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
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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
}
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 -