Rank-based tests for interactions in a two-way design when there are ties

Among the non-parametric approaches to defining and testing the hypothesis of no interactions in a two-way design, the method by Patel and Hoel is appealing because it is a function of the ranks and therefore invariant under monotonic increasing transformations of the data, and it does not require the assumption of no main effects for the approach to be valid. Recently, Wilcox compared three different methods for implementing their method, each method differing in how the appropriate standard error is estimated. In terms of Type I errors, only one of the methods gave satisfactory results in simulations, but it assumes there are no tied values. Cliff has considered the problem of tied values in detail, and his results are easily extended to the problem considered by Wilcox. This paper describes this extension and compares it to the method recommended by Wilcox.