Abstract
We study simultaneous rank procedures for unbalanced designs with independent observations. The hypotheses are formulated in terms of purely nonparametric treatment effects. In this context, we derive rank-based multiple contrast test procedures and simultaneous confidence intervals which take the correlation between the test statistics into account. Hereby, the individual test decisions and the simultaneous confidence intervals are compatible. This means, whenever an individual hypothesis has been rejected by the multiple contrast test, the corresponding simultaneous confidence interval does not include the null, i.e. the hypothetical value of no treatment effect. The procedures allow for testing arbitrary purely nonparametric multiple linear hypotheses (e.g. many-to-one, all-pairs, changepoint, or even average comparisons). We do not assume homogeneous variances of the data; in particular, the distributions can have different shapes even under the null hypothesis. Thus, a solution to the multiple nonparametric Behrens-Fisher problem is presented in this unified framework.
Highlights
Multiple comparisons as well as for the computation of compatible SCI
Parametric multiple contrast test procedures (MCTP) with accompanying compatible SCI for linear contrasts in terms of the expectations of homoscedastic normal samples were derived by Mukerjee, Robertson, and Wright (1987) [28] and Bretz, Genz, and Hothorn (2001) [3]
We apply the MCTP and compatible SCI proposed in the previous sections to a dataset with ordinal data analyzed by Akritas et al (1997) [1]
Summary
Let Xik be the kth (independent) replicate in the ith group among the total a groups. In the context of nonparametric models, the normalized version of the distribution function Fi(x) was first mentioned by Levy (1925) [27] Later on, it was used by Ruymgaart (1980) [34], Akritas, Arnold, and Brunner (1997) [1], Munzel (1999) [29], Gao et al (2008) [17], among others, to derive asymptotic results for rank statistics including the case of ties in a unified way. In the nonparametric setup discussed above, Akritas et al (1997) [1] propose to formulate hypotheses by the distribution functions as H0F : CF = 0, where C denotes an arbitrary contrast matrix, i.e. C1a = 0, and derive global test procedures for H0F. We note that the hypothesis in the classical Behrens-Fisher model is contained in this general setup as a special case This is seen from the fact that pij = 1/2 if Fi and Fj are both symmetric distributions with the same center of symmetry. For a detailed discussion of the hypotheses formulated above we refer to Akritas et al (1997) [1] and Brunner and Munzel (2000) [8]
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