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Several Samples

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Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs

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Abstract

In this section, we introduce nonparametric methods for designs with one fixed factor A whose levels are denoted by i = 1, …, a. At each level i, observations are taken at n i independent subjects (experimental units). Mathematically, we can describe this by random variables \(X_{i1}, \ldots , X_{in_i}\). Observations taken at different subjects within the same factor level i are considered replications. Therefore, they are modeled using the same distribution function F i. That is, X ik ∼ F i(x), i = 1, …, a, k = 1, …, n i. A design of this type is called one-factor design or independent a sample problem. Designs with a = 2 levels constitute an important special case and were considered in more detail in the previous section. However, there are many situations where it is not sufficient to consider only two treatments or factor levels. For example, when examining the toxicity of a substance, which is administered in different dose levels, or the efficacy of a new drug is compared to placebo and to an existing standard drug (gold standard design). In this section, several of the results for a = 2 are being generalized to a > 2 samples. In addition, tests for patterned alternatives, as well as multiple comparisons and simultaneous confidence intervals, are discussed here.

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Author information

Authors and Affiliations

  1. Department of Medical Statistics, University of G¨ottingen, University Medical Center, Göttingen, Germany

    Edgar Brunner

  2. Department of Mathematics, University of Salzburg, Salzburg, Austria

    Arne C. Bathke

  3. Institute of Biometry and Clinical Epidemiology, Charité – University Medical School, Berlin, Germany

    Frank Konietschke

Authors
  1. Edgar Brunner
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  2. Arne C. Bathke
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  3. Frank Konietschke
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Brunner, E., Bathke, A.C., Konietschke, F. (2018). Several Samples. In: Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs . Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-02914-2_4

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