher problem in repeated measures designs is considered. That is, even under the null hypothesis, the marginal distribution functions in the different groups may have different shapes, and are not assumed to be equal. Moreover, the continuity of the marginal distribution functions is not required so that data with ties and, particularly, ordered categorical data are covered by this model. A multiple relative treatment effect is defined which can be estimated by using the mid-ranks of the observations within pairwise samples. The asymptotic distribution of this estimator is derived, along with a consistent estimator of its asymptotic covariance matrix. In addition, a multiple contrast test and related simultaneous confidence intervals for the relative marginal effects are derived and compared to rank-based Wald-type and ANOVA-type statistics. Simulations show that the ANOVA-type statistic and the multiple contrast test appear to maintain the pre-assigned level of the test quite accurately (even for rather small sample sizes) while the Wald-type statistic leads, as expected, to somewhat liberal decisions. Regarding the power, none of the statistics is uniformly superior. A real data set illustrates the application.

KW - Behrens-Fisher problem

KW - Nonparametric hypothesis

KW - Ordered categorical data

KW - Rank test

KW - Repeated measures design

KW - Ties

UR - http://www.scopus.com/inward/record.url?scp=77950864271&partnerID=8YFLogxK

U2 - 10.1016/j.csda.2010.02.019

DO - 10.1016/j.csda.2010.02.019

M3 - Article

AN - SCOPUS:77950864271

VL - 54

SP - 1895

EP - 1905

JO - Computational Statistics and Data Analysis

JF - Computational Statistics and Data Analysis

SN - 0167-9473

IS - 8

ER -