New Nonparametric Rank Tests for Interactions in Factorial Designs with Repeated Measures
Techniques have been proposed for the nonparametric analysis of interactions in factorial designs. They rank the observations and then perform parametric tests on ranks. The aligned rank tests belong to one class of these. Aligning implies that some estimate of a location (e.g., for the effect on a certain level of a given factor), such as the mean or median of the observation, is subtracted from each observation. These data, thus aligned according to the desired main and/or interaction effects, are then ranked and parametric tests are performed on these aligned ranks. A second class of such tests, named the new rank based methods, first rank the (not aligned) observations, and then the relative treatment effects are defined in reference to the distribution of the variables measured and estimated through elaborate calculations on these ranks.
The alignment methodology was introduced by Hodges and Lehmann (1962) and extended to two-way layouts by Sen (1968). McSweeney (1967) developed a test (M test) for interaction using the aligned ranks in the two-way layout. The aligned rank tests were publicized by: Hettmansperger (1984), Puri and Sen (1985), Sawilowsky (1990), and Higgins and Tashtoush (1994). More recently, Beasley and Zumbo (2003; 2009) added the aligned Friedman rank test for interactions in split-plot or repeated measures designs. Reviews of the aligned rank tests have been provided by Sawilowsky (1990), Higgins and Tashtoush (1994), Toothaker and Newman (1994), Kelley and Sawilowsky (1997), Richter and Payton (1999), Peterson (2002), and Rodriguez, Álvarez, and Remirez (2009). Salazar-Álvarez, Tercero-Gómez, Temblador-Pérez, and Conover (2014) recently reviewed nonparametric test for interactions. They overlooked the – in my opinion – important contributions by Beasley and Zumbo (2003; 2009) on designs with repeated measures. The general conclusion of these reviews was that the aligned rank tests are valid nonparametric alternatives for the parametric tests for the interaction, especially when sample sizes are small (Sawilowsky, 1990) or the departure from normality of the distribution of the observations is extreme (e.g., heavy tailed; Kelley & Sawilowsky, 1997).
Pioneers on the new rank-based methods were Akritas (1990), Akritas and Arnold (1994), Akritas, Arnold, and Brunner (1997), and Akritas and Brunner (1997). Brunner and Puri (2001) reviewed these methods.
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