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Two-Factor Crossed Designs

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

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Abstract

In many experiments, responses are influenced by more than one factor. A detailed discussion of factors and their properties can be found in Chap. 1 in Sect. 1.2.1. For simplicity, often two relevant factors are chosen, and their effect on the response is analyzed. In this section, methods for the analysis of such designs are presented. First, in Sect. 5.1, the basic ideas are illustrated using two examples. In Sect. 5.2, nonparametric effects and their relation to the effects in a linear model are explained. Section 5.3 shows some rather general results. Here, hypotheses regarding nonparametric effects are described, as well as statistics for testing them. Some particular computational aspects and software are considered in detail in Sect. 5.5. Confidence intervals and methods for patterned alternatives are briefly considered in Sect. 5.6, along with some explanations on how to use the general software considered in the preceding Sect. 5.5. We also consider procedures using stratified rankings in Sect. 5.7, and we demonstrate problems related to the non-transitivity of the relative effects if stratified ranks are used. Further, some issues regarding the well-known van Elteren test (Van Elteren, Bull Int Stat Inst 37:351–361, 1960), as well as the procedures by Mack and Skillings (J Am Stat Assoc 75:947–951, 1980) and by Boos and Brownie (Biometrics 48:61–72, 1992) are discussed in this section. The special case of a 2 × 2 design is treated separately in Sect. 5.8. Application of the procedures is demonstrated in Sect. 5.5.4 by means of a 5 × 2 design, and in Sect. 5.8 using a 2 × 2 design.

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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). Two-Factor Crossed Designs. 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_5

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