Testing and estimation of purely nonparametric effects in repeated measures designs
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.
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- Vicente Núñez-Antón & Juan Rodríguez-Póo & Philippe Vieu, 1999. "Longitudinal data with nonstationary errors: a nonparametric three-stage approach," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(1), pages 201-231, June.
- Brunner, Edgar & Munzel, Ulrich & Puri, Madan L., 1999. "Rank-Score Tests in Factorial Designs with Repeated Measures," Journal of Multivariate Analysis, Elsevier, vol. 70(2), pages 286-317, August.
- Bathke, Arne C. & Schabenberger, Oliver & Tobias, Randall D. & Madden, Laurence V., 2009. "Greenhouseâ€“Geisser Adjustment and the ANOVA-Type Statistic: Cousins or Twins?," The American Statistician, American Statistical Association, vol. 63(3), pages 239-246.
- Munzel, Ullrich, 1999. "Linear rank score statistics when ties are present," Statistics & Probability Letters, Elsevier, vol. 41(4), pages 389-395, February.
- Edgar Brunner & Madan Puri, 2001. "Nonparametric methods in factorial designs," Statistical Papers, Springer, vol. 42(1), pages 1-52, January.
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